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Theorem poimirlem4 35404
Description: Lemma for poimir 35433 connecting the admissible faces on the back face of the (𝑀 + 1)-cube to admissible simplices in the 𝑀-cube. (Contributed by Brendan Leahy, 21-Aug-2020.)
Hypotheses
Ref Expression
poimir.0 (𝜑𝑁 ∈ ℕ)
poimirlem4.1 (𝜑𝐾 ∈ ℕ)
poimirlem4.2 (𝜑𝑀 ∈ ℕ0)
poimirlem4.3 (𝜑𝑀 < 𝑁)
Assertion
Ref Expression
poimirlem4 (𝜑 → {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))})
Distinct variable groups:   𝑓,𝑖,𝑗,𝑝,𝑠   𝜑,𝑗   𝑗,𝑀   𝑗,𝑁   𝜑,𝑖,𝑝,𝑠   𝐵,𝑓,𝑖,𝑗,𝑠   𝑓,𝐾,𝑖,𝑗,𝑝,𝑠   𝑓,𝑀,𝑖,𝑝,𝑠   𝑓,𝑁,𝑖,𝑝,𝑠
Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑝)

Proof of Theorem poimirlem4
Dummy variables 𝑘 𝑛 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poimir.0 . . . . . . . 8 (𝜑𝑁 ∈ ℕ)
21adantr 484 . . . . . . 7 ((𝜑𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → 𝑁 ∈ ℕ)
3 poimirlem4.1 . . . . . . . 8 (𝜑𝐾 ∈ ℕ)
43adantr 484 . . . . . . 7 ((𝜑𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → 𝐾 ∈ ℕ)
5 poimirlem4.2 . . . . . . . 8 (𝜑𝑀 ∈ ℕ0)
65adantr 484 . . . . . . 7 ((𝜑𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → 𝑀 ∈ ℕ0)
7 poimirlem4.3 . . . . . . . 8 (𝜑𝑀 < 𝑁)
87adantr 484 . . . . . . 7 ((𝜑𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → 𝑀 < 𝑁)
9 xp1st 7746 . . . . . . . . 9 (𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (1st𝑡) ∈ ((0..^𝐾) ↑m (1...𝑀)))
10 elmapi 8459 . . . . . . . . 9 ((1st𝑡) ∈ ((0..^𝐾) ↑m (1...𝑀)) → (1st𝑡):(1...𝑀)⟶(0..^𝐾))
119, 10syl 17 . . . . . . . 8 (𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (1st𝑡):(1...𝑀)⟶(0..^𝐾))
1211adantl 485 . . . . . . 7 ((𝜑𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → (1st𝑡):(1...𝑀)⟶(0..^𝐾))
13 xp2nd 7747 . . . . . . . . 9 (𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (2nd𝑡) ∈ {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})
14 fvex 6687 . . . . . . . . . 10 (2nd𝑡) ∈ V
15 f1oeq1 6606 . . . . . . . . . 10 (𝑓 = (2nd𝑡) → (𝑓:(1...𝑀)–1-1-onto→(1...𝑀) ↔ (2nd𝑡):(1...𝑀)–1-1-onto→(1...𝑀)))
1614, 15elab 3573 . . . . . . . . 9 ((2nd𝑡) ∈ {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ↔ (2nd𝑡):(1...𝑀)–1-1-onto→(1...𝑀))
1713, 16sylib 221 . . . . . . . 8 (𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (2nd𝑡):(1...𝑀)–1-1-onto→(1...𝑀))
1817adantl 485 . . . . . . 7 ((𝜑𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → (2nd𝑡):(1...𝑀)–1-1-onto→(1...𝑀))
192, 4, 6, 8, 12, 18poimirlem3 35403 . . . . . 6 ((𝜑𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘f + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 → (⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘f + (((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ (((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ (((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1)))))
20 fvex 6687 . . . . . . . . . . . . . . . 16 (1st𝑡) ∈ V
21 snex 5298 . . . . . . . . . . . . . . . 16 {⟨(𝑀 + 1), 0⟩} ∈ V
2220, 21unex 7487 . . . . . . . . . . . . . . 15 ((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∈ V
23 snex 5298 . . . . . . . . . . . . . . . 16 {⟨(𝑀 + 1), (𝑀 + 1)⟩} ∈ V
2414, 23unex 7487 . . . . . . . . . . . . . . 15 ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ V
2522, 24op1std 7724 . . . . . . . . . . . . . 14 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (1st𝑠) = ((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}))
2622, 24op2ndd 7725 . . . . . . . . . . . . . . . . 17 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (2nd𝑠) = ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
2726imaeq1d 5902 . . . . . . . . . . . . . . . 16 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → ((2nd𝑠) “ (1...𝑗)) = (((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)))
2827xpeq1d 5554 . . . . . . . . . . . . . . 15 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (((2nd𝑠) “ (1...𝑗)) × {1}) = ((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}))
2926imaeq1d 5902 . . . . . . . . . . . . . . . 16 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → ((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) = (((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))))
3029xpeq1d 5554 . . . . . . . . . . . . . . 15 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = ((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))
3128, 30uneq12d 4054 . . . . . . . . . . . . . 14 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) = (((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})))
3225, 31oveq12d 7188 . . . . . . . . . . . . 13 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → ((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) = (((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘f + (((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))))
3332uneq1d 4052 . . . . . . . . . . . 12 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ((((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘f + (((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
3433csbeq1d 3794 . . . . . . . . . . 11 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 = ((((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘f + (((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵)
3534eqeq2d 2749 . . . . . . . . . 10 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵𝑖 = ((((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘f + (((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵))
3635rexbidv 3207 . . . . . . . . 9 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘f + (((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵))
3736ralbidv 3109 . . . . . . . 8 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘f + (((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵))
3825fveq1d 6676 . . . . . . . . 9 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → ((1st𝑠)‘(𝑀 + 1)) = (((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)))
3938eqeq1d 2740 . . . . . . . 8 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (((1st𝑠)‘(𝑀 + 1)) = 0 ↔ (((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0))
4026fveq1d 6676 . . . . . . . . 9 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → ((2nd𝑠)‘(𝑀 + 1)) = (((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)))
4140eqeq1d 2740 . . . . . . . 8 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1) ↔ (((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1)))
4237, 39, 413anbi123d 1437 . . . . . . 7 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1)) ↔ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘f + (((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ (((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ (((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1))))
4342elrab 3588 . . . . . 6 (⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))} ↔ (⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘f + (((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ (((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ (((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1))))
4419, 43syl6ibr 255 . . . . 5 ((𝜑𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘f + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 → ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))}))
4544ralrimiva 3096 . . . 4 (𝜑 → ∀𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘f + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 → ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))}))
46 fveq2 6674 . . . . . . . . . . 11 (𝑠 = 𝑡 → (1st𝑠) = (1st𝑡))
47 fveq2 6674 . . . . . . . . . . . . . 14 (𝑠 = 𝑡 → (2nd𝑠) = (2nd𝑡))
4847imaeq1d 5902 . . . . . . . . . . . . 13 (𝑠 = 𝑡 → ((2nd𝑠) “ (1...𝑗)) = ((2nd𝑡) “ (1...𝑗)))
4948xpeq1d 5554 . . . . . . . . . . . 12 (𝑠 = 𝑡 → (((2nd𝑠) “ (1...𝑗)) × {1}) = (((2nd𝑡) “ (1...𝑗)) × {1}))
5047imaeq1d 5902 . . . . . . . . . . . . 13 (𝑠 = 𝑡 → ((2nd𝑠) “ ((𝑗 + 1)...𝑀)) = ((2nd𝑡) “ ((𝑗 + 1)...𝑀)))
5150xpeq1d 5554 . . . . . . . . . . . 12 (𝑠 = 𝑡 → (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}) = (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))
5249, 51uneq12d 4054 . . . . . . . . . . 11 (𝑠 = 𝑡 → ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0})) = ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0})))
5346, 52oveq12d 7188 . . . . . . . . . 10 (𝑠 = 𝑡 → ((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) = ((1st𝑡) ∘f + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))))
5453uneq1d 4052 . . . . . . . . 9 (𝑠 = 𝑡 → (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) = (((1st𝑡) ∘f + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})))
5554csbeq1d 3794 . . . . . . . 8 (𝑠 = 𝑡(((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 = (((1st𝑡) ∘f + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵)
5655eqeq2d 2749 . . . . . . 7 (𝑠 = 𝑡 → (𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵𝑖 = (((1st𝑡) ∘f + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵))
5756rexbidv 3207 . . . . . 6 (𝑠 = 𝑡 → (∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘f + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵))
5857ralbidv 3109 . . . . 5 (𝑠 = 𝑡 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘f + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵))
5958ralrab 3593 . . . 4 (∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))} ↔ ∀𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘f + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 → ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))}))
6045, 59sylibr 237 . . 3 (𝜑 → ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))})
61 xp1st 7746 . . . . . . . . . . . . . . 15 (𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (1st𝑘) ∈ ((0..^𝐾) ↑m (1...(𝑀 + 1))))
