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Theorem poimirlem3 33044
Description: Lemma for poimir 33074 to add an interior point to an admissible face on the back face of the 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)
poimirlem3.4 (𝜑𝑇:(1...𝑀)⟶(0..^𝐾))
poimirlem3.5 (𝜑𝑈:(1...𝑀)–1-1-onto→(1...𝑀))
Assertion
Ref Expression
poimirlem3 (𝜑 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 → (⟨(𝑇 ∪ {⟨(𝑀 + 1), 0⟩}), (𝑈 ∪ {⟨(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 = (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1)))))
Distinct variable groups:   𝑓,𝑖,𝑗,𝑝   𝜑,𝑗   𝑗,𝑀   𝑗,𝑁   𝑇,𝑗   𝑈,𝑗   𝜑,𝑖,𝑝   𝐵,𝑓,𝑖,𝑗   𝑓,𝐾,𝑖,𝑗,𝑝   𝑓,𝑀,𝑖,𝑝   𝑓,𝑁,𝑖,𝑝   𝑇,𝑓,𝑖,𝑝   𝑈,𝑓,𝑖,𝑝
Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑝)

Proof of Theorem poimirlem3
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 poimirlem3.4 . . . . . . . . . . . . . . 15 (𝜑𝑇:(1...𝑀)⟶(0..^𝐾))
2 ffn 6002 . . . . . . . . . . . . . . 15 (𝑇:(1...𝑀)⟶(0..^𝐾) → 𝑇 Fn (1...𝑀))
31, 2syl 17 . . . . . . . . . . . . . 14 (𝜑𝑇 Fn (1...𝑀))
43adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0...𝑀)) → 𝑇 Fn (1...𝑀))
5 1ex 9979 . . . . . . . . . . . . . . . . 17 1 ∈ V
6 fnconstg 6050 . . . . . . . . . . . . . . . . 17 (1 ∈ V → ((𝑈 “ (1...𝑗)) × {1}) Fn (𝑈 “ (1...𝑗)))
75, 6ax-mp 5 . . . . . . . . . . . . . . . 16 ((𝑈 “ (1...𝑗)) × {1}) Fn (𝑈 “ (1...𝑗))
8 c0ex 9978 . . . . . . . . . . . . . . . . 17 0 ∈ V
9 fnconstg 6050 . . . . . . . . . . . . . . . . 17 (0 ∈ V → ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) Fn (𝑈 “ ((𝑗 + 1)...𝑀)))
108, 9ax-mp 5 . . . . . . . . . . . . . . . 16 ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) Fn (𝑈 “ ((𝑗 + 1)...𝑀))
117, 10pm3.2i 471 . . . . . . . . . . . . . . 15 (((𝑈 “ (1...𝑗)) × {1}) Fn (𝑈 “ (1...𝑗)) ∧ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) Fn (𝑈 “ ((𝑗 + 1)...𝑀)))
12 poimirlem3.5 . . . . . . . . . . . . . . . . 17 (𝜑𝑈:(1...𝑀)–1-1-onto→(1...𝑀))
13 dff1o3 6100 . . . . . . . . . . . . . . . . . 18 (𝑈:(1...𝑀)–1-1-onto→(1...𝑀) ↔ (𝑈:(1...𝑀)–onto→(1...𝑀) ∧ Fun 𝑈))
1413simprbi 480 . . . . . . . . . . . . . . . . 17 (𝑈:(1...𝑀)–1-1-onto→(1...𝑀) → Fun 𝑈)
15 imain 5932 . . . . . . . . . . . . . . . . 17 (Fun 𝑈 → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑀))))
1612, 14, 153syl 18 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑀))))
17 elfznn0 12374 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℕ0)
1817nn0red 11296 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℝ)
1918ltp1d 10898 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (0...𝑀) → 𝑗 < (𝑗 + 1))
20 fzdisj 12310 . . . . . . . . . . . . . . . . . . 19 (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑀)) = ∅)
2119, 20syl 17 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (0...𝑀) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑀)) = ∅)
2221imaeq2d 5425 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (0...𝑀) → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = (𝑈 “ ∅))
23 ima0 5440 . . . . . . . . . . . . . . . . 17 (𝑈 “ ∅) = ∅
2422, 23syl6eq 2671 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (0...𝑀) → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = ∅)
2516, 24sylan9req 2676 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (0...𝑀)) → ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑀))) = ∅)
26 fnun 5955 . . . . . . . . . . . . . . 15 (((((𝑈 “ (1...𝑗)) × {1}) Fn (𝑈 “ (1...𝑗)) ∧ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) Fn (𝑈 “ ((𝑗 + 1)...𝑀))) ∧ ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑀))) = ∅) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})) Fn ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑀))))
2711, 25, 26sylancr 694 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (0...𝑀)) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})) Fn ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑀))))
28 imaundi 5504 . . . . . . . . . . . . . . . 16 (𝑈 “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))) = ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑀)))
29 nn0p1nn 11276 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ ℕ0 → (𝑗 + 1) ∈ ℕ)
30 nnuz 11667 . . . . . . . . . . . . . . . . . . . . . 22 ℕ = (ℤ‘1)
3129, 30syl6eleq 2708 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ ℕ0 → (𝑗 + 1) ∈ (ℤ‘1))
3217, 31syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈ (ℤ‘1))
33 elfzuz3 12281 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0...𝑀) → 𝑀 ∈ (ℤ𝑗))
34 fzsplit2 12308 . . . . . . . . . . . . . . . . . . . 20 (((𝑗 + 1) ∈ (ℤ‘1) ∧ 𝑀 ∈ (ℤ𝑗)) → (1...𝑀) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑀)))
3532, 33, 34syl2anc 692 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (0...𝑀) → (1...𝑀) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑀)))
3635eqcomd 2627 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (0...𝑀) → ((1...𝑗) ∪ ((𝑗 + 1)...𝑀)) = (1...𝑀))
3736imaeq2d 5425 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (0...𝑀) → (𝑈 “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))) = (𝑈 “ (1...𝑀)))
38 f1ofo 6101 . . . . . . . . . . . . . . . . . 18 (𝑈:(1...𝑀)–1-1-onto→(1...𝑀) → 𝑈:(1...𝑀)–onto→(1...𝑀))
39 foima 6077 . . . . . . . . . . . . . . . . . 18 (𝑈:(1...𝑀)–onto→(1...𝑀) → (𝑈 “ (1...𝑀)) = (1...𝑀))
4012, 38, 393syl 18 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑈 “ (1...𝑀)) = (1...𝑀))
4137, 40sylan9eqr 2677 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (0...𝑀)) → (𝑈 “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))) = (1...𝑀))
4228, 41syl5eqr 2669 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (0...𝑀)) → ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑀))) = (1...𝑀))
4342fneq2d 5940 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (0...𝑀)) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})) Fn ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑀))) ↔ (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀)))
4427, 43mpbid 222 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0...𝑀)) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀))
45 ovex 6632 . . . . . . . . . . . . . 14 (1...𝑀) ∈ V
4645a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0...𝑀)) → (1...𝑀) ∈ V)
47 inidm 3800 . . . . . . . . . . . . 13 ((1...𝑀) ∩ (1...𝑀)) = (1...𝑀)
48 eqidd 2622 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → (𝑇𝑛) = (𝑇𝑛))
49 eqidd 2622 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛) = ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))
504, 44, 46, 46, 47, 48, 49offval 6857 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0...𝑀)) → (𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) = (𝑛 ∈ (1...𝑀) ↦ ((𝑇𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))))
51 poimirlem4.2 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 ∈ ℕ0)
52 nn0p1nn 11276 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ ℕ0 → (𝑀 + 1) ∈ ℕ)
