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Theorem poimirlem23 32405
Description: Lemma for poimir 32415, two ways of expressing the property that a face is not on the back face of the cube. (Contributed by Brendan Leahy, 21-Aug-2020.)
Hypotheses
Ref Expression
poimir.0 (𝜑𝑁 ∈ ℕ)
poimirlem23.1 (𝜑𝑇:(1...𝑁)⟶(0..^𝐾))
poimirlem23.2 (𝜑𝑈:(1...𝑁)–1-1-onto→(1...𝑁))
poimirlem23.3 (𝜑𝑉 ∈ (0...𝑁))
Assertion
Ref Expression
poimirlem23 (𝜑 → (∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝𝑁) ≠ 0 ↔ ¬ (𝑉 = 𝑁 ∧ ((𝑇𝑁) = 0 ∧ (𝑈𝑁) = 𝑁))))
Distinct variable groups:   𝑗,𝑝,𝑦,𝜑   𝑗,𝑁,𝑦   𝑇,𝑗,𝑦   𝑈,𝑗,𝑦   𝑗,𝑉,𝑦   𝜑,𝑝   𝑗,𝐾,𝑝   𝑁,𝑝   𝑇,𝑝   𝑈,𝑝   𝑦,𝐾   𝑉,𝑝

Proof of Theorem poimirlem23
StepHypRef Expression
1 ovex 6555 . . . . . 6 (𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
21csbex 4716 . . . . 5 if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
32rgenw 2907 . . . 4 𝑦 ∈ (0...(𝑁 − 1))if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
4 eqid 2609 . . . . 5 (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
5 fveq1 6087 . . . . . . 7 (𝑝 = if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) → (𝑝𝑁) = (if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁))
65neeq1d 2840 . . . . . 6 (𝑝 = if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) → ((𝑝𝑁) ≠ 0 ↔ (if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) ≠ 0))
7 df-ne 2781 . . . . . 6 ((if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) ≠ 0 ↔ ¬ (if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0)
86, 7syl6bb 274 . . . . 5 (𝑝 = if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) → ((𝑝𝑁) ≠ 0 ↔ ¬ (if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0))
94, 8rexrnmpt 6262 . . . 4 (∀𝑦 ∈ (0...(𝑁 − 1))if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V → (∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝𝑁) ≠ 0 ↔ ∃𝑦 ∈ (0...(𝑁 − 1)) ¬ (if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0))
103, 9ax-mp 5 . . 3 (∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝𝑁) ≠ 0 ↔ ∃𝑦 ∈ (0...(𝑁 − 1)) ¬ (if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0)
11 rexnal 2977 . . 3 (∃𝑦 ∈ (0...(𝑁 − 1)) ¬ (if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ¬ ∀𝑦 ∈ (0...(𝑁 − 1))(if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0)
1210, 11bitri 262 . 2 (∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝𝑁) ≠ 0 ↔ ¬ ∀𝑦 ∈ (0...(𝑁 − 1))(if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0)
13 poimir.0 . . . . . . . . . . 11 (𝜑𝑁 ∈ ℕ)
1413nnzd 11313 . . . . . . . . . 10 (𝜑𝑁 ∈ ℤ)
15 poimirlem23.3 . . . . . . . . . . 11 (𝜑𝑉 ∈ (0...𝑁))
16 elfzelz 12168 . . . . . . . . . . 11 (𝑉 ∈ (0...𝑁) → 𝑉 ∈ ℤ)
1715, 16syl 17 . . . . . . . . . 10 (𝜑𝑉 ∈ ℤ)
18 zlem1lt 11262 . . . . . . . . . 10 ((𝑁 ∈ ℤ ∧ 𝑉 ∈ ℤ) → (𝑁𝑉 ↔ (𝑁 − 1) < 𝑉))
1914, 17, 18syl2anc 690 . . . . . . . . 9 (𝜑 → (𝑁𝑉 ↔ (𝑁 − 1) < 𝑉))
20 elfzle2 12171 . . . . . . . . . . 11 (𝑉 ∈ (0...𝑁) → 𝑉𝑁)
2115, 20syl 17 . . . . . . . . . 10 (𝜑𝑉𝑁)
2217zred 11314 . . . . . . . . . . . 12 (𝜑𝑉 ∈ ℝ)
2313nnred 10882 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℝ)
2422, 23letri3d 10030 . . . . . . . . . . 11 (𝜑 → (𝑉 = 𝑁 ↔ (𝑉𝑁𝑁𝑉)))
2524biimprd 236 . . . . . . . . . 10 (𝜑 → ((𝑉𝑁𝑁𝑉) → 𝑉 = 𝑁))
2621, 25mpand 706 . . . . . . . . 9 (𝜑 → (𝑁𝑉𝑉 = 𝑁))
2719, 26sylbird 248 . . . . . . . 8 (𝜑 → ((𝑁 − 1) < 𝑉𝑉 = 𝑁))
2827necon3ad 2794 . . . . . . 7 (𝜑 → (𝑉𝑁 → ¬ (𝑁 − 1) < 𝑉))
29 nnm1nn0 11181 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
3013, 29syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑁 − 1) ∈ ℕ0)
31 nn0fz0 12261 . . . . . . . . . . . 12 ((𝑁 − 1) ∈ ℕ0 ↔ (𝑁 − 1) ∈ (0...(𝑁 − 1)))
3230, 31sylib 206 . . . . . . . . . . 11 (𝜑 → (𝑁 − 1) ∈ (0...(𝑁 − 1)))
3332adantr 479 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → (𝑁 − 1) ∈ (0...(𝑁 − 1)))
34 iffalse 4044 . . . . . . . . . . . . . . . 16 (¬ (𝑁 − 1) < 𝑉 → if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) = ((𝑁 − 1) + 1))
3513nncnd 10883 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ ℂ)
36 npcan1 10306 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
3735, 36syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
3834, 37sylan9eqr 2665 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) = 𝑁)
3938csbeq1d 3505 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = 𝑁 / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
40 oveq2 6535 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑁 → (1...𝑗) = (1...𝑁))
4140imaeq2d 5372 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑁 → (𝑈 “ (1...𝑗)) = (𝑈 “ (1...𝑁)))
4241xpeq1d 5052 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑁 → ((𝑈 “ (1...𝑗)) × {1}) = ((𝑈 “ (1...𝑁)) × {1}))
43 oveq1 6534 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = 𝑁 → (𝑗 + 1) = (𝑁 + 1))
4443oveq1d 6542 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑁 → ((𝑗 + 1)...𝑁) = ((𝑁 + 1)...𝑁))
4544imaeq2d 5372 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑁 → (𝑈 “ ((𝑗 + 1)...𝑁)) = (𝑈 “ ((𝑁 + 1)...𝑁)))
4645xpeq1d 5052 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑁 → ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) = ((𝑈 “ ((𝑁 + 1)...𝑁)) × {0}))
4742, 46uneq12d 3729 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑁 → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) = (((𝑈 “ (1...𝑁)) × {1}) ∪ ((𝑈 “ ((𝑁 + 1)...𝑁)) × {0})))
48 poimirlem23.2 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑈:(1...𝑁)–1-1-onto→(1...𝑁))
49 f1ofo 6042 . . . . . . . . . . . . . . . . . . . . . 22 (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈:(1...𝑁)–onto→(1...𝑁))