62 elmapi 8459 . . . . . . . . . . . . . . 15 ((1st𝑘) ∈ ((0..^𝐾) ↑m (1...(𝑀 + 1))) → (1st𝑘):(1...(𝑀 + 1))⟶(0..^𝐾))
6361, 62syl 17 . . . . . . . . . . . . . 14 (𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (1st𝑘):(1...(𝑀 + 1))⟶(0..^𝐾))
64 fzssp1 13041 . . . . . . . . . . . . . 14 (1...𝑀) ⊆ (1...(𝑀 + 1))
65 fssres 6544 . . . . . . . . . . . . . 14 (((1st𝑘):(1...(𝑀 + 1))⟶(0..^𝐾) ∧ (1...𝑀) ⊆ (1...(𝑀 + 1))) → ((1st𝑘) ↾ (1...𝑀)):(1...𝑀)⟶(0..^𝐾))
6663, 64, 65sylancl 589 . . . . . . . . . . . . 13 (𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((1st𝑘) ↾ (1...𝑀)):(1...𝑀)⟶(0..^𝐾))
67 ovex 7203 . . . . . . . . . . . . . 14 (0..^𝐾) ∈ V
68 ovex 7203 . . . . . . . . . . . . . 14 (1...𝑀) ∈ V
6967, 68elmap 8481 . . . . . . . . . . . . 13 (((1st𝑘) ↾ (1...𝑀)) ∈ ((0..^𝐾) ↑m (1...𝑀)) ↔ ((1st𝑘) ↾ (1...𝑀)):(1...𝑀)⟶(0..^𝐾))
7066, 69sylibr 237 . . . . . . . . . . . 12 (𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((1st𝑘) ↾ (1...𝑀)) ∈ ((0..^𝐾) ↑m (1...𝑀)))
7170ad2antlr 727 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((1st𝑘) ↾ (1...𝑀)) ∈ ((0..^𝐾) ↑m (1...𝑀)))
72 xp2nd 7747 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (2nd𝑘) ∈ {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})
73 fvex 6687 . . . . . . . . . . . . . . . . . 18 (2nd𝑘) ∈ V
74 f1oeq1 6606 . . . . . . . . . . . . . . . . . 18 (𝑓 = (2nd𝑘) → (𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) ↔ (2nd𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))))
7573, 74elab 3573 . . . . . . . . . . . . . . . . 17 ((2nd𝑘) ∈ {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))} ↔ (2nd𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)))
7672, 75sylib 221 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (2nd𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)))
77 f1of1 6617 . . . . . . . . . . . . . . . 16 ((2nd𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) → (2nd𝑘):(1...(𝑀 + 1))–1-1→(1...(𝑀 + 1)))
7876, 77syl 17 . . . . . . . . . . . . . . 15 (𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (2nd𝑘):(1...(𝑀 + 1))–1-1→(1...(𝑀 + 1)))
79 f1ores 6632 . . . . . . . . . . . . . . 15 (((2nd𝑘):(1...(𝑀 + 1))–1-1→(1...(𝑀 + 1)) ∧ (1...𝑀) ⊆ (1...(𝑀 + 1))) → ((2nd𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((2nd𝑘) “ (1...𝑀)))
8078, 64, 79sylancl 589 . . . . . . . . . . . . . 14 (𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((2nd𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((2nd𝑘) “ (1...𝑀)))
8180ad2antlr 727 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((2nd𝑘) “ (1...𝑀)))
82 dff1o3 6624 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) ↔ ((2nd𝑘):(1...(𝑀 + 1))–onto→(1...(𝑀 + 1)) ∧ Fun (2nd𝑘)))
8382simprbi 500 . . . . . . . . . . . . . . . . . 18 ((2nd𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) → Fun (2nd𝑘))
84 imadif 6423 . . . . . . . . . . . . . . . . . 18 (Fun (2nd𝑘) → ((2nd𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = (((2nd𝑘) “ (1...(𝑀 + 1))) ∖ ((2nd𝑘) “ {(𝑀 + 1)})))
8576, 83, 843syl 18 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((2nd𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = (((2nd𝑘) “ (1...(𝑀 + 1))) ∖ ((2nd𝑘) “ {(𝑀 + 1)})))
8685ad2antlr 727 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = (((2nd𝑘) “ (1...(𝑀 + 1))) ∖ ((2nd𝑘) “ {(𝑀 + 1)})))
87 f1ofo 6625 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) → (2nd𝑘):(1...(𝑀 + 1))–onto→(1...(𝑀 + 1)))
88 foima 6597 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑘):(1...(𝑀 + 1))–onto→(1...(𝑀 + 1)) → ((2nd𝑘) “ (1...(𝑀 + 1))) = (1...(𝑀 + 1)))
8976, 87, 883syl 18 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((2nd𝑘) “ (1...(𝑀 + 1))) = (1...(𝑀 + 1)))
9089ad2antlr 727 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd𝑘) “ (1...(𝑀 + 1))) = (1...(𝑀 + 1)))
91 f1ofn 6619 . . . . . . . . . . . . . . . . . . . 20 ((2nd𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) → (2nd𝑘) Fn (1...(𝑀 + 1)))
9276, 91syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (2nd𝑘) Fn (1...(𝑀 + 1)))
93 nn0p1nn 12015 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ ℕ0 → (𝑀 + 1) ∈ ℕ)
945, 93syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑀 + 1) ∈ ℕ)
95 elfz1end 13028 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 + 1) ∈ ℕ ↔ (𝑀 + 1) ∈ (1...(𝑀 + 1)))
9694, 95sylib 221 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑀 + 1) ∈ (1...(𝑀 + 1)))
97 fnsnfv 6747 . . . . . . . . . . . . . . . . . . 19 (((2nd𝑘) Fn (1...(𝑀 + 1)) ∧ (𝑀 + 1) ∈ (1...(𝑀 + 1))) → {((2nd𝑘)‘(𝑀 + 1))} = ((2nd𝑘) “ {(𝑀 + 1)}))
9892, 96, 97syl2anr 600 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → {((2nd𝑘)‘(𝑀 + 1))} = ((2nd𝑘) “ {(𝑀 + 1)}))
99 sneq 4526 . . . . . . . . . . . . . . . . . 18 (((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → {((2nd𝑘)‘(𝑀 + 1))} = {(𝑀 + 1)})
10098, 99sylan9req 2794 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd𝑘) “ {(𝑀 + 1)}) = {(𝑀 + 1)})
10190, 100difeq12d 4014 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (((2nd𝑘) “ (1...(𝑀 + 1))) ∖ ((2nd𝑘) “ {(𝑀 + 1)})) = ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}))
10286, 101eqtrd 2773 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}))
103 1z 12093 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ ℤ
104 nn0uz 12362 . . . . . . . . . . . . . . . . . . . . . . 23 0 = (ℤ‘0)
105 1m1e0 11788 . . . . . . . . . . . . . . . . . . . . . . . 24 (1 − 1) = 0
106105fveq2i 6677 . . . . . . . . . . . . . . . . . . . . . . 23 (ℤ‘(1 − 1)) = (ℤ‘0)
107104, 106eqtr4i 2764 . . . . . . . . . . . . . . . . . . . . . 22 0 = (ℤ‘(1 − 1))
1085, 107eleqtrdi 2843 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑀 ∈ (ℤ‘(1 − 1)))
109 fzsuc2 13056 . . . . . . . . . . . . . . . . . . . . 21 ((1 ∈ ℤ ∧ 𝑀 ∈ (ℤ‘(1 − 1))) → (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)}))
110103, 108, 109sylancr 590 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)}))
111110difeq1d 4012 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = (((1...𝑀) ∪ {(𝑀 + 1)}) ∖ {(𝑀 + 1)}))
112 difun2 4370 . . . . . . . . . . . . . . . . . . 19 (((1...𝑀) ∪ {(𝑀 + 1)}) ∖ {(𝑀 + 1)}) = ((1...𝑀) ∖ {(𝑀 + 1)})
113111, 112eqtrdi 2789 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = ((1...𝑀) ∖ {(𝑀 + 1)}))
1145nn0red 12037 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑀 ∈ ℝ)
115 ltp1 11558 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 ∈ ℝ → 𝑀 < (𝑀 + 1))
116 peano2re 10891 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ)
117 ltnle 10798 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑀 ∈ ℝ ∧ (𝑀 + 1) ∈ ℝ) → (𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀))
118116, 117mpdan 687 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 ∈ ℝ → (𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀))
119115, 118mpbid 235 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ ℝ → ¬ (𝑀 + 1) ≤ 𝑀)
120114, 119syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ¬ (𝑀 + 1) ≤ 𝑀)
121 elfzle2 13002 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 + 1) ∈ (1...𝑀) → (𝑀 + 1) ≤ 𝑀)
122120, 121nsyl 142 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ¬ (𝑀 + 1) ∈ (1...𝑀))
123 difsn 4686 . . . . . . . . . . . . . . . . . . 19 (¬ (𝑀 + 1) ∈ (1...𝑀) → ((1...𝑀) ∖ {(𝑀 + 1)}) = (1...𝑀))
124122, 123syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1...𝑀) ∖ {(𝑀 + 1)}) = (1...𝑀))
125113, 124eqtrd 2773 . . . . . . . . . . . . . . . . 17 (𝜑 → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = (1...𝑀))
126125imaeq2d 5903 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((2nd𝑘) “ (1...𝑀)))
127126ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((2nd𝑘) “ (1...𝑀)))
128125ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = (1...𝑀))
129102, 127, 1283eqtr3d 2781 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd𝑘) “ (1...𝑀)) = (1...𝑀))
130129f1oeq3d 6615 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (((2nd𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((2nd𝑘) “ (1...𝑀)) ↔ ((2nd𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→(1...𝑀)))
13181, 130mpbid 235 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→(1...𝑀))
13273resex 5873 . . . . . . . . . . . . 13 ((2nd𝑘) ↾ (1...𝑀)) ∈ V
133 f1oeq1 6606 . . . . . . . . . . . . 13 (𝑓 = ((2nd𝑘) ↾ (1...𝑀)) → (𝑓:(1...𝑀)–1-1-onto→(1...𝑀) ↔ ((2nd𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→(1...𝑀)))
134132, 133elab 3573 . . . . . . . . . . . 12 (((2nd𝑘) ↾ (1...𝑀)) ∈ {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ↔ ((2nd𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→(1...𝑀))
135131, 134sylibr 237 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd𝑘) ↾ (1...𝑀)) ∈ {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})
13671, 135opelxpd 5563 . . . . . . . . . 10 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))
1371363ad2antr3 1191 . . . . . . . . 9 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))
138 3anass 1096 . . . . . . . . . . 11 ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ↔ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))))
139138biancomi 466 . . . . . . . . . 10 ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ↔ ((((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵))
14094nnzd 12167 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝑀 + 1) ∈ ℤ)
141 uzid 12339 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑀 + 1) ∈ ℤ → (𝑀 + 1) ∈ (ℤ‘(𝑀 + 1)))
142 peano2uz 12383 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑀 + 1) ∈ (ℤ‘(𝑀 + 1)) → ((𝑀 + 1) + 1) ∈ (ℤ‘(𝑀 + 1)))
143140, 141, 1423syl 18 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((𝑀 + 1) + 1) ∈ (ℤ‘(𝑀 + 1)))
