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Theorem poimirlem2 34775
Description: Lemma for poimir 34806- consecutive vertices differ in at most one dimension. (Contributed by Brendan Leahy, 21-Aug-2020.)
Hypotheses
Ref Expression
poimir.0 (𝜑𝑁 ∈ ℕ)
poimirlem2.1 (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗(𝑇f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))))
poimirlem2.2 (𝜑𝑇:(1...𝑁)⟶ℤ)
poimirlem2.3 (𝜑𝑈:(1...𝑁)–1-1-onto→(1...𝑁))
poimirlem2.4 (𝜑𝑉 ∈ (1...(𝑁 − 1)))
poimirlem2.5 (𝜑𝑀 ∈ ((0...𝑁) ∖ {𝑉}))
Assertion
Ref Expression
poimirlem2 (𝜑 → ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛))
Distinct variable groups:   𝑗,𝑛,𝑦,𝜑   𝑗,𝐹,𝑛,𝑦   𝑗,𝑀,𝑛,𝑦   𝑗,𝑁,𝑛,𝑦   𝑇,𝑗,𝑛,𝑦   𝑈,𝑗,𝑛,𝑦   𝑗,𝑉,𝑛,𝑦

Proof of Theorem poimirlem2
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 poimirlem2.3 . . . . . . . . . . . . . . . 16 (𝜑𝑈:(1...𝑁)–1-1-onto→(1...𝑁))
2 dff1o3 6614 . . . . . . . . . . . . . . . . 17 (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (𝑈:(1...𝑁)–onto→(1...𝑁) ∧ Fun 𝑈))
32simprbi 497 . . . . . . . . . . . . . . . 16 (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → Fun 𝑈)
41, 3syl 17 . . . . . . . . . . . . . . 15 (𝜑 → Fun 𝑈)
5 imadif 6431 . . . . . . . . . . . . . . 15 (Fun 𝑈 → (𝑈 “ ((1...𝑁) ∖ {(𝑉 + 1)})) = ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {(𝑉 + 1)})))
64, 5syl 17 . . . . . . . . . . . . . 14 (𝜑 → (𝑈 “ ((1...𝑁) ∖ {(𝑉 + 1)})) = ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {(𝑉 + 1)})))
7 poimirlem2.4 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑉 ∈ (1...(𝑁 − 1)))
8 fzp1elp1 12948 . . . . . . . . . . . . . . . . . . . 20 (𝑉 ∈ (1...(𝑁 − 1)) → (𝑉 + 1) ∈ (1...((𝑁 − 1) + 1)))
97, 8syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑉 + 1) ∈ (1...((𝑁 − 1) + 1)))
10 poimir.0 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑁 ∈ ℕ)
1110nncnd 11642 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑁 ∈ ℂ)
12 npcan1 11053 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
1311, 12syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
1413oveq2d 7161 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁))
159, 14eleqtrd 2912 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑉 + 1) ∈ (1...𝑁))
16 fzsplit 12921 . . . . . . . . . . . . . . . . . 18 ((𝑉 + 1) ∈ (1...𝑁) → (1...𝑁) = ((1...(𝑉 + 1)) ∪ (((𝑉 + 1) + 1)...𝑁)))
1715, 16syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (1...𝑁) = ((1...(𝑉 + 1)) ∪ (((𝑉 + 1) + 1)...𝑁)))
1817difeq1d 4095 . . . . . . . . . . . . . . . 16 (𝜑 → ((1...𝑁) ∖ {(𝑉 + 1)}) = (((1...(𝑉 + 1)) ∪ (((𝑉 + 1) + 1)...𝑁)) ∖ {(𝑉 + 1)}))
19 difundir 4254 . . . . . . . . . . . . . . . . 17 (((1...(𝑉 + 1)) ∪ (((𝑉 + 1) + 1)...𝑁)) ∖ {(𝑉 + 1)}) = (((1...(𝑉 + 1)) ∖ {(𝑉 + 1)}) ∪ ((((𝑉 + 1) + 1)...𝑁) ∖ {(𝑉 + 1)}))
20 elfzuz 12892 . . . . . . . . . . . . . . . . . . . . . 22 (𝑉 ∈ (1...(𝑁 − 1)) → 𝑉 ∈ (ℤ‘1))
217, 20syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑉 ∈ (ℤ‘1))
22 fzsuc 12942 . . . . . . . . . . . . . . . . . . . . 21 (𝑉 ∈ (ℤ‘1) → (1...(𝑉 + 1)) = ((1...𝑉) ∪ {(𝑉 + 1)}))