5351, 52syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑀 + 1) ∈ ℕ)
5453nnzd 11425 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑀 + 1) ∈ ℤ)
55 uzid 11646 . . . . . . . . . . . . . . . . . 18 ((𝑀 + 1) ∈ ℤ → (𝑀 + 1) ∈ (ℤ‘(𝑀 + 1)))
56 peano2uz 11685 . . . . . . . . . . . . . . . . . 18 ((𝑀 + 1) ∈ (ℤ‘(𝑀 + 1)) → ((𝑀 + 1) + 1) ∈ (ℤ‘(𝑀 + 1)))
5754, 55, 563syl 18 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑀 + 1) + 1) ∈ (ℤ‘(𝑀 + 1)))
58 poimirlem4.3 . . . . . . . . . . . . . . . . . 18 (𝜑𝑀 < 𝑁)
5951nn0zd 11424 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑀 ∈ ℤ)
60 poimir.0 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑁 ∈ ℕ)
6160nnzd 11425 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ ℤ)
62 zltp1le 11371 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁))
63 peano2z 11362 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ ℤ → (𝑀 + 1) ∈ ℤ)
64 eluz 11645 . . . . . . . . . . . . . . . . . . . . 21 (((𝑀 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ‘(𝑀 + 1)) ↔ (𝑀 + 1) ≤ 𝑁))
6563, 64sylan 488 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ‘(𝑀 + 1)) ↔ (𝑀 + 1) ≤ 𝑁))
6662, 65bitr4d 271 . . . . . . . . . . . . . . . . . . 19 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁𝑁 ∈ (ℤ‘(𝑀 + 1))))
6759, 61, 66syl2anc 692 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑀 < 𝑁𝑁 ∈ (ℤ‘(𝑀 + 1))))
6858, 67mpbid 222 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ (ℤ‘(𝑀 + 1)))
69 fzsplit2 12308 . . . . . . . . . . . . . . . . 17 ((((𝑀 + 1) + 1) ∈ (ℤ‘(𝑀 + 1)) ∧ 𝑁 ∈ (ℤ‘(𝑀 + 1))) → ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)))
7057, 68, 69syl2anc 692 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)))
71 fzsn 12325 . . . . . . . . . . . . . . . . . 18 ((𝑀 + 1) ∈ ℤ → ((𝑀 + 1)...(𝑀 + 1)) = {(𝑀 + 1)})
7254, 71syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑀 + 1)...(𝑀 + 1)) = {(𝑀 + 1)})
7372uneq1d 3744 . . . . . . . . . . . . . . . 16 (𝜑 → (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)) = ({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)))
7470, 73eqtrd 2655 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑀 + 1)...𝑁) = ({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)))
7574xpeq1d 5098 . . . . . . . . . . . . . 14 (𝜑 → (((𝑀 + 1)...𝑁) × {0}) = (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0}))
76 xpundir 5133 . . . . . . . . . . . . . . 15 (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0}) = (({(𝑀 + 1)} × {0}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))
77 ovex 6632 . . . . . . . . . . . . . . . . 17 (𝑀 + 1) ∈ V
7877, 8xpsn 6361 . . . . . . . . . . . . . . . 16 ({(𝑀 + 1)} × {0}) = {⟨(𝑀 + 1), 0⟩}
7978uneq1i 3741 . . . . . . . . . . . . . . 15 (({(𝑀 + 1)} × {0}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))
8076, 79eqtri 2643 . . . . . . . . . . . . . 14 (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0}) = ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))
8175, 80syl6eq 2671 . . . . . . . . . . . . 13 (𝜑 → (((𝑀 + 1)...𝑁) × {0}) = ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
8281adantr 481 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0...𝑀)) → (((𝑀 + 1)...𝑁) × {0}) = ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
8350, 82uneq12d 3746 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0...𝑀)) → ((𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...𝑀) ↦ ((𝑇𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))) ∪ ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))))
84 unass 3748 . . . . . . . . . . 11 (((𝑛 ∈ (1...𝑀) ↦ ((𝑇𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...𝑀) ↦ ((𝑇𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))) ∪ ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
8583, 84syl6eqr 2673 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0...𝑀)) → ((𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) = (((𝑛 ∈ (1...𝑀) ↦ ((𝑇𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
8651nn0red 11296 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑀 ∈ ℝ)
8786ltp1d 10898 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 < (𝑀 + 1))
8853nnred 10979 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑀 + 1) ∈ ℝ)
8986, 88ltnled 10128 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀))
9087, 89mpbid 222 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ¬ (𝑀 + 1) ≤ 𝑀)
91 elfzle2 12287 . . . . . . . . . . . . . . . . . . 19 ((𝑀 + 1) ∈ (1...𝑀) → (𝑀 + 1) ≤ 𝑀)
9290, 91nsyl 135 . . . . . . . . . . . . . . . . . 18 (𝜑 → ¬ (𝑀 + 1) ∈ (1...𝑀))
93 disjsn 4216 . . . . . . . . . . . . . . . . . 18 (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ↔ ¬ (𝑀 + 1) ∈ (1...𝑀))
9492, 93sylibr 224 . . . . . . . . . . . . . . . . 17 (𝜑 → ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅)
95 eqid 2621 . . . . . . . . . . . . . . . . . . 19 {⟨(𝑀 + 1), 0⟩} = {⟨(𝑀 + 1), 0⟩}
9677, 8fsn 6356 . . . . . . . . . . . . . . . . . . 19 ({⟨(𝑀 + 1), 0⟩}:{(𝑀 + 1)}⟶{0} ↔ {⟨(𝑀 + 1), 0⟩} = {⟨(𝑀 + 1), 0⟩})
9795, 96mpbir 221 . . . . . . . . . . . . . . . . . 18 {⟨(𝑀 + 1), 0⟩}:{(𝑀 + 1)}⟶{0}
98 fun 6023 . . . . . . . . . . . . . . . . . 18 (((𝑇:(1...𝑀)⟶(0..^𝐾) ∧ {⟨(𝑀 + 1), 0⟩}:{(𝑀 + 1)}⟶{0}) ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅) → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):((1...𝑀) ∪ {(𝑀 + 1)})⟶((0..^𝐾) ∪ {0}))
9997, 98mpanl2 716 . . . . . . . . . . . . . . . . 17 ((𝑇:(1...𝑀)⟶(0..^𝐾) ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅) → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):((1...𝑀) ∪ {(𝑀 + 1)})⟶((0..^𝐾) ∪ {0}))
1001, 94, 99syl2anc 692 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):((1...𝑀) ∪ {(𝑀 + 1)})⟶((0..^𝐾) ∪ {0}))
101 1z 11351 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℤ
102 nn0uz 11666 . . . . . . . . . . . . . . . . . . . . 21 0 = (ℤ‘0)
103 1m1e0 11033 . . . . . . . . . . . . . . . . . . . . . 22 (1 − 1) = 0
104103fveq2i 6151 . . . . . . . . . . . . . . . . . . . . 21 (ℤ‘(1 − 1)) = (ℤ‘0)
105102, 104eqtr4i 2646 . . . . . . . . . . . . . . . . . . . 20 0 = (ℤ‘(1 − 1))
10651, 105syl6eleq 2708 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑀 ∈ (ℤ‘(1 − 1)))
107 fzsuc2 12340 . . . . . . . . . . . . . . . . . . 19 ((1 ∈ ℤ ∧ 𝑀 ∈ (ℤ‘(1 − 1))) → (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)}))
108101, 106, 107sylancr 694 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)}))
109108eqcomd 2627 . . . . . . . . . . . . . . . . 17 (𝜑 → ((1...𝑀) ∪ {(𝑀 + 1)}) = (1...(𝑀 + 1)))
110 poimirlem4.1 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐾 ∈ ℕ)
111 lbfzo0 12448 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ (0..^𝐾) ↔ 𝐾 ∈ ℕ)
112110, 111sylibr 224 . . . . . . . . . . . . . . . . . . 19 (𝜑 → 0 ∈ (0..^𝐾))
113112snssd 4309 . . . . . . . . . . . . . . . . . 18 (𝜑 → {0} ⊆ (0..^𝐾))