1445nn0zd 12166 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝑀 ∈ ℤ)
1451nnzd 12167 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝑁 ∈ ℤ)
146 zltp1le 12113 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁))
147 peano2z 12104 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑀 ∈ ℤ → (𝑀 + 1) ∈ ℤ)
148 eluz 12338 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑀 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ‘(𝑀 + 1)) ↔ (𝑀 + 1) ≤ 𝑁))
149147, 148sylan 583 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ‘(𝑀 + 1)) ↔ (𝑀 + 1) ≤ 𝑁))
150146, 149bitr4d 285 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁𝑁 ∈ (ℤ‘(𝑀 + 1))))
151144, 145, 150syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝑀 < 𝑁𝑁 ∈ (ℤ‘(𝑀 + 1))))
1527, 151mpbid 235 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑁 ∈ (ℤ‘(𝑀 + 1)))
153 fzsplit2 13023 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑀 + 1) + 1) ∈ (ℤ‘(𝑀 + 1)) ∧ 𝑁 ∈ (ℤ‘(𝑀 + 1))) → ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)))
154143, 152, 153syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)))
155 fzsn 13040 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑀 + 1) ∈ ℤ → ((𝑀 + 1)...(𝑀 + 1)) = {(𝑀 + 1)})
156140, 155syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((𝑀 + 1)...(𝑀 + 1)) = {(𝑀 + 1)})
157156uneq1d 4052 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)) = ({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)))
158154, 157eqtrd 2773 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((𝑀 + 1)...𝑁) = ({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)))
159158xpeq1d 5554 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (((𝑀 + 1)...𝑁) × {0}) = (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0}))
160159uneq2d 4053 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (((𝑀 + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0})))
161 xpundir 5592 . . . . . . . . . . . . . . . . . . . . . . 23 (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0}) = (({(𝑀 + 1)} × {0}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))
162 ovex 7203 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑀 + 1) ∈ V
163 c0ex 10713 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ∈ V
164162, 163xpsn 6913 . . . . . . . . . . . . . . . . . . . . . . . 24 ({(𝑀 + 1)} × {0}) = {⟨(𝑀 + 1), 0⟩}
165164uneq1i 4049 . . . . . . . . . . . . . . . . . . . . . . 23 (({(𝑀 + 1)} × {0}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))
166161, 165eqtri 2761 . . . . . . . . . . . . . . . . . . . . . 22 (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0}) = ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))
167166uneq2i 4050 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0})) = ((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
168 unass 4056 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
169167, 168eqtr4i 2764 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0})) = (((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))
170160, 169eqtrdi 2789 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (((𝑀 + 1)...𝑁) × {0})) = (((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
171170ad3antrrr 730 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (((𝑀 + 1)...𝑁) × {0})) = (((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
172162a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑀 + 1) ∈ V)
173163a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → 0 ∈ V)
174110eqcomd 2744 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((1...𝑀) ∪ {(𝑀 + 1)}) = (1...(𝑀 + 1)))
175174ad3antrrr 730 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → ((1...𝑀) ∪ {(𝑀 + 1)}) = (1...(𝑀 + 1)))
176 fveq2 6674 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = (𝑀 + 1) → ((1st𝑘)‘𝑛) = ((1st𝑘)‘(𝑀 + 1)))
177 fveq2 6674 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = (𝑀 + 1) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)))
178176, 177oveq12d 7188 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = (𝑀 + 1) → (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛)) = (((1st𝑘)‘(𝑀 + 1)) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1))))
179 simplrl 777 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → ((1st𝑘)‘(𝑀 + 1)) = 0)
180 imain 6424 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Fun (2nd𝑘) → ((2nd𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = (((2nd𝑘) “ (1...𝑗)) ∩ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))))
18176, 83, 1803syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((2nd𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = (((2nd𝑘) “ (1...𝑗)) ∩ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))))
182181ad2antlr 727 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = (((2nd𝑘) “ (1...𝑗)) ∩ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))))
183 elfznn0 13091 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℕ0)
184183nn0red 12037 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℝ)
185184ltp1d 11648 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ (0...𝑀) → 𝑗 < (𝑗 + 1))
186 fzdisj 13025 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1))) = ∅)
187185, 186syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (0...𝑀) → ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1))) = ∅)
188187imaeq2d 5903 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (0...𝑀) → ((2nd𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = ((2nd𝑘) “ ∅))
189 ima0 5919 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((2nd𝑘) “ ∅) = ∅
190188, 189eqtrdi 2789 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (0...𝑀) → ((2nd𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = ∅)
191182, 190sylan9req 2794 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((2nd𝑘) “ (1...𝑗)) ∩ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅)
192 simplr 769 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))
19392ad2antlr 727 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (2nd𝑘) Fn (1...(𝑀 + 1)))
194 nn0p1nn 12015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 ∈ ℕ0 → (𝑗 + 1) ∈ ℕ)
195 nnuz 12363 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ℕ = (ℤ‘1)
196194, 195eleqtrdi 2843 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ ℕ0 → (𝑗 + 1) ∈ (ℤ‘1))
197 fzss1 13037 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑗 + 1) ∈ (ℤ‘1) → ((𝑗 + 1)...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)))
198183, 196, 1973syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (0...𝑀) → ((𝑗 + 1)...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)))
199 elfzuz3 12995 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ (0...𝑀) → 𝑀 ∈ (ℤ𝑗))
200 eluzp1p1 12352 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑀 ∈ (ℤ𝑗) → (𝑀 + 1) ∈ (ℤ‘(𝑗 + 1)))
201 eluzfz2 13006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑀 + 1) ∈ (ℤ‘(𝑗 + 1)) → (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1)))
202199, 200, 2013syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1)))
203198, 202jca 515 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (0...𝑀) → (((𝑗 + 1)...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)) ∧ (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1))))
204 fnfvima 7006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((2nd𝑘) Fn (1...(𝑀 + 1)) ∧ ((𝑗 + 1)...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)) ∧ (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1))) → ((2nd𝑘)‘(𝑀 + 1)) ∈ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
2052043expb 1121 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((2nd𝑘) Fn (1...(𝑀 + 1)) ∧ (((𝑗 + 1)...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)) ∧ (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1)))) → ((2nd𝑘)‘(𝑀 + 1)) ∈ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
206193, 203, 205syl2an 599 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((2nd𝑘)‘(𝑀 + 1)) ∈ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
207192, 206eqeltrrd 2834 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑀 + 1) ∈ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
208 1ex 10715 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1 ∈ V
209 fnconstg 6566 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (1 ∈ V → (((2nd𝑘) “ (1...𝑗)) × {1}) Fn ((2nd𝑘) “ (1...𝑗)))
210208, 209ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((2nd𝑘) “ (1...𝑗)) × {1}) Fn ((2nd𝑘) “ (1...𝑗))
211 fnconstg 6566 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (0 ∈ V → (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
212163, 211ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))
213 fvun2 6760 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((2nd𝑘) “ (1...𝑗)) × {1}) Fn ((2nd𝑘) “ (1...𝑗)) ∧ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) ∧ ((((2nd𝑘) “ (1...𝑗)) ∩ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅ ∧ (𝑀 + 1) ∈ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = ((((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)))
214210, 212, 213mp3an12 1452 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((2nd𝑘) “ (1...𝑗)) ∩ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅ ∧ (𝑀 + 1) ∈ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = ((((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)))
215191, 207, 214syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = ((((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)))
216163fvconst2 6976 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑀 + 1) ∈ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) → ((((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)) = 0)