2321, 22syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (1...(𝑉 + 1)) = ((1...𝑉) ∪ {(𝑉 + 1)}))
2423difeq1d 4095 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((1...(𝑉 + 1)) ∖ {(𝑉 + 1)}) = (((1...𝑉) ∪ {(𝑉 + 1)}) ∖ {(𝑉 + 1)}))
25 difun2 4425 . . . . . . . . . . . . . . . . . . . 20 (((1...𝑉) ∪ {(𝑉 + 1)}) ∖ {(𝑉 + 1)}) = ((1...𝑉) ∖ {(𝑉 + 1)})
26 elfzelz 12896 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑉 ∈ (1...(𝑁 − 1)) → 𝑉 ∈ ℤ)
277, 26syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑉 ∈ ℤ)
2827zred 12075 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑉 ∈ ℝ)
2928ltp1d 11558 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑉 < (𝑉 + 1))
3027peano2zd 12078 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝑉 + 1) ∈ ℤ)
3130zred 12075 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑉 + 1) ∈ ℝ)
3228, 31ltnled 10775 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑉 < (𝑉 + 1) ↔ ¬ (𝑉 + 1) ≤ 𝑉))
3329, 32mpbid 233 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ¬ (𝑉 + 1) ≤ 𝑉)
34 elfzle2 12899 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑉 + 1) ∈ (1...𝑉) → (𝑉 + 1) ≤ 𝑉)
3533, 34nsyl 142 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ¬ (𝑉 + 1) ∈ (1...𝑉))
36 difsn 4723 . . . . . . . . . . . . . . . . . . . . 21 (¬ (𝑉 + 1) ∈ (1...𝑉) → ((1...𝑉) ∖ {(𝑉 + 1)}) = (1...𝑉))
3735, 36syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((1...𝑉) ∖ {(𝑉 + 1)}) = (1...𝑉))
3825, 37syl5eq 2865 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (((1...𝑉) ∪ {(𝑉 + 1)}) ∖ {(𝑉 + 1)}) = (1...𝑉))
3924, 38eqtrd 2853 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1...(𝑉 + 1)) ∖ {(𝑉 + 1)}) = (1...𝑉))
4031ltp1d 11558 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑉 + 1) < ((𝑉 + 1) + 1))
41 peano2re 10801 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑉 + 1) ∈ ℝ → ((𝑉 + 1) + 1) ∈ ℝ)
4231, 41syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((𝑉 + 1) + 1) ∈ ℝ)
4331, 42ltnled 10775 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑉 + 1) < ((𝑉 + 1) + 1) ↔ ¬ ((𝑉 + 1) + 1) ≤ (𝑉 + 1)))
4440, 43mpbid 233 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ¬ ((𝑉 + 1) + 1) ≤ (𝑉 + 1))
45 elfzle1 12898 . . . . . . . . . . . . . . . . . . . 20 ((𝑉 + 1) ∈ (((𝑉 + 1) + 1)...𝑁) → ((𝑉 + 1) + 1) ≤ (𝑉 + 1))
4644, 45nsyl 142 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ¬ (𝑉 + 1) ∈ (((𝑉 + 1) + 1)...𝑁))
47 difsn 4723 . . . . . . . . . . . . . . . . . . 19 (¬ (𝑉 + 1) ∈ (((𝑉 + 1) + 1)...𝑁) → ((((𝑉 + 1) + 1)...𝑁) ∖ {(𝑉 + 1)}) = (((𝑉 + 1) + 1)...𝑁))
4846, 47syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((((𝑉 + 1) + 1)...𝑁) ∖ {(𝑉 + 1)}) = (((𝑉 + 1) + 1)...𝑁))
4939, 48uneq12d 4137 . . . . . . . . . . . . . . . . 17 (𝜑 → (((1...(𝑉 + 1)) ∖ {(𝑉 + 1)}) ∪ ((((𝑉 + 1) + 1)...𝑁) ∖ {(𝑉 + 1)})) = ((1...𝑉) ∪ (((𝑉 + 1) + 1)...𝑁)))
5019, 49syl5eq 2865 . . . . . . . . . . . . . . . 16 (𝜑 → (((1...(𝑉 + 1)) ∪ (((𝑉 + 1) + 1)...𝑁)) ∖ {(𝑉 + 1)}) = ((1...𝑉) ∪ (((𝑉 + 1) + 1)...𝑁)))
5118, 50eqtrd 2853 . . . . . . . . . . . . . . 15 (𝜑 → ((1...𝑁) ∖ {(𝑉 + 1)}) = ((1...𝑉) ∪ (((𝑉 + 1) + 1)...𝑁)))