50 foima 6018 . . . . . . . . . . . . . . . . . . . . . 22 (𝑈:(1...𝑁)–onto→(1...𝑁) → (𝑈 “ (1...𝑁)) = (1...𝑁))
5148, 49, 503syl 18 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑈 “ (1...𝑁)) = (1...𝑁))
5251xpeq1d 5052 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑈 “ (1...𝑁)) × {1}) = ((1...𝑁) × {1}))
5323ltp1d 10803 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑁 < (𝑁 + 1))
5414peano2zd 11317 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝑁 + 1) ∈ ℤ)
55 fzn 12183 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑁 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
5654, 14, 55syl2anc 690 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
5753, 56mpbid 220 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑁 + 1)...𝑁) = ∅)
5857imaeq2d 5372 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑈 “ ((𝑁 + 1)...𝑁)) = (𝑈 “ ∅))
5958xpeq1d 5052 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑈 “ ((𝑁 + 1)...𝑁)) × {0}) = ((𝑈 “ ∅) × {0}))
60 ima0 5387 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑈 “ ∅) = ∅
6160xpeq1i 5049 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑈 “ ∅) × {0}) = (∅ × {0})
62 0xp 5112 . . . . . . . . . . . . . . . . . . . . . 22 (∅ × {0}) = ∅
6361, 62eqtri 2631 . . . . . . . . . . . . . . . . . . . . 21 ((𝑈 “ ∅) × {0}) = ∅
6459, 63syl6eq 2659 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑈 “ ((𝑁 + 1)...𝑁)) × {0}) = ∅)
6552, 64uneq12d 3729 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (((𝑈 “ (1...𝑁)) × {1}) ∪ ((𝑈 “ ((𝑁 + 1)...𝑁)) × {0})) = (((1...𝑁) × {1}) ∪ ∅))
66 un0 3918 . . . . . . . . . . . . . . . . . . 19 (((1...𝑁) × {1}) ∪ ∅) = ((1...𝑁) × {1})
6765, 66syl6eq 2659 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((𝑈 “ (1...𝑁)) × {1}) ∪ ((𝑈 “ ((𝑁 + 1)...𝑁)) × {0})) = ((1...𝑁) × {1}))
6847, 67sylan9eqr 2665 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 = 𝑁) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) = ((1...𝑁) × {1}))
6968oveq2d 6543 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 = 𝑁) → (𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇𝑓 + ((1...𝑁) × {1})))
7013, 69csbied 3525 . . . . . . . . . . . . . . 15 (𝜑𝑁 / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇𝑓 + ((1...𝑁) × {1})))
7170adantr 479 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → 𝑁 / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇𝑓 + ((1...𝑁) × {1})))
7239, 71eqtrd 2643 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇𝑓 + ((1...𝑁) × {1})))
7372fveq1d 6090 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → (if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = ((𝑇𝑓 + ((1...𝑁) × {1}))‘𝑁))
74 elfzonn0 12335 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (0..^𝐾) → 𝑗 ∈ ℕ0)
75 nn0p1nn 11179 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ ℕ0 → (𝑗 + 1) ∈ ℕ)
7674, 75syl 17 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (0..^𝐾) → (𝑗 + 1) ∈ ℕ)
77 elsni 4141 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ {1} → 𝑦 = 1)
7877oveq2d 6543 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ {1} → (𝑗 + 𝑦) = (𝑗 + 1))
7978eleq1d 2671 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ {1} → ((𝑗 + 𝑦) ∈ ℕ ↔ (𝑗 + 1) ∈ ℕ))
8076, 79syl5ibrcom 235 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (0..^𝐾) → (𝑦 ∈ {1} → (𝑗 + 𝑦) ∈ ℕ))
8180imp 443 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ (0..^𝐾) ∧ 𝑦 ∈ {1}) → (𝑗 + 𝑦) ∈ ℕ)
8281adantl 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑗 ∈ (0..^𝐾) ∧ 𝑦 ∈ {1})) → (𝑗 + 𝑦) ∈ ℕ)
83 poimirlem23.1 . . . . . . . . . . . . . . 15 (𝜑𝑇:(1...𝑁)⟶(0..^𝐾))
84 1ex 9891 . . . . . . . . . . . . . . . . 17 1 ∈ V
8584fconst 5989 . . . . . . . . . . . . . . . 16 ((1...𝑁) × {1}):(1...𝑁)⟶{1}
8685a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → ((1...𝑁) × {1}):(1...𝑁)⟶{1})
87 ovex 6555 . . . . . . . . . . . . . . . 16 (1...𝑁) ∈ V
8887a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → (1...𝑁) ∈ V)
89 inidm 3783 . . . . . . . . . . . . . . 15 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
9082, 83, 86, 88, 88, 89off 6787 . . . . . . . . . . . . . 14 (𝜑 → (𝑇𝑓 + ((1...𝑁) × {1})):(1...𝑁)⟶ℕ)
91 elfz1end 12197 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁))
9213, 91sylib 206 . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ (1...𝑁))
9390, 92ffvelrnd 6253 . . . . . . . . . . . . 13 (𝜑 → ((𝑇𝑓 + ((1...𝑁) × {1}))‘𝑁) ∈ ℕ)
9493adantr 479 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → ((𝑇𝑓 + ((1...𝑁) × {1}))‘𝑁) ∈ ℕ)
9573, 94eqeltrd 2687 . . . . . . . . . . 11 ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → (if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) ∈ ℕ)
9695nnne0d 10912 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → (if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) ≠ 0)
97 breq1 4580 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑁 − 1) → (𝑦 < 𝑉 ↔ (𝑁 − 1) < 𝑉))
98 id 22 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑁 − 1) → 𝑦 = (𝑁 − 1))
99 oveq1 6534 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑁 − 1) → (𝑦 + 1) = ((𝑁 − 1) + 1))
10097, 98, 99ifbieq12d 4062 . . . . . . . . . . . . . . 15 (𝑦 = (𝑁 − 1) → if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) = if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)))
101100csbeq1d 3505 . . . . . . . . . . . . . 14 (𝑦 = (𝑁 − 1) → if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
102101fveq1d 6090 . . . . . . . . . . . . 13 (𝑦 = (𝑁 − 1) → (if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = (if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁))