114 ssequn2 3764 . . . . . . . . . . . . . . . . . 18 ({0} ⊆ (0..^𝐾) ↔ ((0..^𝐾) ∪ {0}) = (0..^𝐾))
115113, 114sylib 208 . . . . . . . . . . . . . . . . 17 (𝜑 → ((0..^𝐾) ∪ {0}) = (0..^𝐾))
116109, 115feq23d 5997 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):((1...𝑀) ∪ {(𝑀 + 1)})⟶((0..^𝐾) ∪ {0}) ↔ (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):(1...(𝑀 + 1))⟶(0..^𝐾)))
117100, 116mpbid 222 . . . . . . . . . . . . . . 15 (𝜑 → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):(1...(𝑀 + 1))⟶(0..^𝐾))
118 ffn 6002 . . . . . . . . . . . . . . 15 ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):(1...(𝑀 + 1))⟶(0..^𝐾) → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) Fn (1...(𝑀 + 1)))
119117, 118syl 17 . . . . . . . . . . . . . 14 (𝜑 → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) Fn (1...(𝑀 + 1)))
120119adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0...𝑀)) → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) Fn (1...(𝑀 + 1)))
121 fnconstg 6050 . . . . . . . . . . . . . . . . 17 (1 ∈ V → (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)))
1225, 121ax-mp 5 . . . . . . . . . . . . . . . 16 (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗))
123 fnconstg 6050 . . . . . . . . . . . . . . . . 17 (0 ∈ V → (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))))
1248, 123ax-mp 5 . . . . . . . . . . . . . . . 16 (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))
125122, 124pm3.2i 471 . . . . . . . . . . . . . . 15 ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∧ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))))
12677, 77f1osn 6133 . . . . . . . . . . . . . . . . . . 19 {⟨(𝑀 + 1), (𝑀 + 1)⟩}:{(𝑀 + 1)}–1-1-onto→{(𝑀 + 1)}
127 f1oun 6113 . . . . . . . . . . . . . . . . . . 19 (((𝑈:(1...𝑀)–1-1-onto→(1...𝑀) ∧ {⟨(𝑀 + 1), (𝑀 + 1)⟩}:{(𝑀 + 1)}–1-1-onto→{(𝑀 + 1)}) ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅)) → (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)}))
128126, 127mpanl2 716 . . . . . . . . . . . . . . . . . 18 ((𝑈:(1...𝑀)–1-1-onto→(1...𝑀) ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅)) → (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)}))
12912, 94, 94, 128syl12anc 1321 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)}))
130 dff1o3 6100 . . . . . . . . . . . . . . . . . 18 ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)}) ↔ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–onto→((1...𝑀) ∪ {(𝑀 + 1)}) ∧ Fun (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})))
131130simprbi 480 . . . . . . . . . . . . . . . . 17 ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)}) → Fun (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
132 imain 5932 . . . . . . . . . . . . . . . . 17 (Fun (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∩ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))))
133129, 131, 1323syl 18 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∩ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))))
134 fzdisj 12310 . . . . . . . . . . . . . . . . . . 19 (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1))) = ∅)
13519, 134syl 17 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (0...𝑀) → ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1))) = ∅)
136135imaeq2d 5425 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (0...𝑀) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ∅))
137 ima0 5440 . . . . . . . . . . . . . . . . 17 ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ∅) = ∅
138136, 137syl6eq 2671 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (0...𝑀) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = ∅)
139133, 138sylan9req 2676 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (0...𝑀)) → (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∩ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅)
140 fnun 5955 . . . . . . . . . . . . . . 15 ((((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∧ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))) ∧ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∩ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅) → ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∪ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))))
141125, 139, 140sylancr 694 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (0...𝑀)) → ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∪ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))))
142 f1ofo 6101 . . . . . . . . . . . . . . . . . . 19 ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)}) → (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–onto→((1...𝑀) ∪ {(𝑀 + 1)}))
143 foima 6077 . . . . . . . . . . . . . . . . . . 19 ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–onto→((1...𝑀) ∪ {(𝑀 + 1)}) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑀) ∪ {(𝑀 + 1)})) = ((1...𝑀) ∪ {(𝑀 + 1)}))
144129, 142, 1433syl 18 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑀) ∪ {(𝑀 + 1)})) = ((1...𝑀) ∪ {(𝑀 + 1)}))
145108imaeq2d 5425 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...(𝑀 + 1))) = ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑀) ∪ {(𝑀 + 1)})))
146144, 145, 1083eqtr4d 2665 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...(𝑀 + 1))) = (1...(𝑀 + 1)))
147 peano2uz 11685 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ (ℤ𝑗) → (𝑀 + 1) ∈ (ℤ𝑗))
14833, 147syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈ (ℤ𝑗))
149 fzsplit2 12308 . . . . . . . . . . . . . . . . . . 19 (((𝑗 + 1) ∈ (ℤ‘1) ∧ (𝑀 + 1) ∈ (ℤ𝑗)) → (1...(𝑀 + 1)) = ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1))))
15032, 148, 149syl2anc 692 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (0...𝑀) → (1...(𝑀 + 1)) = ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1))))
151150imaeq2d 5425 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (0...𝑀) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...(𝑀 + 1))) = ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1)))))
152146, 151sylan9req 2676 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (0...𝑀)) → (1...(𝑀 + 1)) = ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1)))))
153 imaundi 5504 . . . . . . . . . . . . . . . 16 ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1)))) = (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∪ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))))
154152, 153syl6eq 2671 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (0...𝑀)) → (1...(𝑀 + 1)) = (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∪ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))))
155154fneq2d 5940 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (0...𝑀)) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (1...(𝑀 + 1)) ↔ ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∪ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))))))