5251imaeq2d 5922 . . . . . . . . . . . . . 14 (𝜑 → (𝑈 “ ((1...𝑁) ∖ {(𝑉 + 1)})) = (𝑈 “ ((1...𝑉) ∪ (((𝑉 + 1) + 1)...𝑁))))
536, 52eqtr3d 2855 . . . . . . . . . . . . 13 (𝜑 → ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {(𝑉 + 1)})) = (𝑈 “ ((1...𝑉) ∪ (((𝑉 + 1) + 1)...𝑁))))
54 imaundi 6001 . . . . . . . . . . . . 13 (𝑈 “ ((1...𝑉) ∪ (((𝑉 + 1) + 1)...𝑁))) = ((𝑈 “ (1...𝑉)) ∪ (𝑈 “ (((𝑉 + 1) + 1)...𝑁)))
5553, 54syl6eq 2869 . . . . . . . . . . . 12 (𝜑 → ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {(𝑉 + 1)})) = ((𝑈 “ (1...𝑉)) ∪ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))))
5655eleq2d 2895 . . . . . . . . . . 11 (𝜑 → (𝑛 ∈ ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {(𝑉 + 1)})) ↔ 𝑛 ∈ ((𝑈 “ (1...𝑉)) ∪ (𝑈 “ (((𝑉 + 1) + 1)...𝑁)))))
57 eldif 3943 . . . . . . . . . . 11 (𝑛 ∈ ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {(𝑉 + 1)})) ↔ (𝑛 ∈ (𝑈 “ (1...𝑁)) ∧ ¬ 𝑛 ∈ (𝑈 “ {(𝑉 + 1)})))
58 elun 4122 . . . . . . . . . . 11 (𝑛 ∈ ((𝑈 “ (1...𝑉)) ∪ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) ↔ (𝑛 ∈ (𝑈 “ (1...𝑉)) ∨ 𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))))
5956, 57, 583bitr3g 314 . . . . . . . . . 10 (𝜑 → ((𝑛 ∈ (𝑈 “ (1...𝑁)) ∧ ¬ 𝑛 ∈ (𝑈 “ {(𝑉 + 1)})) ↔ (𝑛 ∈ (𝑈 “ (1...𝑉)) ∨ 𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁)))))
6059adantr 481 . . . . . . . . 9 ((𝜑𝑀 < 𝑉) → ((𝑛 ∈ (𝑈 “ (1...𝑁)) ∧ ¬ 𝑛 ∈ (𝑈 “ {(𝑉 + 1)})) ↔ (𝑛 ∈ (𝑈 “ (1...𝑉)) ∨ 𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁)))))
61 imassrn 5933 . . . . . . . . . . . . . . . 16 (𝑈 “ (1...𝑉)) ⊆ ran 𝑈
62 f1of 6608 . . . . . . . . . . . . . . . . . 18 (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈:(1...𝑁)⟶(1...𝑁))
631, 62syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝑈:(1...𝑁)⟶(1...𝑁))
6463frnd 6514 . . . . . . . . . . . . . . . 16 (𝜑 → ran 𝑈 ⊆ (1...𝑁))
6561, 64sstrid 3975 . . . . . . . . . . . . . . 15 (𝜑 → (𝑈 “ (1...𝑉)) ⊆ (1...𝑁))
6665sselda 3964 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) → 𝑛 ∈ (1...𝑁))
67 poimirlem2.2 . . . . . . . . . . . . . . . . . 18 (𝜑𝑇:(1...𝑁)⟶ℤ)
6867ffnd 6508 . . . . . . . . . . . . . . . . 17 (𝜑𝑇 Fn (1...𝑁))
6968adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) → 𝑇 Fn (1...𝑁))
70 1ex 10625 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ V
71 fnconstg 6560 . . . . . . . . . . . . . . . . . . . . 21 (1 ∈ V → ((𝑈 “ (1...𝑉)) × {1}) Fn (𝑈 “ (1...𝑉)))
7270, 71ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((𝑈 “ (1...𝑉)) × {1}) Fn (𝑈 “ (1...𝑉))
73 c0ex 10623 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ V
74 fnconstg 6560 . . . . . . . . . . . . . . . . . . . . 21 (0 ∈ V → ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}) Fn (𝑈 “ ((𝑉 + 1)...𝑁)))
7573, 74ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}) Fn (𝑈 “ ((𝑉 + 1)...𝑁))
7672, 75pm3.2i 471 . . . . . . . . . . . . . . . . . . 19 (((𝑈 “ (1...𝑉)) × {1}) Fn (𝑈 “ (1...𝑉)) ∧ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}) Fn (𝑈 “ ((𝑉 + 1)...𝑁)))
77 imain 6432 . . . . . . . . . . . . . . . . . . . . 21 (Fun 𝑈 → (𝑈 “ ((1...𝑉) ∩ ((𝑉 + 1)...𝑁))) = ((𝑈 “ (1...𝑉)) ∩ (𝑈 “ ((𝑉 + 1)...𝑁))))