217207, 216syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)) = 0)
218215, 217eqtrd 2773 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = 0)
219218adantlrl 720 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = 0)
220179, 219oveq12d 7188 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (((1st𝑘)‘(𝑀 + 1)) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1))) = (0 + 0))
221 00id 10893 . . . . . . . . . . . . . . . . . . . . . 22 (0 + 0) = 0
222220, 221eqtrdi 2789 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (((1st𝑘)‘(𝑀 + 1)) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1))) = 0)
223178, 222sylan9eqr 2795 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 = (𝑀 + 1)) → (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛)) = 0)
224172, 173, 175, 223fmptapd 6943 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) = (𝑛 ∈ (1...(𝑀 + 1)) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))))
225224uneq1d 4052 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...(𝑀 + 1)) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
226171, 225eqtrd 2773 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (((𝑀 + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...(𝑀 + 1)) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
227 elmapfn 8475 . . . . . . . . . . . . . . . . . . . . . . 23 ((1st𝑘) ∈ ((0..^𝐾) ↑m (1...(𝑀 + 1))) → (1st𝑘) Fn (1...(𝑀 + 1)))
22861, 227syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (1st𝑘) Fn (1...(𝑀 + 1)))
229 fnssres 6459 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑘) Fn (1...(𝑀 + 1)) ∧ (1...𝑀) ⊆ (1...(𝑀 + 1))) → ((1st𝑘) ↾ (1...𝑀)) Fn (1...𝑀))
230228, 64, 229sylancl 589 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((1st𝑘) ↾ (1...𝑀)) Fn (1...𝑀))
231230ad3antlr 731 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((1st𝑘) ↾ (1...𝑀)) Fn (1...𝑀))
232 fnconstg 6566 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (0 ∈ V → (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) Fn ((2nd𝑘) “ ((𝑗 + 1)...𝑀)))
233163, 232ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) Fn ((2nd𝑘) “ ((𝑗 + 1)...𝑀))
234210, 233pm3.2i 474 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((2nd𝑘) “ (1...𝑗)) × {1}) Fn ((2nd𝑘) “ (1...𝑗)) ∧ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) Fn ((2nd𝑘) “ ((𝑗 + 1)...𝑀)))
235 imain 6424 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (Fun (2nd𝑘) → ((2nd𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = (((2nd𝑘) “ (1...𝑗)) ∩ ((2nd𝑘) “ ((𝑗 + 1)...𝑀))))
23676, 83, 2353syl 18 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((2nd𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = (((2nd𝑘) “ (1...𝑗)) ∩ ((2nd𝑘) “ ((𝑗 + 1)...𝑀))))
237 fzdisj 13025 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑀)) = ∅)
238185, 237syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (0...𝑀) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑀)) = ∅)
239238imaeq2d 5903 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (0...𝑀) → ((2nd𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = ((2nd𝑘) “ ∅))
240239, 189eqtrdi 2789 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0...𝑀) → ((2nd𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = ∅)
241236, 240sylan9req 2794 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (((2nd𝑘) “ (1...𝑗)) ∩ ((2nd𝑘) “ ((𝑗 + 1)...𝑀))) = ∅)
242 fnun 6449 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((2nd𝑘) “ (1...𝑗)) × {1}) Fn ((2nd𝑘) “ (1...𝑗)) ∧ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) Fn ((2nd𝑘) “ ((𝑗 + 1)...𝑀))) ∧ (((2nd𝑘) “ (1...𝑗)) ∩ ((2nd𝑘) “ ((𝑗 + 1)...𝑀))) = ∅) → ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (((2nd𝑘) “ (1...𝑗)) ∪ ((2nd𝑘) “ ((𝑗 + 1)...𝑀))))
243234, 241, 242sylancr 590 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (((2nd𝑘) “ (1...𝑗)) ∪ ((2nd𝑘) “ ((𝑗 + 1)...𝑀))))
244243ad4ant24 754 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (((2nd𝑘) “ (1...𝑗)) ∪ ((2nd𝑘) “ ((𝑗 + 1)...𝑀))))
245101adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((2nd𝑘) “ (1...(𝑀 + 1))) ∖ ((2nd𝑘) “ {(𝑀 + 1)})) = ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}))
24685ad3antlr 731 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((2nd𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = (((2nd𝑘) “ (1...(𝑀 + 1))) ∖ ((2nd𝑘) “ {(𝑀 + 1)})))
247183, 194syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈ ℕ)
248247, 195eleqtrdi 2843 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈ (ℤ‘1))
249 fzsplit2 13023 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑗 + 1) ∈ (ℤ‘1) ∧ 𝑀 ∈ (ℤ𝑗)) → (1...𝑀) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑀)))
250248, 199, 249syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (0...𝑀) → (1...𝑀) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑀)))
251128, 250sylan9eq 2793 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑀)))
252251imaeq2d 5903 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((2nd𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((2nd𝑘) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))))
253246, 252eqtr3d 2775 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((2nd𝑘) “ (1...(𝑀 + 1))) ∖ ((2nd𝑘) “ {(𝑀 + 1)})) = ((2nd𝑘) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))))
254125ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = (1...𝑀))
255245, 253, 2543eqtr3rd 2782 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (1...𝑀) = ((2nd𝑘) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))))
256 imaundi 5982 . . . . . . . . . . . . . . . . . . . . . . . 24 ((2nd𝑘) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))) = (((2nd𝑘) “ (1...𝑗)) ∪ ((2nd𝑘) “ ((𝑗 + 1)...𝑀)))
257255, 256eqtrdi 2789 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (1...𝑀) = (((2nd𝑘) “ (1...𝑗)) ∪ ((2nd𝑘) “ ((𝑗 + 1)...𝑀))))
258257fneq2d 6432 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀) ↔ ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (((2nd𝑘) “ (1...𝑗)) ∪ ((2nd𝑘) “ ((𝑗 + 1)...𝑀)))))
259244, 258mpbird 260 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀))
260 fzss2 13038 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑀 ∈ (ℤ𝑗) → (1...𝑗) ⊆ (1...𝑀))
261 resima2 5860 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((1...𝑗) ⊆ (1...𝑀) → (((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) = ((2nd𝑘) “ (1...𝑗)))
262199, 260, 2613syl 18 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0...𝑀) → (((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) = ((2nd𝑘) “ (1...𝑗)))
263262xpeq1d 5554 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (0...𝑀) → ((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) = (((2nd𝑘) “ (1...𝑗)) × {1}))
264183, 196syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈ (ℤ‘1))
265 fzss1 13037 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑗 + 1) ∈ (ℤ‘1) → ((𝑗 + 1)...𝑀) ⊆ (1...𝑀))
266 resima2 5860 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑗 + 1)...𝑀) ⊆ (1...𝑀) → (((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) = ((2nd𝑘) “ ((𝑗 + 1)...𝑀)))
267264, 265, 2663syl 18 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0...𝑀) → (((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) = ((2nd𝑘) “ ((𝑗 + 1)...𝑀)))
268267xpeq1d 5554 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (0...𝑀) → ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}) = (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}))
269263, 268uneq12d 4054 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (0...𝑀) → (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0})) = ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})))
270269adantl 485 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0})) = ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})))
271270fneq1d 6431 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀) ↔ ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀)))
272259, 271mpbird 260 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀))
273 fzfid 13432 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (1...𝑀) ∈ Fin)
274 inidm 4109 . . . . . . . . . . . . . . . . . . . 20 ((1...𝑀) ∩ (1...𝑀)) = (1...𝑀)
275 fvres 6693 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ (1...𝑀) → (((1st𝑘) ↾ (1...𝑀))‘𝑛) = ((1st𝑘)‘𝑛))
276275adantl 485 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → (((1st𝑘) ↾ (1...𝑀))‘𝑛) = ((1st𝑘)‘𝑛))
277 disjsn 4602 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ↔ ¬ (𝑀 + 1) ∈ (1...𝑀))
278122, 277sylibr 237 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅)
279278ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅)
280259, 279jca 515 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀) ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅))
281 fnconstg 6566 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 ∈ V → ({(𝑀 + 1)} × {0}) Fn {(𝑀 + 1)})
282163, 281ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . 24 ({(𝑀 + 1)} × {0}) Fn {(𝑀 + 1)}
283 fvun1 6759 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀) ∧ ({(𝑀 + 1)} × {0}) Fn {(𝑀 + 1)} ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ 𝑛 ∈ (1...𝑀))) → ((((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0}))‘𝑛) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))
284282, 283mp3an2 1450 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀) ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ 𝑛 ∈ (1...𝑀))) → ((((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0}))‘𝑛) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))