103102neeq1d 2840 . . . . . . . . . . . 12 (𝑦 = (𝑁 − 1) → ((if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) ≠ 0 ↔ (if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) ≠ 0))
1047, 103syl5bbr 272 . . . . . . . . . . 11 (𝑦 = (𝑁 − 1) → (¬ (if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ (if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) ≠ 0))
105104rspcev 3281 . . . . . . . . . 10 (((𝑁 − 1) ∈ (0...(𝑁 − 1)) ∧ (if((𝑁 − 1) < 𝑉, (𝑁 − 1), ((𝑁 − 1) + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) ≠ 0) → ∃𝑦 ∈ (0...(𝑁 − 1)) ¬ (if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0)
10633, 96, 105syl2anc 690 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → ∃𝑦 ∈ (0...(𝑁 − 1)) ¬ (if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0)
107106, 11sylib 206 . . . . . . . 8 ((𝜑 ∧ ¬ (𝑁 − 1) < 𝑉) → ¬ ∀𝑦 ∈ (0...(𝑁 − 1))(if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0)
108107ex 448 . . . . . . 7 (𝜑 → (¬ (𝑁 − 1) < 𝑉 → ¬ ∀𝑦 ∈ (0...(𝑁 − 1))(if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0))
10928, 108syld 45 . . . . . 6 (𝜑 → (𝑉𝑁 → ¬ ∀𝑦 ∈ (0...(𝑁 − 1))(if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0))
110109necon4ad 2800 . . . . 5 (𝜑 → (∀𝑦 ∈ (0...(𝑁 − 1))(if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 → 𝑉 = 𝑁))
111110pm4.71rd 664 . . . 4 (𝜑 → (∀𝑦 ∈ (0...(𝑁 − 1))(if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ (𝑉 = 𝑁 ∧ ∀𝑦 ∈ (0...(𝑁 − 1))(if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0)))
11230nn0zd 11312 . . . . . . . . . . . . 13 (𝜑 → (𝑁 − 1) ∈ ℤ)
113 uzid 11534 . . . . . . . . . . . . 13 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
114 peano2uz 11573 . . . . . . . . . . . . 13 ((𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
115112, 113, 1143syl 18 . . . . . . . . . . . 12 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
11637, 115eqeltrrd 2688 . . . . . . . . . . 11 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
117 fzss2 12207 . . . . . . . . . . 11 (𝑁 ∈ (ℤ‘(𝑁 − 1)) → (0...(𝑁 − 1)) ⊆ (0...𝑁))
118116, 117syl 17 . . . . . . . . . 10 (𝜑 → (0...(𝑁 − 1)) ⊆ (0...𝑁))
119118sselda 3567 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...(𝑁 − 1))) → 𝑗 ∈ (0...𝑁))
12092adantr 479 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0...𝑁)) → 𝑁 ∈ (1...𝑁))
121 ffn 5944 . . . . . . . . . . . . . . 15 (𝑇:(1...𝑁)⟶(0..^𝐾) → 𝑇 Fn (1...𝑁))
12283, 121syl 17 . . . . . . . . . . . . . 14 (𝜑𝑇 Fn (1...𝑁))
123122adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0...𝑁)) → 𝑇 Fn (1...𝑁))
12484fconst 5989 . . . . . . . . . . . . . . . . 17 ((𝑈 “ (1...𝑗)) × {1}):(𝑈 “ (1...𝑗))⟶{1}
125 c0ex 9890 . . . . . . . . . . . . . . . . . 18 0 ∈ V
126125fconst 5989 . . . . . . . . . . . . . . . . 17 ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}):(𝑈 “ ((𝑗 + 1)...𝑁))⟶{0}
127124, 126pm3.2i 469 . . . . . . . . . . . . . . . 16 (((𝑈 “ (1...𝑗)) × {1}):(𝑈 “ (1...𝑗))⟶{1} ∧ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}):(𝑈 “ ((𝑗 + 1)...𝑁))⟶{0})
128 dff1o3 6041 . . . . . . . . . . . . . . . . . . 19 (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (𝑈:(1...𝑁)–onto→(1...𝑁) ∧ Fun 𝑈))
129128simprbi 478 . . . . . . . . . . . . . . . . . 18 (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → Fun 𝑈)
130 imain 5874 . . . . . . . . . . . . . . . . . 18 (Fun 𝑈 → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))))
13148, 129, 1303syl 18 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))))
132 elfzelz 12168 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℤ)
133132zred 11314 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℝ)
134133ltp1d 10803 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0...𝑁) → 𝑗 < (𝑗 + 1))
135 fzdisj 12194 . . . . . . . . . . . . . . . . . . . 20 (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅)
136134, 135syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (0...𝑁) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅)
137136imaeq2d 5372 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (0...𝑁) → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = (𝑈 “ ∅))
138137, 60syl6eq 2659 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (0...𝑁) → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ∅)
139131, 138sylan9req 2664 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (0...𝑁)) → ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))) = ∅)
140 fun 5965 . . . . . . . . . . . . . . . 16 (((((𝑈 “ (1...𝑗)) × {1}):(𝑈 “ (1...𝑗))⟶{1} ∧ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}):(𝑈 “ ((𝑗 + 1)...𝑁))⟶{0}) ∧ ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))) = ∅) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0}))
141127, 139, 140sylancr 693 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (0...𝑁)) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0}))
142 elfznn0 12257 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℕ0)
143142, 75syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ (0...𝑁) → (𝑗 + 1) ∈ ℕ)
144 nnuz 11555 . . . . . . . . . . . . . . . . . . . . 21 ℕ = (ℤ‘1)
145143, 144syl6eleq 2697 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0...𝑁) → (𝑗 + 1) ∈ (ℤ‘1))
146 elfzuz3 12165 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0...𝑁) → 𝑁 ∈ (ℤ𝑗))
147 fzsplit2 12192 . . . . . . . . . . . . . . . . . . . 20 (((𝑗 + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ𝑗)) → (1...𝑁) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑁)))