784, 77syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑈 “ ((1...𝑉) ∩ ((𝑉 + 1)...𝑁))) = ((𝑈 “ (1...𝑉)) ∩ (𝑈 “ ((𝑉 + 1)...𝑁))))
79 fzdisj 12922 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑉 < (𝑉 + 1) → ((1...𝑉) ∩ ((𝑉 + 1)...𝑁)) = ∅)
8029, 79syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1...𝑉) ∩ ((𝑉 + 1)...𝑁)) = ∅)
8180imaeq2d 5922 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑈 “ ((1...𝑉) ∩ ((𝑉 + 1)...𝑁))) = (𝑈 “ ∅))
82 ima0 5938 . . . . . . . . . . . . . . . . . . . . 21 (𝑈 “ ∅) = ∅
8381, 82syl6eq 2869 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑈 “ ((1...𝑉) ∩ ((𝑉 + 1)...𝑁))) = ∅)
8478, 83eqtr3d 2855 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑈 “ (1...𝑉)) ∩ (𝑈 “ ((𝑉 + 1)...𝑁))) = ∅)
85 fnun 6456 . . . . . . . . . . . . . . . . . . 19 (((((𝑈 “ (1...𝑉)) × {1}) Fn (𝑈 “ (1...𝑉)) ∧ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}) Fn (𝑈 “ ((𝑉 + 1)...𝑁))) ∧ ((𝑈 “ (1...𝑉)) ∩ (𝑈 “ ((𝑉 + 1)...𝑁))) = ∅) → (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})) Fn ((𝑈 “ (1...𝑉)) ∪ (𝑈 “ ((𝑉 + 1)...𝑁))))
8676, 84, 85sylancr 587 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})) Fn ((𝑈 “ (1...𝑉)) ∪ (𝑈 “ ((𝑉 + 1)...𝑁))))
87 imaundi 6001 . . . . . . . . . . . . . . . . . . . 20 (𝑈 “ ((1...𝑉) ∪ ((𝑉 + 1)...𝑁))) = ((𝑈 “ (1...𝑉)) ∪ (𝑈 “ ((𝑉 + 1)...𝑁)))
8810nnzd 12074 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝑁 ∈ ℤ)
89 peano2zm 12013 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
9088, 89syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (𝑁 − 1) ∈ ℤ)
91 uzid 12246 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
9290, 91syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
93 peano2uz 12289 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
9492, 93syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
9513, 94eqeltrrd 2911 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
96 fzss2 12935 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ (ℤ‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
9795, 96syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
9897, 7sseldd 3965 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑉 ∈ (1...𝑁))
99 fzsplit 12921 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑉 ∈ (1...𝑁) → (1...𝑁) = ((1...𝑉) ∪ ((𝑉 + 1)...𝑁)))
10098, 99syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (1...𝑁) = ((1...𝑉) ∪ ((𝑉 + 1)...𝑁)))
101100imaeq2d 5922 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑈 “ (1...𝑁)) = (𝑈 “ ((1...𝑉) ∪ ((𝑉 + 1)...𝑁))))
102 f1ofo 6615 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈:(1...𝑁)–onto→(1...𝑁))
1031, 102syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑈:(1...𝑁)–onto→(1...𝑁))
104 foima 6588 . . . . . . . . . . . . . . . . . . . . . 22 (𝑈:(1...𝑁)–onto→(1...𝑁) → (𝑈 “ (1...𝑁)) = (1...𝑁))
105103, 104syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑈 “ (1...𝑁)) = (1...𝑁))
106101, 105eqtr3d 2855 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑈 “ ((1...𝑉) ∪ ((𝑉 + 1)...𝑁))) = (1...𝑁))