156141, 155mpbird 247 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0...𝑀)) → ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (1...(𝑀 + 1)))
157 ovex 6632 . . . . . . . . . . . . . 14 (1...(𝑀 + 1)) ∈ V
158157a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0...𝑀)) → (1...(𝑀 + 1)) ∈ V)
159 inidm 3800 . . . . . . . . . . . . 13 ((1...(𝑀 + 1)) ∩ (1...(𝑀 + 1))) = (1...(𝑀 + 1))
160 eqidd 2622 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...(𝑀 + 1))) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) = ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛))
161 eqidd 2622 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...(𝑀 + 1))) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛) = (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))
162120, 156, 158, 158, 159, 160, 161offval 6857 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0...𝑀)) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) = (𝑛 ∈ (1...(𝑀 + 1)) ↦ (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) + (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))))
16377a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0...𝑀)) → (𝑀 + 1) ∈ V)
1648a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0...𝑀)) → 0 ∈ V)
165109adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0...𝑀)) → ((1...𝑀) ∪ {(𝑀 + 1)}) = (1...(𝑀 + 1)))
166 fveq2 6148 . . . . . . . . . . . . . . . . 17 (𝑛 = (𝑀 + 1) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) = ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)))
16777snid 4179 . . . . . . . . . . . . . . . . . . . 20 (𝑀 + 1) ∈ {(𝑀 + 1)}
16877, 8fnsn 5904 . . . . . . . . . . . . . . . . . . . . 21 {⟨(𝑀 + 1), 0⟩} Fn {(𝑀 + 1)}
169 fvun2 6227 . . . . . . . . . . . . . . . . . . . . 21 ((𝑇 Fn (1...𝑀) ∧ {⟨(𝑀 + 1), 0⟩} Fn {(𝑀 + 1)} ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ (𝑀 + 1) ∈ {(𝑀 + 1)})) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = ({⟨(𝑀 + 1), 0⟩}‘(𝑀 + 1)))
170168, 169mp3an2 1409 . . . . . . . . . . . . . . . . . . . 20 ((𝑇 Fn (1...𝑀) ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ (𝑀 + 1) ∈ {(𝑀 + 1)})) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = ({⟨(𝑀 + 1), 0⟩}‘(𝑀 + 1)))
171167, 170mpanr2 719 . . . . . . . . . . . . . . . . . . 19 ((𝑇 Fn (1...𝑀) ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = ({⟨(𝑀 + 1), 0⟩}‘(𝑀 + 1)))
1723, 94, 171syl2anc 692 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = ({⟨(𝑀 + 1), 0⟩}‘(𝑀 + 1)))
17377, 8fvsn 6400 . . . . . . . . . . . . . . . . . 18 ({⟨(𝑀 + 1), 0⟩}‘(𝑀 + 1)) = 0
174172, 173syl6eq 2671 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0)
175166, 174sylan9eqr 2677 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 = (𝑀 + 1)) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) = 0)
176175adantlr 750 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑛 = (𝑀 + 1)) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) = 0)
177 fveq2 6148 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑀 + 1) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛) = (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)))
178 imadmrn 5435 . . . . . . . . . . . . . . . . . . . . . . . 24 (({(𝑀 + 1)} × {(𝑀 + 1)}) “ dom ({(𝑀 + 1)} × {(𝑀 + 1)})) = ran ({(𝑀 + 1)} × {(𝑀 + 1)})
17977, 77xpsn 6361 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ({(𝑀 + 1)} × {(𝑀 + 1)}) = {⟨(𝑀 + 1), (𝑀 + 1)⟩}
180179imaeq1i 5422 . . . . . . . . . . . . . . . . . . . . . . . . 25 (({(𝑀 + 1)} × {(𝑀 + 1)}) “ dom ({(𝑀 + 1)} × {(𝑀 + 1)})) = ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ dom ({(𝑀 + 1)} × {(𝑀 + 1)}))
181 dmxpid 5305 . . . . . . . . . . . . . . . . . . . . . . . . . 26 dom ({(𝑀 + 1)} × {(𝑀 + 1)}) = {(𝑀 + 1)}
182181imaeq2i 5423 . . . . . . . . . . . . . . . . . . . . . . . . 25 ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ dom ({(𝑀 + 1)} × {(𝑀 + 1)})) = ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ {(𝑀 + 1)})
183180, 182eqtri 2643 . . . . . . . . . . . . . . . . . . . . . . . 24 (({(𝑀 + 1)} × {(𝑀 + 1)}) “ dom ({(𝑀 + 1)} × {(𝑀 + 1)})) = ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ {(𝑀 + 1)})
184 rnxpid 5526 . . . . . . . . . . . . . . . . . . . . . . . 24 ran ({(𝑀 + 1)} × {(𝑀 + 1)}) = {(𝑀 + 1)}
185178, 183, 1843eqtr3ri 2652 . . . . . . . . . . . . . . . . . . . . . . 23 {(𝑀 + 1)} = ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ {(𝑀 + 1)})
186 eluzp1p1 11657 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑀 ∈ (ℤ𝑗) → (𝑀 + 1) ∈ (ℤ‘(𝑗 + 1)))
187 eluzfz2 12291 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑀 + 1) ∈ (ℤ‘(𝑗 + 1)) → (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1)))
18833, 186, 1873syl 18 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1)))
189188snssd 4309 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (0...𝑀) → {(𝑀 + 1)} ⊆ ((𝑗 + 1)...(𝑀 + 1)))
190 imass2 5460 . . . . . . . . . . . . . . . . . . . . . . . 24 ({(𝑀 + 1)} ⊆ ((𝑗 + 1)...(𝑀 + 1)) → ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ {(𝑀 + 1)}) ⊆ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))))
191189, 190syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (0...𝑀) → ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ {(𝑀 + 1)}) ⊆ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))))
192185, 191syl5eqss 3628 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ (0...𝑀) → {(𝑀 + 1)} ⊆ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))))
193 ssel 3577 . . . . . . . . . . . . . . . . . . . . . 22 ({(𝑀 + 1)} ⊆ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) → ((𝑀 + 1) ∈ {(𝑀 + 1)} → (𝑀 + 1) ∈ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1)))))
194192, 167, 193mpisyl 21 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))))
195 elun2 3759 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀 + 1) ∈ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) → (𝑀 + 1) ∈ ((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) ∪ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1)))))
196194, 195syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈ ((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) ∪ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1)))))
197 imaundir 5505 . . . . . . . . . . . . . . . . . . . 20 ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) = ((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) ∪ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))))
198196, 197syl6eleqr 2709 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))))
199198adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (0...𝑀)) → (𝑀 + 1) ∈ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))))
200 fvun2 6227 . . . . . . . . . . . . . . . . . . 19 (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∧ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) ∧ ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∩ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅ ∧ (𝑀 + 1) ∈ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))))) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)))