148145, 146, 147syl2anc 690 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (0...𝑁) → (1...𝑁) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑁)))
149148imaeq2d 5372 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (0...𝑁) → (𝑈 “ (1...𝑁)) = (𝑈 “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))))
150 imaundi 5450 . . . . . . . . . . . . . . . . . 18 (𝑈 “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑁))) = ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁)))
151149, 150syl6req 2660 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (0...𝑁) → ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁))) = (𝑈 “ (1...𝑁)))
152151, 51sylan9eqr 2665 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (0...𝑁)) → ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁))) = (1...𝑁))
153152feq2d 5930 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (0...𝑁)) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑁)))⟶({1} ∪ {0}) ↔ (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0})))
154141, 153mpbid 220 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (0...𝑁)) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}))
155 ffn 5944 . . . . . . . . . . . . . 14 ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) Fn (1...𝑁))
156154, 155syl 17 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0...𝑁)) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) Fn (1...𝑁))
15787a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0...𝑁)) → (1...𝑁) ∈ V)
158 eqidd 2610 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑁 ∈ (1...𝑁)) → (𝑇𝑁) = (𝑇𝑁))
159 eqidd 2610 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑁 ∈ (1...𝑁)) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁))
160123, 156, 157, 157, 89, 158, 159ofval 6781 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (0...𝑁)) ∧ 𝑁 ∈ (1...𝑁)) → ((𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = ((𝑇𝑁) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁)))
161120, 160mpdan 698 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0...𝑁)) → ((𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = ((𝑇𝑁) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁)))
162161eqeq1d 2611 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0...𝑁)) → (((𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ((𝑇𝑁) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁)) = 0))
16383, 92ffvelrnd 6253 . . . . . . . . . . . . . 14 (𝜑 → (𝑇𝑁) ∈ (0..^𝐾))
164 elfzonn0 12335 . . . . . . . . . . . . . 14 ((𝑇𝑁) ∈ (0..^𝐾) → (𝑇𝑁) ∈ ℕ0)
165163, 164syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑇𝑁) ∈ ℕ0)
166165nn0red 11199 . . . . . . . . . . . 12 (𝜑 → (𝑇𝑁) ∈ ℝ)
167166adantr 479 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑇𝑁) ∈ ℝ)
168165nn0ge0d 11201 . . . . . . . . . . . 12 (𝜑 → 0 ≤ (𝑇𝑁))
169168adantr 479 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0...𝑁)) → 0 ≤ (𝑇𝑁))
170 1re 9895 . . . . . . . . . . . . . 14 1 ∈ ℝ
171 snssi 4279 . . . . . . . . . . . . . 14 (1 ∈ ℝ → {1} ⊆ ℝ)
172170, 171ax-mp 5 . . . . . . . . . . . . 13 {1} ⊆ ℝ
173 0re 9896 . . . . . . . . . . . . . 14 0 ∈ ℝ
174 snssi 4279 . . . . . . . . . . . . . 14 (0 ∈ ℝ → {0} ⊆ ℝ)
175173, 174ax-mp 5 . . . . . . . . . . . . 13 {0} ⊆ ℝ
176172, 175unssi 3749 . . . . . . . . . . . 12 ({1} ∪ {0}) ⊆ ℝ
177154, 120ffvelrnd 6253 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (0...𝑁)) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ ({1} ∪ {0}))
178176, 177sseldi 3565 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0...𝑁)) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ ℝ)
179 elun 3714 . . . . . . . . . . . . 13 (((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ ({1} ∪ {0}) ↔ (((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ {1} ∨ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ {0}))
180 0le1 10400 . . . . . . . . . . . . . . 15 0 ≤ 1
181 elsni 4141 . . . . . . . . . . . . . . 15 (((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ {1} → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 1)
182180, 181syl5breqr 4615 . . . . . . . . . . . . . 14 (((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ {1} → 0 ≤ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁))
183 0le0 10957 . . . . . . . . . . . . . . 15 0 ≤ 0
184 elsni 4141 . . . . . . . . . . . . . . 15 (((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ {0} → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)
185183, 184syl5breqr 4615 . . . . . . . . . . . . . 14 (((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ {0} → 0 ≤ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁))
186182, 185jaoi 392 . . . . . . . . . . . . 13 ((((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ {1} ∨ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ {0}) → 0 ≤ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁))
187179, 186sylbi 205 . . . . . . . . . . . 12 (((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ ({1} ∪ {0}) → 0 ≤ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁))
188177, 187syl 17 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (0...𝑁)) → 0 ≤ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁))
189 add20 10389 . . . . . . . . . . 11 ((((𝑇𝑁) ∈ ℝ ∧ 0 ≤ (𝑇𝑁)) ∧ (((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ∈ ℝ ∧ 0 ≤ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁))) → (((𝑇𝑁) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁)) = 0 ↔ ((𝑇𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)))