10787, 106syl5eqr 2867 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑈 “ (1...𝑉)) ∪ (𝑈 “ ((𝑉 + 1)...𝑁))) = (1...𝑁))
108107fneq2d 6440 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})) Fn ((𝑈 “ (1...𝑉)) ∪ (𝑈 “ ((𝑉 + 1)...𝑁))) ↔ (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})) Fn (1...𝑁)))
10986, 108mpbid 233 . . . . . . . . . . . . . . . . 17 (𝜑 → (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})) Fn (1...𝑁))
110109adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) → (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})) Fn (1...𝑁))
111 fzfid 13329 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) → (1...𝑁) ∈ Fin)
112 inidm 4192 . . . . . . . . . . . . . . . 16 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
113 eqidd 2819 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) ∧ 𝑛 ∈ (1...𝑁)) → (𝑇𝑛) = (𝑇𝑛))
114 fvun1 6747 . . . . . . . . . . . . . . . . . . . 20 ((((𝑈 “ (1...𝑉)) × {1}) Fn (𝑈 “ (1...𝑉)) ∧ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}) Fn (𝑈 “ ((𝑉 + 1)...𝑁)) ∧ (((𝑈 “ (1...𝑉)) ∩ (𝑈 “ ((𝑉 + 1)...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ (1...𝑉)))) → ((((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (1...𝑉)) × {1})‘𝑛))
11572, 75, 114mp3an12 1442 . . . . . . . . . . . . . . . . . . 19 ((((𝑈 “ (1...𝑉)) ∩ (𝑈 “ ((𝑉 + 1)...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ (1...𝑉))) → ((((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (1...𝑉)) × {1})‘𝑛))
11684, 115sylan 580 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) → ((((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (1...𝑉)) × {1})‘𝑛))
11770fvconst2 6958 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (𝑈 “ (1...𝑉)) → (((𝑈 “ (1...𝑉)) × {1})‘𝑛) = 1)
118117adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) → (((𝑈 “ (1...𝑉)) × {1})‘𝑛) = 1)
119116, 118eqtrd 2853 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) → ((((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))‘𝑛) = 1)
120119adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) ∧ 𝑛 ∈ (1...𝑁)) → ((((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))‘𝑛) = 1)
12169, 110, 111, 111, 112, 113, 120ofval 7407 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))‘𝑛) = ((𝑇𝑛) + 1))
122 fnconstg 6560 . . . . . . . . . . . . . . . . . . . . 21 (1 ∈ V → ((𝑈 “ (1...(𝑉 + 1))) × {1}) Fn (𝑈 “ (1...(𝑉 + 1))))
12370, 122ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((𝑈 “ (1...(𝑉 + 1))) × {1}) Fn (𝑈 “ (1...(𝑉 + 1)))
124 fnconstg 6560 . . . . . . . . . . . . . . . . . . . . 21 (0 ∈ V → ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑉 + 1) + 1)...𝑁)))
12573, 124ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑉 + 1) + 1)...𝑁))
126123, 125pm3.2i 471 . . . . . . . . . . . . . . . . . . 19 (((𝑈 “ (1...(𝑉 + 1))) × {1}) Fn (𝑈 “ (1...(𝑉 + 1))) ∧ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑉 + 1) + 1)...𝑁)))
127 imain 6432 . . . . . . . . . . . . . . . . . . . . 21 (Fun 𝑈 → (𝑈 “ ((1...(𝑉 + 1)) ∩ (((𝑉 + 1) + 1)...𝑁))) = ((𝑈 “ (1...(𝑉 + 1))) ∩ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))))