201122, 124, 200mp3an12 1411 . . . . . . . . . . . . . . . . . 18 (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∩ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅ ∧ (𝑀 + 1) ∈ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)))
202139, 199, 201syl2anc 692 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (0...𝑀)) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)))
2038fvconst2 6423 . . . . . . . . . . . . . . . . . . 19 ((𝑀 + 1) ∈ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) → ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)) = 0)
204198, 203syl 17 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (0...𝑀) → ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)) = 0)
205204adantl 482 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (0...𝑀)) → ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)) = 0)
206202, 205eqtrd 2655 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (0...𝑀)) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = 0)
207177, 206sylan9eqr 2677 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑛 = (𝑀 + 1)) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛) = 0)
208176, 207oveq12d 6622 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑛 = (𝑀 + 1)) → (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) + (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛)) = (0 + 0))
209 00id 10155 . . . . . . . . . . . . . 14 (0 + 0) = 0
210208, 209syl6eq 2671 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑛 = (𝑀 + 1)) → (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) + (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛)) = 0)
211163, 164, 165, 210fmptapd 6391 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0...𝑀)) → ((𝑛 ∈ (1...𝑀) ↦ (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) + (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) = (𝑛 ∈ (1...(𝑀 + 1)) ↦ (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) + (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))))
2123, 94jca 554 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑇 Fn (1...𝑀) ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅))
213 fvun1 6226 . . . . . . . . . . . . . . . . . . 19 ((𝑇 Fn (1...𝑀) ∧ {⟨(𝑀 + 1), 0⟩} Fn {(𝑀 + 1)} ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ 𝑛 ∈ (1...𝑀))) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) = (𝑇𝑛))
214168, 213mp3an2 1409 . . . . . . . . . . . . . . . . . 18 ((𝑇 Fn (1...𝑀) ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ 𝑛 ∈ (1...𝑀))) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) = (𝑇𝑛))
215214anassrs 679 . . . . . . . . . . . . . . . . 17 (((𝑇 Fn (1...𝑀) ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅) ∧ 𝑛 ∈ (1...𝑀)) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) = (𝑇𝑛))
216212, 215sylan 488 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...𝑀)) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) = (𝑇𝑛))
217216adantlr 750 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) = (𝑇𝑛))
218 fvres 6164 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (1...𝑀) → ((((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) ↾ (1...𝑀))‘𝑛) = (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))
219218eqcomd 2627 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (1...𝑀) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛) = ((((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) ↾ (1...𝑀))‘𝑛))
220 resundir 5370 . . . . . . . . . . . . . . . . . 18 (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) ↾ (1...𝑀)) = (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ↾ (1...𝑀)) ∪ ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)))
221 relxp 5188 . . . . . . . . . . . . . . . . . . . . . . 23 Rel ((𝑈 “ (1...𝑗)) × {1})
222 dmxpss 5524 . . . . . . . . . . . . . . . . . . . . . . . . 25 dom ((𝑈 “ (1...𝑗)) × {1}) ⊆ (𝑈 “ (1...𝑗))
223 imassrn 5436 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑈 “ (1...𝑗)) ⊆ ran 𝑈
224222, 223sstri 3592 . . . . . . . . . . . . . . . . . . . . . . . 24 dom ((𝑈 “ (1...𝑗)) × {1}) ⊆ ran 𝑈
225 f1of 6094 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑈:(1...𝑀)–1-1-onto→(1...𝑀) → 𝑈:(1...𝑀)⟶(1...𝑀))
226 frn 6010 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑈:(1...𝑀)⟶(1...𝑀) → ran 𝑈 ⊆ (1...𝑀))
22712, 225, 2263syl 18 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ran 𝑈 ⊆ (1...𝑀))
228224, 227syl5ss 3594 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → dom ((𝑈 “ (1...𝑗)) × {1}) ⊆ (1...𝑀))
229 relssres 5396 . . . . . . . . . . . . . . . . . . . . . . 23 ((Rel ((𝑈 “ (1...𝑗)) × {1}) ∧ dom ((𝑈 “ (1...𝑗)) × {1}) ⊆ (1...𝑀)) → (((𝑈 “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ((𝑈 “ (1...𝑗)) × {1}))
230221, 228, 229sylancr 694 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (((𝑈 “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ((𝑈 “ (1...𝑗)) × {1}))
231230adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (0...𝑀)) → (((𝑈 “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ((𝑈 “ (1...𝑗)) × {1}))
232 imassrn 5436 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ⊆ ran {⟨(𝑀 + 1), (𝑀 + 1)⟩}
23377rnsnop 5575 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ran {⟨(𝑀 + 1), (𝑀 + 1)⟩} = {(𝑀 + 1)}
234232, 233sseqtri 3616 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ⊆ {(𝑀 + 1)}
235 ssrin 3816 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ⊆ {(𝑀 + 1)} → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ∩ (1...𝑀)) ⊆ ({(𝑀 + 1)} ∩ (1...𝑀)))
236234, 235ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . 25 (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ∩ (1...𝑀)) ⊆ ({(𝑀 + 1)} ∩ (1...𝑀))
237 incom 3783 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ({(𝑀 + 1)} ∩ (1...𝑀)) = ((1...𝑀) ∩ {(𝑀 + 1)})
238237, 94syl5eq 2667 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ({(𝑀 + 1)} ∩ (1...𝑀)) = ∅)
239236, 238syl5sseq 3632 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ∩ (1...𝑀)) ⊆ ∅)
240 ss0 3946 . . . . . . . . . . . . . . . . . . . . . . . 24 ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ∩ (1...𝑀)) ⊆ ∅ → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ∩ (1...𝑀)) = ∅)
241239, 240syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ∩ (1...𝑀)) = ∅)
242 fnconstg 6050 . . . . . . . . . . . . . . . . . . . . . . . 24 (1 ∈ V → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) Fn ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)))
243 fnresdisj 5959 . . . . . . . . . . . . . . . . . . . . . . . 24 ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) Fn ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) → ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ∩ (1...𝑀)) = ∅ ↔ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ∅))