190167, 169, 178, 188, 189syl22anc 1318 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0...𝑁)) → (((𝑇𝑁) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁)) = 0 ↔ ((𝑇𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)))
191162, 190bitrd 266 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...𝑁)) → (((𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ((𝑇𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)))
192119, 191syldan 485 . . . . . . . 8 ((𝜑𝑗 ∈ (0...(𝑁 − 1))) → (((𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ((𝑇𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)))
193192ralbidva 2967 . . . . . . 7 (𝜑 → (∀𝑗 ∈ (0...(𝑁 − 1))((𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ∀𝑗 ∈ (0...(𝑁 − 1))((𝑇𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)))
194193adantr 479 . . . . . 6 ((𝜑𝑉 = 𝑁) → (∀𝑗 ∈ (0...(𝑁 − 1))((𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ∀𝑗 ∈ (0...(𝑁 − 1))((𝑇𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)))
195 breq2 4581 . . . . . . . . . . . . . 14 (𝑉 = 𝑁 → (𝑦 < 𝑉𝑦 < 𝑁))
196195ifbid 4057 . . . . . . . . . . . . 13 (𝑉 = 𝑁 → if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) = if(𝑦 < 𝑁, 𝑦, (𝑦 + 1)))
197 elfzelz 12168 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℤ)
198197zred 11314 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ)
199198adantl 480 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ∈ ℝ)
20030nn0red 11199 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁 − 1) ∈ ℝ)
201200adantr 479 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ ℝ)
20223adantr 479 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℝ)
203 elfzle2 12171 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ≤ (𝑁 − 1))
204203adantl 480 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ≤ (𝑁 − 1))
20523ltm1d 10805 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁 − 1) < 𝑁)
206205adantr 479 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) < 𝑁)
207199, 201, 202, 204, 206lelttrd 10046 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < 𝑁)
208207iftrued 4043 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < 𝑁, 𝑦, (𝑦 + 1)) = 𝑦)
209196, 208sylan9eqr 2665 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑉 = 𝑁) → if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) = 𝑦)
210209an32s 841 . . . . . . . . . . 11 (((𝜑𝑉 = 𝑁) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) = 𝑦)
211210csbeq1d 3505 . . . . . . . . . 10 (((𝜑𝑉 = 𝑁) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = 𝑦 / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
212211fveq1d 6090 . . . . . . . . 9 (((𝜑𝑉 = 𝑁) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = (𝑦 / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁))
213212eqeq1d 2611 . . . . . . . 8 (((𝜑𝑉 = 𝑁) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ (𝑦 / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0))
214213ralbidva 2967 . . . . . . 7 ((𝜑𝑉 = 𝑁) → (∀𝑦 ∈ (0...(𝑁 − 1))(if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ∀𝑦 ∈ (0...(𝑁 − 1))(𝑦 / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0))
215 nfv 1829 . . . . . . . 8 𝑦((𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0
216 nfcsb1v 3514 . . . . . . . . . 10 𝑗𝑦 / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))
217 nfcv 2750 . . . . . . . . . 10 𝑗𝑁
218216, 217nffv 6095 . . . . . . . . 9 𝑗(𝑦 / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁)
219218nfeq1 2763 . . . . . . . 8 𝑗(𝑦 / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0
220 csbeq1a 3507 . . . . . . . . . 10 (𝑗 = 𝑦 → (𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = 𝑦 / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
221220fveq1d 6090 . . . . . . . . 9 (𝑗 = 𝑦 → ((𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = (𝑦 / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁))
222221eqeq1d 2611 . . . . . . . 8 (𝑗 = 𝑦 → (((𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ (𝑦 / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0))
223215, 219, 222cbvral 3142 . . . . . . 7 (∀𝑗 ∈ (0...(𝑁 − 1))((𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ∀𝑦 ∈ (0...(𝑁 − 1))(𝑦 / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0)
224214, 223syl6bbr 276 . . . . . 6 ((𝜑𝑉 = 𝑁) → (∀𝑦 ∈ (0...(𝑁 − 1))(if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ∀𝑗 ∈ (0...(𝑁 − 1))((𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0))
225 ne0i 3879 . . . . . . . . . 10 ((𝑁 − 1) ∈ (0...(𝑁 − 1)) → (0...(𝑁 − 1)) ≠ ∅)
226 r19.3rzv 4015 . . . . . . . . . 10 ((0...(𝑁 − 1)) ≠ ∅ → ((𝑇𝑁) = 0 ↔ ∀𝑗 ∈ (0...(𝑁 − 1))(𝑇𝑁) = 0))
22732, 225, 2263syl 18 . . . . . . . . 9 (𝜑 → ((𝑇𝑁) = 0 ↔ ∀𝑗 ∈ (0...(𝑁 − 1))(𝑇𝑁) = 0))
228 elfzelz 12168 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ (0...(𝑁 − 1)) → 𝑗 ∈ ℤ)
229228zred 11314 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0...(𝑁 − 1)) → 𝑗 ∈ ℝ)
230229ltp1d 10803 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (0...(𝑁 − 1)) → 𝑗 < (𝑗 + 1))
231230, 135syl 17 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (0...(𝑁 − 1)) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑁)) = ∅)
232231imaeq2d 5372 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (0...(𝑁 − 1)) → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = (𝑈 “ ∅))
233232, 60syl6eq 2659 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (0...(𝑁 − 1)) → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑁))) = ∅)
234131, 233sylan9req 2664 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (0...(𝑁 − 1))) → ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))) = ∅)