285284anassrs 471 . . . . . . . . . . . . . . . . . . . . . 22 (((((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀) ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅) ∧ 𝑛 ∈ (1...𝑀)) → ((((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0}))‘𝑛) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))
286280, 285sylan 583 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → ((((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0}))‘𝑛) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))
287247nnzd 12167 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈ ℤ)
288183nn0cnd 12038 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℂ)
289 pncan1 11142 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑗 ∈ ℂ → ((𝑗 + 1) − 1) = 𝑗)
290288, 289syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑗 ∈ (0...𝑀) → ((𝑗 + 1) − 1) = 𝑗)
291290fveq2d 6678 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑗 ∈ (0...𝑀) → (ℤ‘((𝑗 + 1) − 1)) = (ℤ𝑗))
292199, 291eleqtrrd 2836 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑗 ∈ (0...𝑀) → 𝑀 ∈ (ℤ‘((𝑗 + 1) − 1)))
293 fzsuc2 13056 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑗 + 1) ∈ ℤ ∧ 𝑀 ∈ (ℤ‘((𝑗 + 1) − 1))) → ((𝑗 + 1)...(𝑀 + 1)) = (((𝑗 + 1)...𝑀) ∪ {(𝑀 + 1)}))
294287, 292, 293syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑗 ∈ (0...𝑀) → ((𝑗 + 1)...(𝑀 + 1)) = (((𝑗 + 1)...𝑀) ∪ {(𝑀 + 1)}))
295294imaeq2d 5903 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑗 ∈ (0...𝑀) → ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) = ((2nd𝑘) “ (((𝑗 + 1)...𝑀) ∪ {(𝑀 + 1)})))
296 imaundi 5982 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((2nd𝑘) “ (((𝑗 + 1)...𝑀) ∪ {(𝑀 + 1)})) = (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) ∪ ((2nd𝑘) “ {(𝑀 + 1)}))
297295, 296eqtrdi 2789 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 ∈ (0...𝑀) → ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) = (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) ∪ ((2nd𝑘) “ {(𝑀 + 1)})))
298297xpeq1d 5554 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ (0...𝑀) → (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = ((((2nd𝑘) “ ((𝑗 + 1)...𝑀)) ∪ ((2nd𝑘) “ {(𝑀 + 1)})) × {0}))
299 xpundir 5592 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((2nd𝑘) “ ((𝑗 + 1)...𝑀)) ∪ ((2nd𝑘) “ {(𝑀 + 1)})) × {0}) = ((((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) ∪ (((2nd𝑘) “ {(𝑀 + 1)}) × {0}))
300298, 299eqtrdi 2789 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (0...𝑀) → (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = ((((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) ∪ (((2nd𝑘) “ {(𝑀 + 1)}) × {0})))
301300uneq2d 4053 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (0...𝑀) → ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) = ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) ∪ (((2nd𝑘) “ {(𝑀 + 1)}) × {0}))))
302 unass 4056 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ (((2nd𝑘) “ {(𝑀 + 1)}) × {0})) = ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) ∪ (((2nd𝑘) “ {(𝑀 + 1)}) × {0})))
303301, 302eqtr4di 2791 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (0...𝑀) → ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ (((2nd𝑘) “ {(𝑀 + 1)}) × {0})))
304303adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ (((2nd𝑘) “ {(𝑀 + 1)}) × {0})))
30598xpeq1d 5554 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ({((2nd𝑘)‘(𝑀 + 1))} × {0}) = (((2nd𝑘) “ {(𝑀 + 1)}) × {0}))
306305uneq2d 4053 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({((2nd𝑘)‘(𝑀 + 1))} × {0})) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ (((2nd𝑘) “ {(𝑀 + 1)}) × {0})))
307306adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({((2nd𝑘)‘(𝑀 + 1))} × {0})) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ (((2nd𝑘) “ {(𝑀 + 1)}) × {0})))
308304, 307eqtr4d 2776 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({((2nd𝑘)‘(𝑀 + 1))} × {0})))
30999xpeq1d 5554 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → ({((2nd𝑘)‘(𝑀 + 1))} × {0}) = ({(𝑀 + 1)} × {0}))
310309uneq2d 4053 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({((2nd𝑘)‘(𝑀 + 1))} × {0})) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0})))
311308, 310sylan9eq 2793 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ 𝑗 ∈ (0...𝑀)) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0})))
312311an32s 652 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0})))
313312fveq1d 6676 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛) = ((((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0}))‘𝑛))
314313adantr 484 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛) = ((((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0}))‘𝑛))
315269fveq1d 6676 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ (0...𝑀) → ((((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))
316315ad2antlr 727 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → ((((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))
317286, 314, 3163eqtr4rd 2784 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → ((((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))
318231, 272, 273, 273, 274, 276, 317offval 7433 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((1st𝑘) ↾ (1...𝑀)) ∘f + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) = (𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))))
319318uneq1d 4052 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((((1st𝑘) ↾ (1...𝑀)) ∘f + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (((𝑀 + 1)...𝑁) × {0})))
320319adantlrl 720 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → ((((1st𝑘) ↾ (1...𝑀)) ∘f + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (((𝑀 + 1)...𝑁) × {0})))
321228adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (1st𝑘) Fn (1...(𝑀 + 1)))
322210, 212pm3.2i 474 . . . . . . . . . . . . . . . . . . . . . 22 ((((2nd𝑘) “ (1...𝑗)) × {1}) Fn ((2nd𝑘) “ (1...𝑗)) ∧ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
323181, 190sylan9req 2794 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (((2nd𝑘) “ (1...𝑗)) ∩ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅)
324 fnun 6449 . . . . . . . . . . . . . . . . . . . . . 22 ((((((2nd𝑘) “ (1...𝑗)) × {1}) Fn ((2nd𝑘) “ (1...𝑗)) ∧ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) ∧ (((2nd𝑘) “ (1...𝑗)) ∩ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅) → ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (((2nd𝑘) “ (1...𝑗)) ∪ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))))
325322, 323, 324sylancr 590 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (((2nd𝑘) “ (1...𝑗)) ∪ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))))
326 peano2uz 12383 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑀 ∈ (ℤ𝑗) → (𝑀 + 1) ∈ (ℤ𝑗))
327199, 326syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈ (ℤ𝑗))
328 fzsplit2 13023 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑗 + 1) ∈ (ℤ‘1) ∧ (𝑀 + 1) ∈ (ℤ𝑗)) → (1...(𝑀 + 1)) = ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1))))
329264, 327, 328syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0...𝑀) → (1...(𝑀 + 1)) = ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1))))
330329imaeq2d 5903 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (0...𝑀) → ((2nd𝑘) “ (1...(𝑀 + 1))) = ((2nd𝑘) “ ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1)))))
331 imaundi 5982 . . . . . . . . . . . . . . . . . . . . . . . 24 ((2nd𝑘) “ ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1)))) = (((2nd𝑘) “ (1...𝑗)) ∪ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
332330, 331eqtr2di 2790 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (0...𝑀) → (((2nd𝑘) “ (1...𝑗)) ∪ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = ((2nd𝑘) “ (1...(𝑀 + 1))))
333332, 89sylan9eqr 2795 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (((2nd𝑘) “ (1...𝑗)) ∪ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = (1...(𝑀 + 1)))
334333fneq2d 6432 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (((2nd𝑘) “ (1...𝑗)) ∪ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) ↔ ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (1...(𝑀 + 1))))
335325, 334mpbid 235 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (1...(𝑀 + 1)))
336 fzfid 13432 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (1...(𝑀 + 1)) ∈ Fin)
337 inidm 4109 . . . . . . . . . . . . . . . . . . . 20 ((1...(𝑀 + 1)) ∩ (1...(𝑀 + 1))) = (1...(𝑀 + 1))
338 eqidd 2739 . . . . . . . . . . . . . . . . . . . 20 (((𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...(𝑀 + 1))) → ((1st𝑘)‘𝑛) = ((1st𝑘)‘𝑛))
339 eqidd 2739 . . . . . . . . . . . . . . . . . . . 20 (((𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...(𝑀 + 1))) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))
340321, 335, 336, 336, 337, 338, 339offval 7433 . . . . . . . . . . . . . . . . . . 19 ((𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → ((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) = (𝑛 ∈ (1...(𝑀 + 1)) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))))
341340uneq1d 4052 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...(𝑀 + 1)) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
342341ad4ant24 754 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...(𝑀 + 1)) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
343226, 320, 3423eqtr4rd 2784 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ((((1st𝑘) ↾ (1...𝑀)) ∘f + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})))
344343csbeq1d 3794 . . . . . . . . . . . . . . 15 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 = ((((1st𝑘) ↾ (1...𝑀)) ∘f + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵)