2445, 242, 243mp2b 10 . . . . . . . . . . . . . . . . . . . . . . 23 ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ∩ (1...𝑀)) = ∅ ↔ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ∅)
245241, 244sylib 208 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ∅)
246245adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (0...𝑀)) → ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ∅)
247231, 246uneq12d 3746 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0...𝑀)) → ((((𝑈 “ (1...𝑗)) × {1}) ↾ (1...𝑀)) ∪ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) ↾ (1...𝑀))) = (((𝑈 “ (1...𝑗)) × {1}) ∪ ∅))
248 imaundir 5505 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) = ((𝑈 “ (1...𝑗)) ∪ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)))
249248xpeq1i 5095 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) = (((𝑈 “ (1...𝑗)) ∪ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗))) × {1})
250 xpundir 5133 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑈 “ (1...𝑗)) ∪ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗))) × {1}) = (((𝑈 “ (1...𝑗)) × {1}) ∪ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}))
251249, 250eqtri 2643 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) = (((𝑈 “ (1...𝑗)) × {1}) ∪ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}))
252251reseq1i 5352 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ((((𝑈 “ (1...𝑗)) × {1}) ∪ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1})) ↾ (1...𝑀))
253 resundir 5370 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑈 “ (1...𝑗)) × {1}) ∪ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1})) ↾ (1...𝑀)) = ((((𝑈 “ (1...𝑗)) × {1}) ↾ (1...𝑀)) ∪ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) ↾ (1...𝑀)))
254252, 253eqtr2i 2644 . . . . . . . . . . . . . . . . . . . 20 ((((𝑈 “ (1...𝑗)) × {1}) ↾ (1...𝑀)) ∪ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) ↾ (1...𝑀))) = ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ↾ (1...𝑀))
255 un0 3939 . . . . . . . . . . . . . . . . . . . 20 (((𝑈 “ (1...𝑗)) × {1}) ∪ ∅) = ((𝑈 “ (1...𝑗)) × {1})
256247, 254, 2553eqtr3g 2678 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0...𝑀)) → ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ((𝑈 “ (1...𝑗)) × {1}))
257 f1odm 6098 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑈:(1...𝑀)–1-1-onto→(1...𝑀) → dom 𝑈 = (1...𝑀))
25812, 257syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → dom 𝑈 = (1...𝑀))
259258ineq2d 3792 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → (((𝑗 + 1)...(𝑀 + 1)) ∩ dom 𝑈) = (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑀)))
260259reseq2d 5356 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (𝑈 ↾ (((𝑗 + 1)...(𝑀 + 1)) ∩ dom 𝑈)) = (𝑈 ↾ (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑀))))
261 f1orel 6097 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑈:(1...𝑀)–1-1-onto→(1...𝑀) → Rel 𝑈)
262 resindm 5403 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Rel 𝑈 → (𝑈 ↾ (((𝑗 + 1)...(𝑀 + 1)) ∩ dom 𝑈)) = (𝑈 ↾ ((𝑗 + 1)...(𝑀 + 1))))
26312, 261, 2623syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (𝑈 ↾ (((𝑗 + 1)...(𝑀 + 1)) ∩ dom 𝑈)) = (𝑈 ↾ ((𝑗 + 1)...(𝑀 + 1))))
264260, 263eqtr3d 2657 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (𝑈 ↾ (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑀))) = (𝑈 ↾ ((𝑗 + 1)...(𝑀 + 1))))
26535ineq2d 3792 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (0...𝑀) → (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑀)) = (((𝑗 + 1)...(𝑀 + 1)) ∩ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))))
266 fzssp1 12326 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑗 + 1)...𝑀) ⊆ ((𝑗 + 1)...(𝑀 + 1))
267 sseqin2 3795 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑗 + 1)...𝑀) ⊆ ((𝑗 + 1)...(𝑀 + 1)) ↔ (((𝑗 + 1)...(𝑀 + 1)) ∩ ((𝑗 + 1)...𝑀)) = ((𝑗 + 1)...𝑀))
268266, 267mpbi 220 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑗 + 1)...(𝑀 + 1)) ∩ ((𝑗 + 1)...𝑀)) = ((𝑗 + 1)...𝑀)
269268a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 ∈ (0...𝑀) → (((𝑗 + 1)...(𝑀 + 1)) ∩ ((𝑗 + 1)...𝑀)) = ((𝑗 + 1)...𝑀))
270 incom 3783 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑗)) = ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))
271270, 135syl5eq 2667 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 ∈ (0...𝑀) → (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑗)) = ∅)
272269, 271uneq12d 3746 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ (0...𝑀) → ((((𝑗 + 1)...(𝑀 + 1)) ∩ ((𝑗 + 1)...𝑀)) ∪ (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑗))) = (((𝑗 + 1)...𝑀) ∪ ∅))
273 uncom 3735 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑗 + 1)...(𝑀 + 1)) ∩ ((𝑗 + 1)...𝑀)) ∪ (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑗))) = ((((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑗)) ∪ (((𝑗 + 1)...(𝑀 + 1)) ∩ ((𝑗 + 1)...𝑀)))
274 indi 3849 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑗 + 1)...(𝑀 + 1)) ∩ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))) = ((((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑗)) ∪ (((𝑗 + 1)...(𝑀 + 1)) ∩ ((𝑗 + 1)...𝑀)))
275273, 274eqtr4i 2646 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑗 + 1)...(𝑀 + 1)) ∩ ((𝑗 + 1)...𝑀)) ∪ (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑗))) = (((𝑗 + 1)...(𝑀 + 1)) ∩ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀)))
276 un0 3939 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑗 + 1)...𝑀) ∪ ∅) = ((𝑗 + 1)...𝑀)
277272, 275, 2763eqtr3g 2678 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (0...𝑀) → (((𝑗 + 1)...(𝑀 + 1)) ∩ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))) = ((𝑗 + 1)...𝑀))
278265, 277eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (0...𝑀) → (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑀)) = ((𝑗 + 1)...𝑀))
279278reseq2d 5356 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (0...𝑀) → (𝑈 ↾ (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑀))) = (𝑈 ↾ ((𝑗 + 1)...𝑀)))
280264, 279sylan9req 2676 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑗 ∈ (0...𝑀)) → (𝑈 ↾ ((𝑗 + 1)...(𝑀 + 1))) = (𝑈 ↾ ((𝑗 + 1)...𝑀)))
281280rneqd 5313 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ (0...𝑀)) → ran (𝑈 ↾ ((𝑗 + 1)...(𝑀 + 1))) = ran (𝑈 ↾ ((𝑗 + 1)...𝑀)))
282 df-ima 5087 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) = ran (𝑈 ↾ ((𝑗 + 1)...(𝑀 + 1)))
283 df-ima 5087 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑈 “ ((𝑗 + 1)...𝑀)) = ran (𝑈 ↾ ((𝑗 + 1)...𝑀))
284281, 282, 2833eqtr4g 2680 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (0...𝑀)) → (𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) = (𝑈 “ ((𝑗 + 1)...𝑀)))
285284xpeq1d 5098 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (0...𝑀)) → ((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))