235234adantlr 746 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑈𝑁) = 𝑁) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))) = ∅)
236 simplr 787 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑈𝑁) = 𝑁) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑈𝑁) = 𝑁)
237 f1ofn 6036 . . . . . . . . . . . . . . . . . . 19 (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈 Fn (1...𝑁))
23848, 237syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑𝑈 Fn (1...𝑁))
239238adantr 479 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (0...(𝑁 − 1))) → 𝑈 Fn (1...𝑁))
240 elfznn0 12257 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ (0...(𝑁 − 1)) → 𝑗 ∈ ℕ0)
241240, 75syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0...(𝑁 − 1)) → (𝑗 + 1) ∈ ℕ)
242241, 144syl6eleq 2697 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (0...(𝑁 − 1)) → (𝑗 + 1) ∈ (ℤ‘1))
243 fzss1 12206 . . . . . . . . . . . . . . . . . . 19 ((𝑗 + 1) ∈ (ℤ‘1) → ((𝑗 + 1)...𝑁) ⊆ (1...𝑁))
244242, 243syl 17 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (0...(𝑁 − 1)) → ((𝑗 + 1)...𝑁) ⊆ (1...𝑁))
245244adantl 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (0...(𝑁 − 1))) → ((𝑗 + 1)...𝑁) ⊆ (1...𝑁))
24637adantr 479 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁)
247 elfzuz3 12165 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ𝑗))
248 eluzp1p1 11545 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 − 1) ∈ (ℤ𝑗) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑗 + 1)))
249247, 248syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑗 + 1)))
250249adantl 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑗 + 1)))
251246, 250eqeltrrd 2688 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ‘(𝑗 + 1)))
252 eluzfz2 12175 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (ℤ‘(𝑗 + 1)) → 𝑁 ∈ ((𝑗 + 1)...𝑁))
253251, 252syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ((𝑗 + 1)...𝑁))
254 fnfvima 6378 . . . . . . . . . . . . . . . . 17 ((𝑈 Fn (1...𝑁) ∧ ((𝑗 + 1)...𝑁) ⊆ (1...𝑁) ∧ 𝑁 ∈ ((𝑗 + 1)...𝑁)) → (𝑈𝑁) ∈ (𝑈 “ ((𝑗 + 1)...𝑁)))
255239, 245, 253, 254syl3anc 1317 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (0...(𝑁 − 1))) → (𝑈𝑁) ∈ (𝑈 “ ((𝑗 + 1)...𝑁)))
256255adantlr 746 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑈𝑁) = 𝑁) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑈𝑁) ∈ (𝑈 “ ((𝑗 + 1)...𝑁)))
257236, 256eqeltrrd 2688 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑈𝑁) = 𝑁) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (𝑈 “ ((𝑗 + 1)...𝑁)))
258 fnconstg 5991 . . . . . . . . . . . . . . . 16 (1 ∈ V → ((𝑈 “ (1...𝑗)) × {1}) Fn (𝑈 “ (1...𝑗)))
25984, 258ax-mp 5 . . . . . . . . . . . . . . 15 ((𝑈 “ (1...𝑗)) × {1}) Fn (𝑈 “ (1...𝑗))
260 fnconstg 5991 . . . . . . . . . . . . . . . 16 (0 ∈ V → ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) Fn (𝑈 “ ((𝑗 + 1)...𝑁)))
261125, 260ax-mp 5 . . . . . . . . . . . . . . 15 ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) Fn (𝑈 “ ((𝑗 + 1)...𝑁))
262 fvun2 6165 . . . . . . . . . . . . . . 15 ((((𝑈 “ (1...𝑗)) × {1}) Fn (𝑈 “ (1...𝑗)) ∧ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) Fn (𝑈 “ ((𝑗 + 1)...𝑁)) ∧ (((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))) = ∅ ∧ 𝑁 ∈ (𝑈 “ ((𝑗 + 1)...𝑁)))) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = (((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})‘𝑁))
263259, 261, 262mp3an12 1405 . . . . . . . . . . . . . 14 ((((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑁))) = ∅ ∧ 𝑁 ∈ (𝑈 “ ((𝑗 + 1)...𝑁))) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = (((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})‘𝑁))
264235, 257, 263syl2anc 690 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑈𝑁) = 𝑁) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = (((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})‘𝑁))
265125fvconst2 6352 . . . . . . . . . . . . . 14 (𝑁 ∈ (𝑈 “ ((𝑗 + 1)...𝑁)) → (((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})‘𝑁) = 0)
266257, 265syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑈𝑁) = 𝑁) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})‘𝑁) = 0)
267264, 266eqtrd 2643 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑈𝑁) = 𝑁) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)
268267ralrimiva 2948 . . . . . . . . . . 11 ((𝜑 ∧ (𝑈𝑁) = 𝑁) → ∀𝑗 ∈ (0...(𝑁 − 1))((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)
269268ex 448 . . . . . . . . . 10 (𝜑 → ((𝑈𝑁) = 𝑁 → ∀𝑗 ∈ (0...(𝑁 − 1))((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0))
27032adantr 479 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑈𝑁) ≠ 𝑁) → (𝑁 − 1) ∈ (0...(𝑁 − 1)))
271 ax-1ne0 9861 . . . . . . . . . . . . . . 15 1 ≠ 0
272 imain 5874 . . . . . . . . . . . . . . . . . . . . 21 (Fun 𝑈 → (𝑈 “ ((1...(𝑁 − 1)) ∩ (((𝑁 − 1) + 1)...𝑁))) = ((𝑈 “ (1...(𝑁 − 1))) ∩ (𝑈 “ (((𝑁 − 1) + 1)...𝑁))))
27348, 129, 2723syl 18 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑈 “ ((1...(𝑁 − 1)) ∩ (((𝑁 − 1) + 1)...𝑁))) = ((𝑈 “ (1...(𝑁 − 1))) ∩ (𝑈 “ (((𝑁 − 1) + 1)...𝑁))))
274205, 37breqtrrd 4605 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑁 − 1) < ((𝑁 − 1) + 1))
275 fzdisj 12194 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 − 1) < ((𝑁 − 1) + 1) → ((1...(𝑁 − 1)) ∩ (((𝑁 − 1) + 1)...𝑁)) = ∅)
276274, 275syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1...(𝑁 − 1)) ∩ (((𝑁 − 1) + 1)...𝑁)) = ∅)
277276imaeq2d 5372 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑈 “ ((1...(𝑁 − 1)) ∩ (((𝑁 − 1) + 1)...𝑁))) = (𝑈 “ ∅))