345344eqeq2d 2749 . . . . . . . . . . . . . 14 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑖 = (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵𝑖 = ((((1st𝑘) ↾ (1...𝑀)) ∘f + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵))
346345rexbidva 3206 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → (∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑘) ↾ (1...𝑀)) ∘f + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵))
347346ralbidv 3109 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑘) ↾ (1...𝑀)) ∘f + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵))
348347biimpd 232 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 → ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑘) ↾ (1...𝑀)) ∘f + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵))
349348impr 458 . . . . . . . . . 10 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵)) → ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑘) ↾ (1...𝑀)) ∘f + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵)
350139, 349sylan2b 597 . . . . . . . . 9 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑘) ↾ (1...𝑀)) ∘f + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵)
351 1st2nd2 7753 . . . . . . . . . . . 12 (𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → 𝑘 = ⟨(1st𝑘), (2nd𝑘)⟩)
352351ad2antlr 727 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → 𝑘 = ⟨(1st𝑘), (2nd𝑘)⟩)
353 fnsnsplit 6956 . . . . . . . . . . . . . . . 16 (((1st𝑘) Fn (1...(𝑀 + 1)) ∧ (𝑀 + 1) ∈ (1...(𝑀 + 1))) → (1st𝑘) = (((1st𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((1st𝑘)‘(𝑀 + 1))⟩}))
354228, 96, 353syl2anr 600 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → (1st𝑘) = (((1st𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((1st𝑘)‘(𝑀 + 1))⟩}))
355354adantr 484 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((1st𝑘)‘(𝑀 + 1)) = 0) → (1st𝑘) = (((1st𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((1st𝑘)‘(𝑀 + 1))⟩}))
356125reseq2d 5825 . . . . . . . . . . . . . . . 16 (𝜑 → ((1st𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((1st𝑘) ↾ (1...𝑀)))
357356adantr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ((1st𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((1st𝑘) ↾ (1...𝑀)))
358 opeq2 4761 . . . . . . . . . . . . . . . 16 (((1st𝑘)‘(𝑀 + 1)) = 0 → ⟨(𝑀 + 1), ((1st𝑘)‘(𝑀 + 1))⟩ = ⟨(𝑀 + 1), 0⟩)
359358sneqd 4528 . . . . . . . . . . . . . . 15 (((1st𝑘)‘(𝑀 + 1)) = 0 → {⟨(𝑀 + 1), ((1st𝑘)‘(𝑀 + 1))⟩} = {⟨(𝑀 + 1), 0⟩})
360 uneq12 4048 . . . . . . . . . . . . . . 15 ((((1st𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((1st𝑘) ↾ (1...𝑀)) ∧ {⟨(𝑀 + 1), ((1st𝑘)‘(𝑀 + 1))⟩} = {⟨(𝑀 + 1), 0⟩}) → (((1st𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((1st𝑘)‘(𝑀 + 1))⟩}) = (((1st𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}))
361357, 359, 360syl2an 599 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((1st𝑘)‘(𝑀 + 1)) = 0) → (((1st𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((1st𝑘)‘(𝑀 + 1))⟩}) = (((1st𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}))
362355, 361eqtrd 2773 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((1st𝑘)‘(𝑀 + 1)) = 0) → (1st𝑘) = (((1st𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}))
363362adantrr 717 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → (1st𝑘) = (((1st𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}))
364 fnsnsplit 6956 . . . . . . . . . . . . . . . 16 (((2nd𝑘) Fn (1...(𝑀 + 1)) ∧ (𝑀 + 1) ∈ (1...(𝑀 + 1))) → (2nd𝑘) = (((2nd𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((2nd𝑘)‘(𝑀 + 1))⟩}))
36592, 96, 364syl2anr 600 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → (2nd𝑘) = (((2nd𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((2nd𝑘)‘(𝑀 + 1))⟩}))
366365adantr 484 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (2nd𝑘) = (((2nd𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((2nd𝑘)‘(𝑀 + 1))⟩}))
367125reseq2d 5825 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((2nd𝑘) ↾ (1...𝑀)))
368367adantr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ((2nd𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((2nd𝑘) ↾ (1...𝑀)))
369 opeq2 4761 . . . . . . . . . . . . . . . 16 (((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → ⟨(𝑀 + 1), ((2nd𝑘)‘(𝑀 + 1))⟩ = ⟨(𝑀 + 1), (𝑀 + 1)⟩)
370369sneqd 4528 . . . . . . . . . . . . . . 15 (((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → {⟨(𝑀 + 1), ((2nd𝑘)‘(𝑀 + 1))⟩} = {⟨(𝑀 + 1), (𝑀 + 1)⟩})
371 uneq12 4048 . . . . . . . . . . . . . . 15 ((((2nd𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((2nd𝑘) ↾ (1...𝑀)) ∧ {⟨(𝑀 + 1), ((2nd𝑘)‘(𝑀 + 1))⟩} = {⟨(𝑀 + 1), (𝑀 + 1)⟩}) → (((2nd𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((2nd𝑘)‘(𝑀 + 1))⟩}) = (((2nd𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
372368, 370, 371syl2an 599 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (((2nd𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((2nd𝑘)‘(𝑀 + 1))⟩}) = (((2nd𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
373366, 372eqtrd 2773 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (2nd𝑘) = (((2nd𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
374373adantrl 716 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → (2nd𝑘) = (((2nd𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
375363, 374opeq12d 4769 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → ⟨(1st𝑘), (2nd𝑘)⟩ = ⟨(((1st𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}), (((2nd𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)
376352, 375eqtrd 2773 . . . . . . . . . 10 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → 𝑘 = ⟨(((1st𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}), (((2nd𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)
3773763adantr1 1170 . . . . . . . . 9 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → 𝑘 = ⟨(((1st𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}), (((2nd𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)
378 fvex 6687 . . . . . . . . . . . . . . . . . . 19 (1st𝑘) ∈ V
379378resex 5873 . . . . . . . . . . . . . . . . . 18 ((1st𝑘) ↾ (1...𝑀)) ∈ V
380379, 132op1std 7724 . . . . . . . . . . . . . . . . 17 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → (1st𝑡) = ((1st𝑘) ↾ (1...𝑀)))
381379, 132op2ndd 7725 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → (2nd𝑡) = ((2nd𝑘) ↾ (1...𝑀)))
382381imaeq1d 5902 . . . . . . . . . . . . . . . . . . 19 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → ((2nd𝑡) “ (1...𝑗)) = (((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)))
383382xpeq1d 5554 . . . . . . . . . . . . . . . . . 18 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → (((2nd𝑡) “ (1...𝑗)) × {1}) = ((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}))
384381imaeq1d 5902 . . . . . . . . . . . . . . . . . . 19 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → ((2nd𝑡) “ ((𝑗 + 1)...𝑀)) = (((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)))
385384xpeq1d 5554 . . . . . . . . . . . . . . . . . 18 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}) = ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))
386383, 385uneq12d 4054 . . . . . . . . . . . . . . . . 17 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0})) = (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0})))
387380, 386oveq12d 7188 . . . . . . . . . . . . . . . 16 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → ((1st𝑡) ∘f + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) = (((1st𝑘) ↾ (1...𝑀)) ∘f + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))))
388387uneq1d 4052 . . . . . . . . . . . . . . 15 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → (((1st𝑡) ∘f + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) = ((((1st𝑘) ↾ (1...𝑀)) ∘f + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})))
389388csbeq1d 3794 . . . . . . . . . . . . . 14 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → (((1st𝑡) ∘f + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 = ((((1st𝑘) ↾ (1...𝑀)) ∘f + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵)
390389eqeq2d 2749 . . . . . . . . . . . . 13 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → (𝑖 = (((1st𝑡) ∘f + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵𝑖 = ((((1st𝑘) ↾ (1...𝑀)) ∘f + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵))
391390rexbidv 3207 . . . . . . . . . . . 12 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → (∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘f + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑘) ↾ (1...𝑀)) ∘f + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵))
392391ralbidv 3109 . . . . . . . . . . 11 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘f + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑘) ↾ (1...𝑀)) ∘f + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵))
393380uneq1d 4052 . . . . . . . . . . . . 13 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → ((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) = (((1st𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}))
394381uneq1d 4052 . . . . . . . . . . . . 13 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = (((2nd𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
395393, 394opeq12d 4769 . . . . . . . . . . . 12 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ = ⟨(((1st𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}), (((2nd𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)
396395eqeq2d 2749 . . . . . . . . . . 11 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → (𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ↔ 𝑘 = ⟨(((1st𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}), (((2nd𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