286285reseq1d 5355 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (0...𝑀)) → (((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) = (((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ↾ (1...𝑀)))
287 relxp 5188 . . . . . . . . . . . . . . . . . . . . . . . 24 Rel ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})
288 dmxpss 5524 . . . . . . . . . . . . . . . . . . . . . . . . . 26 dom ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ⊆ (𝑈 “ ((𝑗 + 1)...𝑀))
289 imassrn 5436 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑈 “ ((𝑗 + 1)...𝑀)) ⊆ ran 𝑈
290288, 289sstri 3592 . . . . . . . . . . . . . . . . . . . . . . . . 25 dom ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ⊆ ran 𝑈
291290, 227syl5ss 3594 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → dom ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ⊆ (1...𝑀))
292 relssres 5396 . . . . . . . . . . . . . . . . . . . . . . . 24 ((Rel ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ∧ dom ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ⊆ (1...𝑀)) → (((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ↾ (1...𝑀)) = ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))
293287, 291, 292sylancr 694 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ↾ (1...𝑀)) = ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))
294293adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (0...𝑀)) → (((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ↾ (1...𝑀)) = ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))
295286, 294eqtrd 2655 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (0...𝑀)) → (((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) = ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))
296 imassrn 5436 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ⊆ ran {⟨(𝑀 + 1), (𝑀 + 1)⟩}
297296, 233sseqtri 3616 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ⊆ {(𝑀 + 1)}
298 ssrin 3816 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ⊆ {(𝑀 + 1)} → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ∩ (1...𝑀)) ⊆ ({(𝑀 + 1)} ∩ (1...𝑀)))
299297, 298ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . 25 (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ∩ (1...𝑀)) ⊆ ({(𝑀 + 1)} ∩ (1...𝑀))
300299, 238syl5sseq 3632 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ∩ (1...𝑀)) ⊆ ∅)
301 ss0 3946 . . . . . . . . . . . . . . . . . . . . . . . 24 ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ∩ (1...𝑀)) ⊆ ∅ → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ∩ (1...𝑀)) = ∅)
302300, 301syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ∩ (1...𝑀)) = ∅)
303 fnconstg 6050 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 ∈ V → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))))
304 fnresdisj 5959 . . . . . . . . . . . . . . . . . . . . . . . 24 ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) → ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ∩ (1...𝑀)) = ∅ ↔ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) = ∅))
3058, 303, 304mp2b 10 . . . . . . . . . . . . . . . . . . . . . . 23 ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ∩ (1...𝑀)) = ∅ ↔ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) = ∅)
306302, 305sylib 208 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) = ∅)
307306adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (0...𝑀)) → ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) = ∅)
308295, 307uneq12d 3746 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (0...𝑀)) → ((((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) ∪ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀))) = (((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ∪ ∅))
309197xpeq1i 5095 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = (((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) ∪ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1)))) × {0})
310 xpundir 5133 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) ∪ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1)))) × {0}) = (((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ∪ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))
311309, 310eqtri 2643 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = (((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ∪ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))
312311reseq1i 5352 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) = ((((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ∪ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) ↾ (1...𝑀))
313 resundir 5370 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ∪ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) ↾ (1...𝑀)) = ((((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) ∪ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)))
314312, 313eqtr2i 2644 . . . . . . . . . . . . . . . . . . . 20 ((((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) ∪ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀))) = ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀))
315 un0 3939 . . . . . . . . . . . . . . . . . . . 20 (((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ∪ ∅) = ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})
316308, 314, 3153eqtr3g 2678 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0...𝑀)) → ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) = ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))
317256, 316uneq12d 3746 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (0...𝑀)) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ↾ (1...𝑀)) ∪ ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀))) = (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})))
318220, 317syl5eq 2667 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (0...𝑀)) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) ↾ (1...𝑀)) = (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})))
319318fveq1d 6150 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (0...𝑀)) → ((((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) ↾ (1...𝑀))‘𝑛) = ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))
320219, 319sylan9eqr 2677 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛) = ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))
321217, 320oveq12d 6622 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) + (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛)) = ((𝑇𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛)))
322321mpteq2dva 4704 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0...𝑀)) → (𝑛 ∈ (1...𝑀) ↦ (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) + (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) = (𝑛 ∈ (1...𝑀) ↦ ((𝑇𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))))