278277, 60syl6eq 2659 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑈 “ ((1...(𝑁 − 1)) ∩ (((𝑁 − 1) + 1)...𝑁))) = ∅)
279273, 278eqtr3d 2645 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑈 “ (1...(𝑁 − 1))) ∩ (𝑈 “ (((𝑁 − 1) + 1)...𝑁))) = ∅)
280279adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑈𝑁) ≠ 𝑁) → ((𝑈 “ (1...(𝑁 − 1))) ∩ (𝑈 “ (((𝑁 − 1) + 1)...𝑁))) = ∅)
28192adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑈𝑁) ≠ 𝑁) → 𝑁 ∈ (1...𝑁))
282 elimasni 5398 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ (𝑈 “ {𝑁}) → 𝑁𝑈𝑁)
283 fnbrfvb 6131 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑈 Fn (1...𝑁) ∧ 𝑁 ∈ (1...𝑁)) → ((𝑈𝑁) = 𝑁𝑁𝑈𝑁))
284238, 92, 283syl2anc 690 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑈𝑁) = 𝑁𝑁𝑈𝑁))
285282, 284syl5ibr 234 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑁 ∈ (𝑈 “ {𝑁}) → (𝑈𝑁) = 𝑁))
286285necon3ad 2794 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑈𝑁) ≠ 𝑁 → ¬ 𝑁 ∈ (𝑈 “ {𝑁})))
287286imp 443 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑈𝑁) ≠ 𝑁) → ¬ 𝑁 ∈ (𝑈 “ {𝑁}))
288281, 287eldifd 3550 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑈𝑁) ≠ 𝑁) → 𝑁 ∈ ((1...𝑁) ∖ (𝑈 “ {𝑁})))
289 imadif 5873 . . . . . . . . . . . . . . . . . . . . . 22 (Fun 𝑈 → (𝑈 “ ((1...𝑁) ∖ {𝑁})) = ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {𝑁})))
29048, 129, 2893syl 18 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑈 “ ((1...𝑁) ∖ {𝑁})) = ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {𝑁})))
291 difun2 3999 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁}) = ((1...(𝑁 − 1)) ∖ {𝑁})
29213, 144syl6eleq 2697 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝑁 ∈ (ℤ‘1))
293 fzm1 12244 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ (ℤ‘1) → (𝑗 ∈ (1...𝑁) ↔ (𝑗 ∈ (1...(𝑁 − 1)) ∨ 𝑗 = 𝑁)))
294292, 293syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝑗 ∈ (1...𝑁) ↔ (𝑗 ∈ (1...(𝑁 − 1)) ∨ 𝑗 = 𝑁)))
295 elun 3714 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑗 ∈ (1...(𝑁 − 1)) ∨ 𝑗 ∈ {𝑁}))
296 velsn 4140 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ {𝑁} ↔ 𝑗 = 𝑁)
297296orbi2i 539 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑗 ∈ (1...(𝑁 − 1)) ∨ 𝑗 ∈ {𝑁}) ↔ (𝑗 ∈ (1...(𝑁 − 1)) ∨ 𝑗 = 𝑁))
298295, 297bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑗 ∈ (1...(𝑁 − 1)) ∨ 𝑗 = 𝑁))
299294, 298syl6rbbr 277 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝑗 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ 𝑗 ∈ (1...𝑁)))
300299eqrdv 2607 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((1...(𝑁 − 1)) ∪ {𝑁}) = (1...𝑁))
301300difeq1d 3688 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (((1...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁}) = ((1...𝑁) ∖ {𝑁}))
302200, 23ltnled 10035 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
303205, 302mpbid 220 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
304 elfzle2 12171 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
305303, 304nsyl 133 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
306 difsn 4268 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑁 ∈ (1...(𝑁 − 1)) → ((1...(𝑁 − 1)) ∖ {𝑁}) = (1...(𝑁 − 1)))
307305, 306syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((1...(𝑁 − 1)) ∖ {𝑁}) = (1...(𝑁 − 1)))
308291, 301, 3073eqtr3a 2667 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
309308imaeq2d 5372 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑈 “ ((1...𝑁) ∖ {𝑁})) = (𝑈 “ (1...(𝑁 − 1))))
31051difeq1d 3688 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {𝑁})) = ((1...𝑁) ∖ (𝑈 “ {𝑁})))
311290, 309, 3103eqtr3rd 2652 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((1...𝑁) ∖ (𝑈 “ {𝑁})) = (𝑈 “ (1...(𝑁 − 1))))
312311adantr 479 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑈𝑁) ≠ 𝑁) → ((1...𝑁) ∖ (𝑈 “ {𝑁})) = (𝑈 “ (1...(𝑁 − 1))))
313288, 312eleqtrd 2689 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑈𝑁) ≠ 𝑁) → 𝑁 ∈ (𝑈 “ (1...(𝑁 − 1))))
314 fnconstg 5991 . . . . . . . . . . . . . . . . . . . 20 (1 ∈ V → ((𝑈 “ (1...(𝑁 − 1))) × {1}) Fn (𝑈 “ (1...(𝑁 − 1))))
31584, 314ax-mp 5 . . . . . . . . . . . . . . . . . . 19 ((𝑈 “ (1...(𝑁 − 1))) × {1}) Fn (𝑈 “ (1...(𝑁 − 1)))
316 fnconstg 5991 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ V → ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑁 − 1) + 1)...𝑁)))
317125, 316ax-mp 5 . . . . . . . . . . . . . . . . . . 19 ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑁 − 1) + 1)...𝑁))
318 fvun1 6164 . . . . . . . . . . . . . . . . . . 19 ((((𝑈 “ (1...(𝑁 − 1))) × {1}) Fn (𝑈 “ (1...(𝑁 − 1))) ∧ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑁 − 1) + 1)...𝑁)) ∧ (((𝑈 “ (1...(𝑁 − 1))) ∩ (𝑈 “ (((𝑁 − 1) + 1)...𝑁))) = ∅ ∧ 𝑁 ∈ (𝑈 “ (1...(𝑁 − 1))))) → ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) = (((𝑈 “ (1...(𝑁 − 1))) × {1})‘𝑁))
319315, 317, 318mp3an12 1405 . . . . . . . . . . . . . . . . . 18 ((((𝑈 “ (1...(𝑁 − 1))) ∩ (𝑈 “ (((𝑁 − 1) + 1)...𝑁))) = ∅ ∧ 𝑁 ∈ (𝑈 “ (1...(𝑁 − 1)))) → ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) = (((𝑈 “ (1...(𝑁 − 1))) × {1})‘𝑁))
320280, 313, 319syl2anc 690 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑈𝑁) ≠ 𝑁) → ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) = (((𝑈 “ (1...(𝑁 − 1))) × {1})‘𝑁))
32184fvconst2 6352 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (𝑈 “ (1...(𝑁 − 1))) → (((𝑈 “ (1...(𝑁 − 1))) × {1})‘𝑁) = 1)
322313, 321syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑈𝑁) ≠ 𝑁) → (((𝑈 “ (1...(𝑁 − 1))) × {1})‘𝑁) = 1)
323320, 322eqtrd 2643 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑈𝑁) ≠ 𝑁) → ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) = 1)