397392, 396anbi12d 634 . . . . . . . . . 10 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘f + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) ↔ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑘) ↾ (1...𝑀)) ∘f + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵𝑘 = ⟨(((1st𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}), (((2nd𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)))
398397rspcev 3526 . . . . . . . . 9 ((⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑘) ↾ (1...𝑀)) ∘f + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵𝑘 = ⟨(((1st𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}), (((2nd𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)) → ∃𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘f + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
399137, 350, 377, 398syl12anc 836 . . . . . . . 8 (((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → ∃𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘f + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
400399ex 416 . . . . . . 7 ((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ∃𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘f + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)))
401 elrabi 3582 . . . . . . . . . . 11 (𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} → 𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))
402 elrabi 3582 . . . . . . . . . . 11 (𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} → 𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))
403401, 402anim12i 616 . . . . . . . . . 10 ((𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ∧ 𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}) → (𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})))
404 eqtr2 2759 . . . . . . . . . . . 12 ((𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)
40522, 24opth 5334 . . . . . . . . . . . . 13 (⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ↔ (((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}) ∧ ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})))
406 difeq1 4006 . . . . . . . . . . . . . . 15 (((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}) → (((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∖ {⟨(𝑀 + 1), 0⟩}) = (((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}) ∖ {⟨(𝑀 + 1), 0⟩}))
407 difun2 4370 . . . . . . . . . . . . . . 15 (((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑡) ∖ {⟨(𝑀 + 1), 0⟩})
408 difun2 4370 . . . . . . . . . . . . . . 15 (((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∖ {⟨(𝑀 + 1), 0⟩})
409406, 407, 4083eqtr3g 2796 . . . . . . . . . . . . . 14 (((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}) → ((1st𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∖ {⟨(𝑀 + 1), 0⟩}))
410 difeq1 4006 . . . . . . . . . . . . . . 15 (((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) → (((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = (((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
411 difun2 4370 . . . . . . . . . . . . . . 15 (((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩})
412 difun2 4370 . . . . . . . . . . . . . . 15 (((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩})
413410, 411, 4123eqtr3g 2796 . . . . . . . . . . . . . 14 (((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) → ((2nd𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
414409, 413anim12i 616 . . . . . . . . . . . . 13 ((((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}) ∧ ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})) → (((1st𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) ∧ ((2nd𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩})))
415405, 414sylbi 220 . . . . . . . . . . . 12 (⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (((1st𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) ∧ ((2nd𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩})))
416404, 415syl 17 . . . . . . . . . . 11 ((𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → (((1st𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) ∧ ((2nd𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩})))
417 elmapfn 8475 . . . . . . . . . . . . . . . . . . 19 ((1st𝑡) ∈ ((0..^𝐾) ↑m (1...𝑀)) → (1st𝑡) Fn (1...𝑀))
418 fnop 6446 . . . . . . . . . . . . . . . . . . . 20 (((1st𝑡) Fn (1...𝑀) ∧ ⟨(𝑀 + 1), 0⟩ ∈ (1st𝑡)) → (𝑀 + 1) ∈ (1...𝑀))
419418ex 416 . . . . . . . . . . . . . . . . . . 19 ((1st𝑡) Fn (1...𝑀) → (⟨(𝑀 + 1), 0⟩ ∈ (1st𝑡) → (𝑀 + 1) ∈ (1...𝑀)))
4209, 417, 4193syl 18 . . . . . . . . . . . . . . . . . 18 (𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (⟨(𝑀 + 1), 0⟩ ∈ (1st𝑡) → (𝑀 + 1) ∈ (1...𝑀)))
421420, 122nsyli 160 . . . . . . . . . . . . . . . . 17 (𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (𝜑 → ¬ ⟨(𝑀 + 1), 0⟩ ∈ (1st𝑡)))
422421impcom 411 . . . . . . . . . . . . . . . 16 ((𝜑𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ¬ ⟨(𝑀 + 1), 0⟩ ∈ (1st𝑡))
423 difsn 4686 . . . . . . . . . . . . . . . 16 (¬ ⟨(𝑀 + 1), 0⟩ ∈ (1st𝑡) → ((1st𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = (1st𝑡))
424422, 423syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ((1st𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = (1st𝑡))
425 xp1st 7746 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (1st𝑛) ∈ ((0..^𝐾) ↑m (1...𝑀)))
426 elmapfn 8475 . . . . . . . . . . . . . . . . . . 19 ((1st𝑛) ∈ ((0..^𝐾) ↑m (1...𝑀)) → (1st𝑛) Fn (1...𝑀))
427 fnop 6446 . . . . . . . . . . . . . . . . . . . 20 (((1st𝑛) Fn (1...𝑀) ∧ ⟨(𝑀 + 1), 0⟩ ∈ (1st𝑛)) → (𝑀 + 1) ∈ (1...𝑀))
428427ex 416 . . . . . . . . . . . . . . . . . . 19 ((1st𝑛) Fn (1...𝑀) → (⟨(𝑀 + 1), 0⟩ ∈ (1st𝑛) → (𝑀 + 1) ∈ (1...𝑀)))
429425, 426, 4283syl 18 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (⟨(𝑀 + 1), 0⟩ ∈ (1st𝑛) → (𝑀 + 1) ∈ (1...𝑀)))
430429, 122nsyli 160 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (𝜑 → ¬ ⟨(𝑀 + 1), 0⟩ ∈ (1st𝑛)))
431430impcom 411 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ¬ ⟨(𝑀 + 1), 0⟩ ∈ (1st𝑛))
432 difsn 4686 . . . . . . . . . . . . . . . 16 (¬ ⟨(𝑀 + 1), 0⟩ ∈ (1st𝑛) → ((1st𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) = (1st𝑛))
433431, 432syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ((1st𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) = (1st𝑛))
434424, 433eqeqan12d 2755 . . . . . . . . . . . . . 14 (((𝜑𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) ∧ (𝜑𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → (((1st𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) ↔ (1st𝑡) = (1st𝑛)))
435434anandis 678 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → (((1st𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) ↔ (1st𝑡) = (1st𝑛)))
436 f1ofn 6619 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑡):(1...𝑀)–1-1-onto→(1...𝑀) → (2nd𝑡) Fn (1...𝑀))
437 fnop 6446 . . . . . . . . . . . . . . . . . . . 20 (((2nd𝑡) Fn (1...𝑀) ∧ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd𝑡)) → (𝑀 + 1) ∈ (1...𝑀))
438437ex 416 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑡) Fn (1...𝑀) → (⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd𝑡) → (𝑀 + 1) ∈ (1...𝑀)))
43917, 436, 4383syl 18 . . . . . . . . . . . . . . . . . 18 (𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd𝑡) → (𝑀 + 1) ∈ (1...𝑀)))
440439, 122nsyli 160 . . . . . . . . . . . . . . . . 17 (𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (𝜑 → ¬ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd𝑡)))
441440impcom 411 . . . . . . . . . . . . . . . 16 ((𝜑𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ¬ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd𝑡))
442 difsn 4686 . . . . . . . . . . . . . . . 16 (¬ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd𝑡) → ((2nd𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = (2nd𝑡))
443441, 442syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ((2nd𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = (2nd𝑡))
444 xp2nd 7747 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (2nd𝑛) ∈ {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})
445 fvex 6687 . . . . . . . . . . . . . . . . . . . . 21 (2nd𝑛) ∈ V
446 f1oeq1 6606 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = (2nd𝑛) → (𝑓:(1...𝑀)–1-1-onto→(1...𝑀) ↔ (2nd𝑛):(1...𝑀)–1-1-onto→(1...𝑀)))
447445, 446elab 3573 . . . . . . . . . . . . . . . . . . . 20 ((2nd𝑛) ∈ {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ↔ (2nd𝑛):(1...𝑀)–1-1-onto→(1...𝑀))
448444, 447sylib 221 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (2nd𝑛):(1...𝑀)–1-1-onto→(1...𝑀))
449 f1ofn 6619 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑛):(1...𝑀)–1-1-onto→(1...𝑀) → (2nd𝑛) Fn (1...𝑀))
450 fnop 6446 . . . . . . . . . . . . . . . . . . . 20 (((2nd𝑛) Fn (1...𝑀) ∧ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd𝑛)) → (𝑀 + 1) ∈ (1...𝑀))
451450ex 416 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑛) Fn (1...𝑀) → (⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd𝑛) → (𝑀 + 1) ∈ (1...𝑀)))
452448, 449, 4513syl 18 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd𝑛) → (𝑀 + 1) ∈ (1...𝑀)))
453452, 122nsyli 160 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (𝜑 → ¬ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd𝑛)))
454453impcom 411 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ¬ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd𝑛))
455 difsn 4686 . . . . . . . . . . . . . . . 16 (¬ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd𝑛) → ((2nd𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = (2nd𝑛))