1284, 127syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑈 “ ((1...(𝑉 + 1)) ∩ (((𝑉 + 1) + 1)...𝑁))) = ((𝑈 “ (1...(𝑉 + 1))) ∩ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))))
129 fzdisj 12922 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑉 + 1) < ((𝑉 + 1) + 1) → ((1...(𝑉 + 1)) ∩ (((𝑉 + 1) + 1)...𝑁)) = ∅)
13040, 129syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1...(𝑉 + 1)) ∩ (((𝑉 + 1) + 1)...𝑁)) = ∅)
131130imaeq2d 5922 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑈 “ ((1...(𝑉 + 1)) ∩ (((𝑉 + 1) + 1)...𝑁))) = (𝑈 “ ∅))
132131, 82syl6eq 2869 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑈 “ ((1...(𝑉 + 1)) ∩ (((𝑉 + 1) + 1)...𝑁))) = ∅)
133128, 132eqtr3d 2855 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑈 “ (1...(𝑉 + 1))) ∩ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) = ∅)
134 fnun 6456 . . . . . . . . . . . . . . . . . . 19 (((((𝑈 “ (1...(𝑉 + 1))) × {1}) Fn (𝑈 “ (1...(𝑉 + 1))) ∧ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) ∧ ((𝑈 “ (1...(𝑉 + 1))) ∩ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) = ∅) → (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑉 + 1))) ∪ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))))
135126, 133, 134sylancr 587 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑉 + 1))) ∪ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))))
136 imaundi 6001 . . . . . . . . . . . . . . . . . . . 20 (𝑈 “ ((1...(𝑉 + 1)) ∪ (((𝑉 + 1) + 1)...𝑁))) = ((𝑈 “ (1...(𝑉 + 1))) ∪ (𝑈 “ (((𝑉 + 1) + 1)...𝑁)))
13717imaeq2d 5922 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑈 “ (1...𝑁)) = (𝑈 “ ((1...(𝑉 + 1)) ∪ (((𝑉 + 1) + 1)...𝑁))))
138137, 105eqtr3d 2855 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑈 “ ((1...(𝑉 + 1)) ∪ (((𝑉 + 1) + 1)...𝑁))) = (1...𝑁))
139136, 138syl5eqr 2867 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑈 “ (1...(𝑉 + 1))) ∪ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) = (1...𝑁))
140139fneq2d 6440 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑉 + 1))) ∪ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) ↔ (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁)))
141135, 140mpbid 233 . . . . . . . . . . . . . . . . 17 (𝜑 → (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁))
142141adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) → (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁))
143 uzid 12246 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑉 ∈ ℤ → 𝑉 ∈ (ℤ𝑉))
14427, 143syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑉 ∈ (ℤ𝑉))
145 peano2uz 12289 . . . . . . . . . . . . . . . . . . . . . 22 (𝑉 ∈ (ℤ𝑉) → (𝑉 + 1) ∈ (ℤ𝑉))
146144, 145syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑉 + 1) ∈ (ℤ𝑉))
147 fzss2 12935 . . . . . . . . . . . . . . . . . . . . 21 ((𝑉 + 1) ∈ (ℤ𝑉) → (1...𝑉) ⊆ (1...(𝑉 + 1)))
148146, 147syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (1...𝑉) ⊆ (1...(𝑉 + 1)))
149 imass2 5958 . . . . . . . . . . . . . . . . . . . 20 ((1...𝑉) ⊆ (1...(𝑉 + 1)) → (𝑈 “ (1...𝑉)) ⊆ (𝑈 “ (1...(𝑉 + 1))))