323322uneq1d 3744 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0...𝑀)) → ((𝑛 ∈ (1...𝑀) ↦ (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) + (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) = ((𝑛 ∈ (1...𝑀) ↦ ((𝑇𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}))
324162, 211, 3233eqtr2d 2661 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0...𝑀)) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) = ((𝑛 ∈ (1...𝑀) ↦ ((𝑇𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}))
325324uneq1d 3744 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0...𝑀)) → (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = (((𝑛 ∈ (1...𝑀) ↦ ((𝑇𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
32685, 325eqtr4d 2658 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...𝑀)) → ((𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) = (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
327326csbeq1d 3521 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑀)) → ((𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 = (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵)
328327eqeq2d 2631 . . . . . . 7 ((𝜑𝑗 ∈ (0...𝑀)) → (𝑖 = ((𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵𝑖 = (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵))
329328rexbidva 3042 . . . . . 6 (𝜑 → (∃𝑗 ∈ (0...𝑀)𝑖 = ((𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∃𝑗 ∈ (0...𝑀)𝑖 = (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵))
330329ralbidv 2980 . . . . 5 (𝜑 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵))
331330biimpd 219 . . . 4 (𝜑 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 → ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵))
332 f1ofn 6095 . . . . . . . 8 (𝑈:(1...𝑀)–1-1-onto→(1...𝑀) → 𝑈 Fn (1...𝑀))
33312, 332syl 17 . . . . . . 7 (𝜑𝑈 Fn (1...𝑀))
33477, 77fnsn 5904 . . . . . . . . 9 {⟨(𝑀 + 1), (𝑀 + 1)⟩} Fn {(𝑀 + 1)}
335 fvun2 6227 . . . . . . . . 9 ((𝑈 Fn (1...𝑀) ∧ {⟨(𝑀 + 1), (𝑀 + 1)⟩} Fn {(𝑀 + 1)} ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ (𝑀 + 1) ∈ {(𝑀 + 1)})) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = ({⟨(𝑀 + 1), (𝑀 + 1)⟩}‘(𝑀 + 1)))
336334, 335mp3an2 1409 . . . . . . . 8 ((𝑈 Fn (1...𝑀) ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ (𝑀 + 1) ∈ {(𝑀 + 1)})) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = ({⟨(𝑀 + 1), (𝑀 + 1)⟩}‘(𝑀 + 1)))
337167, 336mpanr2 719 . . . . . . 7 ((𝑈 Fn (1...𝑀) ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = ({⟨(𝑀 + 1), (𝑀 + 1)⟩}‘(𝑀 + 1)))
338333, 94, 337syl2anc 692 . . . . . 6 (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = ({⟨(𝑀 + 1), (𝑀 + 1)⟩}‘(𝑀 + 1)))
33977, 77fvsn 6400 . . . . . 6 ({⟨(𝑀 + 1), (𝑀 + 1)⟩}‘(𝑀 + 1)) = (𝑀 + 1)
340338, 339syl6eq 2671 . . . . 5 (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1))
341174, 340jca 554 . . . 4 (𝜑 → (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1)))
342331, 341jctird 566 . . 3 (𝜑 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1)))))
343 3anass 1040 . . 3 ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1)) ↔ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1))))
344342, 343syl6ibr 242 . 2 (𝜑 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1))))
3451, 97jctir 560 . . . . . 6 (𝜑 → (𝑇:(1...𝑀)⟶(0..^𝐾) ∧ {⟨(𝑀 + 1), 0⟩}:{(𝑀 + 1)}⟶{0}))
346345, 94, 98syl2anc 692 . . . . 5 (𝜑 → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):((1...𝑀) ∪ {(𝑀 + 1)})⟶((0..^𝐾) ∪ {0}))
347346, 116mpbid 222 . . . 4 (𝜑 → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):(1...(𝑀 + 1))⟶(0..^𝐾))
348 ovex 6632 . . . . 5 (0..^𝐾) ∈ V
349348, 157elmap 7830 . . . 4 ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∈ ((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) ↔ (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):(1...(𝑀 + 1))⟶(0..^𝐾))
350347, 349sylibr 224 . . 3 (𝜑 → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∈ ((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))))
351 f1oexrnex 7062 . . . . . . . 8 ((𝑈:(1...𝑀)–1-1-onto→(1...𝑀) ∧ (1...𝑀) ∈ V) → 𝑈 ∈ V)
35212, 45, 351sylancl 693 . . . . . . 7 (𝜑𝑈 ∈ V)
353 snex 4869 . . . . . . 7 {⟨(𝑀 + 1), (𝑀 + 1)⟩} ∈ V
354 unexg 6912 . . . . . . 7 ((𝑈 ∈ V ∧ {⟨(𝑀 + 1), (𝑀 + 1)⟩} ∈ V) → (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ V)
355352, 353, 354sylancl 693 . . . . . 6 (𝜑 → (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ V)
356 f1oeq1 6084 . . . . . . 7 (𝑓 = (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) → (𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) ↔ (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))))
357356elabg 3334 . . . . . 6 ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ V → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))} ↔ (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))))
358355, 357syl 17 . . . . 5 (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))} ↔ (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))))
359 f1oeq23 6087 . . . . . 6 (((1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)}) ∧ (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)})) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) ↔ (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)})))
360108, 108, 359syl2anc 692 . . . . 5 (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) ↔ (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)})))
361358, 360bitrd 268 . . . 4 (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))} ↔ (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)})))
362129, 361mpbird 247 . . 3 (𝜑 → (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})
363 opelxpi 5108 . . 3 (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∈ ((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) ∧ (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ⟨(𝑇 ∪ {⟨(𝑀 + 1), 0⟩}), (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}))
364350, 362, 363syl2anc 692 . 2 (𝜑 → ⟨(𝑇 ∪ {⟨(𝑀 + 1), 0⟩}), (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}))
365344, 364jctild 565 1 (𝜑 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 → (⟨(𝑇 ∪ {⟨(𝑀 + 1), 0⟩}), (𝑈 ∪ {⟨(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 = (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  {cab 2607  wral 2907  wrex 2908  Vcvv 3186  csb 3514  cun 3553  cin 3554  wss 3555  c0 3891  {csn 4148  cop 4154   class class class wbr 4613  cmpt 4673   × cxp 5072  ccnv 5073  dom cdm 5074  ran crn 5075  cres 5076  cima 5077  Rel wrel 5079  Fun wfun 5841   Fn wfn 5842  wf 5843  ontowfo 5845  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  𝑓 cof 6848  𝑚 cmap 7802  0cc0 9880  1c1 9881   + caddc 9883   < clt 10018  cle 10019  cmin 10210  cn 10964  0cn0 11236  cz 11321  cuz 11631  ...cfz 12268  ..^cfzo 12406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407
This theorem is referenced by:  poimirlem4  33045
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