456454, 455syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ((2nd𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = (2nd𝑛))
457443, 456eqeqan12d 2755 . . . . . . . . . . . . . 14 (((𝜑𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) ∧ (𝜑𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → (((2nd𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ↔ (2nd𝑡) = (2nd𝑛)))
458457anandis 678 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → (((2nd𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ↔ (2nd𝑡) = (2nd𝑛)))
459435, 458anbi12d 634 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → ((((1st𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) ∧ ((2nd𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩})) ↔ ((1st𝑡) = (1st𝑛) ∧ (2nd𝑡) = (2nd𝑛))))
460 xpopth 7755 . . . . . . . . . . . . 13 ((𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → (((1st𝑡) = (1st𝑛) ∧ (2nd𝑡) = (2nd𝑛)) ↔ 𝑡 = 𝑛))
461460adantl 485 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → (((1st𝑡) = (1st𝑛) ∧ (2nd𝑡) = (2nd𝑛)) ↔ 𝑡 = 𝑛))
462459, 461bitrd 282 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → ((((1st𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) ∧ ((2nd𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩})) ↔ 𝑡 = 𝑛))
463416, 462syl5ib 247 . . . . . . . . . 10 ((𝜑 ∧ (𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → ((𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛))
464403, 463sylan2 596 . . . . . . . . 9 ((𝜑 ∧ (𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ∧ 𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵})) → ((𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛))
465464ralrimivva 3103 . . . . . . . 8 (𝜑 → ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ((𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛))
466465adantr 484 . . . . . . 7 ((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ((𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛))
467400, 466jctird 530 . . . . . 6 ((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (∃𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘f + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) ∧ ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ((𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛))))
468 fveq2 6674 . . . . . . . . . . 11 (𝑡 = 𝑛 → (1st𝑡) = (1st𝑛))
469468uneq1d 4052 . . . . . . . . . 10 (𝑡 = 𝑛 → ((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}))
470 fveq2 6674 . . . . . . . . . . 11 (𝑡 = 𝑛 → (2nd𝑡) = (2nd𝑛))
471470uneq1d 4052 . . . . . . . . . 10 (𝑡 = 𝑛 → ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
472469, 471opeq12d 4769 . . . . . . . . 9 (𝑡 = 𝑛 → ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)
473472eqeq2d 2749 . . . . . . . 8 (𝑡 = 𝑛 → (𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ↔ 𝑘 = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
474473reu4 3630 . . . . . . 7 (∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ↔ (∃𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ((𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛)))
47558rexrab 3595 . . . . . . . 8 (∃𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ↔ ∃𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘f + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
476475anbi1i 627 . . . . . . 7 ((∃𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ((𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛)) ↔ (∃𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘f + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) ∧ ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ((𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛)))
477474, 476bitri 278 . . . . . 6 (∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ↔ (∃𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘f + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) ∧ ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ((𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛)))
478467, 477syl6ibr 255 . . . . 5 ((𝜑𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
479478ralrimiva 3096 . . . 4 (𝜑 → ∀𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
480 fveq2 6674 . . . . . . . . . . . 12 (𝑠 = 𝑘 → (1st𝑠) = (1st𝑘))
481 fveq2 6674 . . . . . . . . . . . . . . 15 (𝑠 = 𝑘 → (2nd𝑠) = (2nd𝑘))
482481imaeq1d 5902 . . . . . . . . . . . . . 14 (𝑠 = 𝑘 → ((2nd𝑠) “ (1...𝑗)) = ((2nd𝑘) “ (1...𝑗)))
483482xpeq1d 5554 . . . . . . . . . . . . 13 (𝑠 = 𝑘 → (((2nd𝑠) “ (1...𝑗)) × {1}) = (((2nd𝑘) “ (1...𝑗)) × {1}))
484481imaeq1d 5902 . . . . . . . . . . . . . 14 (𝑠 = 𝑘 → ((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) = ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
485484xpeq1d 5554 . . . . . . . . . . . . 13 (𝑠 = 𝑘 → (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))
486483, 485uneq12d 4054 . . . . . . . . . . . 12 (𝑠 = 𝑘 → ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) = ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})))
487480, 486oveq12d 7188 . . . . . . . . . . 11 (𝑠 = 𝑘 → ((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) = ((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))))
488487uneq1d 4052 . . . . . . . . . 10 (𝑠 = 𝑘 → (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
489488csbeq1d 3794 . . . . . . . . 9 (𝑠 = 𝑘(((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 = (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵)
490489eqeq2d 2749 . . . . . . . 8 (𝑠 = 𝑘 → (𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵𝑖 = (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵))
491490rexbidv 3207 . . . . . . 7 (𝑠 = 𝑘 → (∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵))
492491ralbidv 3109 . . . . . 6 (𝑠 = 𝑘 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵))
493480fveq1d 6676 . . . . . . 7 (𝑠 = 𝑘 → ((1st𝑠)‘(𝑀 + 1)) = ((1st𝑘)‘(𝑀 + 1)))
494493eqeq1d 2740 . . . . . 6 (𝑠 = 𝑘 → (((1st𝑠)‘(𝑀 + 1)) = 0 ↔ ((1st𝑘)‘(𝑀 + 1)) = 0))
495481fveq1d 6676 . . . . . . 7 (𝑠 = 𝑘 → ((2nd𝑠)‘(𝑀 + 1)) = ((2nd𝑘)‘(𝑀 + 1)))
496495eqeq1d 2740 . . . . . 6 (𝑠 = 𝑘 → (((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1) ↔ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)))
497492, 494, 4963anbi123d 1437 . . . . 5 (𝑠 = 𝑘 → ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1)) ↔ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))))
498497ralrab 3593 . . . 4 (∀𝑘 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))}∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ↔ ∀𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘f + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
499479, 498sylibr 237 . . 3 (𝜑 → ∀𝑘 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))}∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)
500 eqid 2738 . . . 4 (𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ↦ ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) = (𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ↦ ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)
501500f1ompt 6885 . . 3 ((𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ↦ ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩):{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))} ↔ (∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))} ∧ ∀𝑘 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))}∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
50260, 499, 501sylanbrc 586 . 2 (𝜑 → (𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ↦ ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩):{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))})
503 ovex 7203 . . . . 5 ((0..^𝐾) ↑m (1...𝑀)) ∈ V
504 ovex 7203 . . . . . 6 ((1...𝑀) ↑m (1...𝑀)) ∈ V
505 f1of 6618 . . . . . . . 8 (𝑓:(1...𝑀)–1-1-onto→(1...𝑀) → 𝑓:(1...𝑀)⟶(1...𝑀))
506505ss2abi 3956 . . . . . . 7 {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ⊆ {𝑓𝑓:(1...𝑀)⟶(1...𝑀)}
50768, 68mapval 8449 . . . . . . 7 ((1...𝑀) ↑m (1...𝑀)) = {𝑓𝑓:(1...𝑀)⟶(1...𝑀)}
508506, 507sseqtrri 3914 . . . . . 6 {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ⊆ ((1...𝑀) ↑m (1...𝑀))
509504, 508ssexi 5190 . . . . 5 {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ∈ V
510503, 509xpex 7494 . . . 4 (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∈ V
511510rabex 5200 . . 3 {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ∈ V
512511f1oen 8576 . 2 ((𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ↦ ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩):{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))} → {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))})
513502, 512syl 17 1 (𝜑 → {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘f + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  {cab 2716  wral 3053  wrex 3054  ∃!wreu 3055  {crab 3057  Vcvv 3398  csb 3790  cdif 3840  cun 3841  cin 3842  wss 3843  c0 4211  {csn 4516  cop 4522   class class class wbr 5030  cmpt 5110   × cxp 5523  ccnv 5524  cres 5527  cima 5528  Fun wfun 6333   Fn wfn 6334  wf 6335  1-1wf1 6336  ontowfo 6337  1-1-ontowf1o 6338  cfv 6339  (class class class)co 7170  f cof 7423  1st c1st 7712  2nd c2nd 7713  m cmap 8437  cen 8552  Fincfn 8555  cc 10613  cr 10614  0cc0 10615  1c1 10616   + caddc 10618   < clt 10753  cle 10754  cmin 10948  cn 11716  0cn0 11976  cz 12062  cuz 12324  ...cfz 12981  ..^cfzo 13124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-of 7425  df-om 7600  df-1st 7714  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-er 8320  df-map 8439  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-nn 11717  df-n0 11977  df-z 12063  df-uz 12325  df-fz 12982  df-fzo 13125
This theorem is referenced by:  poimirlem28  35428
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