324323neeq1d 2840 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑈𝑁) ≠ 𝑁) → (((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) ≠ 0 ↔ 1 ≠ 0))
325271, 324mpbiri 246 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑈𝑁) ≠ 𝑁) → ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) ≠ 0)
326 df-ne 2781 . . . . . . . . . . . . . . . 16 (((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ≠ 0 ↔ ¬ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)
327 oveq2 6535 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (𝑁 − 1) → (1...𝑗) = (1...(𝑁 − 1)))
328327imaeq2d 5372 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑁 − 1) → (𝑈 “ (1...𝑗)) = (𝑈 “ (1...(𝑁 − 1))))
329328xpeq1d 5052 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑁 − 1) → ((𝑈 “ (1...𝑗)) × {1}) = ((𝑈 “ (1...(𝑁 − 1))) × {1}))
330 oveq1 6534 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = (𝑁 − 1) → (𝑗 + 1) = ((𝑁 − 1) + 1))
331330oveq1d 6542 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (𝑁 − 1) → ((𝑗 + 1)...𝑁) = (((𝑁 − 1) + 1)...𝑁))
332331imaeq2d 5372 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑁 − 1) → (𝑈 “ ((𝑗 + 1)...𝑁)) = (𝑈 “ (((𝑁 − 1) + 1)...𝑁)))
333332xpeq1d 5052 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑁 − 1) → ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) = ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))
334329, 333uneq12d 3729 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑁 − 1) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) = (((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0})))
335334fveq1d 6090 . . . . . . . . . . . . . . . . 17 (𝑗 = (𝑁 − 1) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁))
336335neeq1d 2840 . . . . . . . . . . . . . . . 16 (𝑗 = (𝑁 − 1) → (((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) ≠ 0 ↔ ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) ≠ 0))
337326, 336syl5bbr 272 . . . . . . . . . . . . . . 15 (𝑗 = (𝑁 − 1) → (¬ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0 ↔ ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) ≠ 0))
338337rspcev 3281 . . . . . . . . . . . . . 14 (((𝑁 − 1) ∈ (0...(𝑁 − 1)) ∧ ((((𝑈 “ (1...(𝑁 − 1))) × {1}) ∪ ((𝑈 “ (((𝑁 − 1) + 1)...𝑁)) × {0}))‘𝑁) ≠ 0) → ∃𝑗 ∈ (0...(𝑁 − 1)) ¬ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)
339270, 325, 338syl2anc 690 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑈𝑁) ≠ 𝑁) → ∃𝑗 ∈ (0...(𝑁 − 1)) ¬ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)
340339ex 448 . . . . . . . . . . . 12 (𝜑 → ((𝑈𝑁) ≠ 𝑁 → ∃𝑗 ∈ (0...(𝑁 − 1)) ¬ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0))
341 rexnal 2977 . . . . . . . . . . . 12 (∃𝑗 ∈ (0...(𝑁 − 1)) ¬ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0 ↔ ¬ ∀𝑗 ∈ (0...(𝑁 − 1))((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)
342340, 341syl6ib 239 . . . . . . . . . . 11 (𝜑 → ((𝑈𝑁) ≠ 𝑁 → ¬ ∀𝑗 ∈ (0...(𝑁 − 1))((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0))
343342necon4ad 2800 . . . . . . . . . 10 (𝜑 → (∀𝑗 ∈ (0...(𝑁 − 1))((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0 → (𝑈𝑁) = 𝑁))
344269, 343impbid 200 . . . . . . . . 9 (𝜑 → ((𝑈𝑁) = 𝑁 ↔ ∀𝑗 ∈ (0...(𝑁 − 1))((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0))
345227, 344anbi12d 742 . . . . . . . 8 (𝜑 → (((𝑇𝑁) = 0 ∧ (𝑈𝑁) = 𝑁) ↔ (∀𝑗 ∈ (0...(𝑁 − 1))(𝑇𝑁) = 0 ∧ ∀𝑗 ∈ (0...(𝑁 − 1))((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)))
346 r19.26 3045 . . . . . . . 8 (∀𝑗 ∈ (0...(𝑁 − 1))((𝑇𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0) ↔ (∀𝑗 ∈ (0...(𝑁 − 1))(𝑇𝑁) = 0 ∧ ∀𝑗 ∈ (0...(𝑁 − 1))((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0))
347345, 346syl6bbr 276 . . . . . . 7 (𝜑 → (((𝑇𝑁) = 0 ∧ (𝑈𝑁) = 𝑁) ↔ ∀𝑗 ∈ (0...(𝑁 − 1))((𝑇𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)))
348347adantr 479 . . . . . 6 ((𝜑𝑉 = 𝑁) → (((𝑇𝑁) = 0 ∧ (𝑈𝑁) = 𝑁) ↔ ∀𝑗 ∈ (0...(𝑁 − 1))((𝑇𝑁) = 0 ∧ ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))‘𝑁) = 0)))
349194, 224, 3483bitr4d 298 . . . . 5 ((𝜑𝑉 = 𝑁) → (∀𝑦 ∈ (0...(𝑁 − 1))(if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ((𝑇𝑁) = 0 ∧ (𝑈𝑁) = 𝑁)))
350349pm5.32da 670 . . . 4 (𝜑 → ((𝑉 = 𝑁 ∧ ∀𝑦 ∈ (0...(𝑁 − 1))(if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0) ↔ (𝑉 = 𝑁 ∧ ((𝑇𝑁) = 0 ∧ (𝑈𝑁) = 𝑁))))
351111, 350bitrd 266 . . 3 (𝜑 → (∀𝑦 ∈ (0...(𝑁 − 1))(if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ (𝑉 = 𝑁 ∧ ((𝑇𝑁) = 0 ∧ (𝑈𝑁) = 𝑁))))
352351notbid 306 . 2 (𝜑 → (¬ ∀𝑦 ∈ (0...(𝑁 − 1))(if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))‘𝑁) = 0 ↔ ¬ (𝑉 = 𝑁 ∧ ((𝑇𝑁) = 0 ∧ (𝑈𝑁) = 𝑁))))
35312, 352syl5bb 270 1 (𝜑 → (∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝𝑁) ≠ 0 ↔ ¬ (𝑉 = 𝑁 ∧ ((𝑇𝑁) = 0 ∧ (𝑈𝑁) = 𝑁))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382   = wceq 1474  wcel 1976  wne 2779  wral 2895  wrex 2896  Vcvv 3172  csb 3498  cdif 3536  cun 3537  cin 3538  wss 3539  c0 3873  ifcif 4035  {csn 4124   class class class wbr 4577  cmpt 4637   × cxp 5026  ccnv 5027  ran crn 5029  cima 5031  Fun wfun 5784   Fn wfn 5785  wf 5786  ontowfo 5788  1-1-ontowf1o 5789  cfv 5790  (class class class)co 6527  𝑓 cof 6770  cc 9790  cr 9791  0cc0 9792  1c1 9793   + caddc 9795   < clt 9930  cle 9931  cmin 10117  cn 10867  0cn0 11139  cz 11210  cuz 11519  ...cfz 12152  ..^cfzo 12289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6772  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-fzo 12290
This theorem is referenced by:  poimirlem24  32406
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