150148, 149syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑈 “ (1...𝑉)) ⊆ (𝑈 “ (1...(𝑉 + 1))))
151150sselda 3964 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) → 𝑛 ∈ (𝑈 “ (1...(𝑉 + 1))))
152 fvun1 6747 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑈 “ (1...(𝑉 + 1))) × {1}) Fn (𝑈 “ (1...(𝑉 + 1))) ∧ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑉 + 1) + 1)...𝑁)) ∧ (((𝑈 “ (1...(𝑉 + 1))) ∩ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ (1...(𝑉 + 1))))) → ((((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (1...(𝑉 + 1))) × {1})‘𝑛))
153123, 125, 152mp3an12 1442 . . . . . . . . . . . . . . . . . . . 20 ((((𝑈 “ (1...(𝑉 + 1))) ∩ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ (1...(𝑉 + 1)))) → ((((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (1...(𝑉 + 1))) × {1})‘𝑛))
154133, 153sylan 580 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 + 1)))) → ((((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (1...(𝑉 + 1))) × {1})‘𝑛))
15570fvconst2 6958 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (𝑈 “ (1...(𝑉 + 1))) → (((𝑈 “ (1...(𝑉 + 1))) × {1})‘𝑛) = 1)
156155adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 + 1)))) → (((𝑈 “ (1...(𝑉 + 1))) × {1})‘𝑛) = 1)
157154, 156eqtrd 2853 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 + 1)))) → ((((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}))‘𝑛) = 1)
158151, 157syldan 591 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) → ((((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}))‘𝑛) = 1)
159158adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) ∧ 𝑛 ∈ (1...𝑁)) → ((((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}))‘𝑛) = 1)
16069, 142, 111, 111, 112, 113, 159ofval 7407 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇f + (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})))‘𝑛) = ((𝑇𝑛) + 1))
161121, 160eqtr4d 2856 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))‘𝑛) = ((𝑇f + (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})))‘𝑛))
16266, 161mpdan 683 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) → ((𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))‘𝑛) = ((𝑇f + (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})))‘𝑛))
163 imassrn 5933 . . . . . . . . . . . . . . . 16 (𝑈 “ (((𝑉 + 1) + 1)...𝑁)) ⊆ ran 𝑈
164163, 64sstrid 3975 . . . . . . . . . . . . . . 15 (𝜑 → (𝑈 “ (((𝑉 + 1) + 1)...𝑁)) ⊆ (1...𝑁))
165164sselda 3964 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) → 𝑛 ∈ (1...𝑁))
16668adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) → 𝑇 Fn (1...𝑁))
167109adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) → (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})) Fn (1...𝑁))
168 fzfid 13329 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) → (1...𝑁) ∈ Fin)
169 eqidd 2819 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → (𝑇𝑛) = (𝑇𝑛))
170 uzid 12246 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑉 + 1) ∈ ℤ → (𝑉 + 1) ∈ (ℤ‘(𝑉 + 1)))
17130, 170syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑉 + 1) ∈ (ℤ‘(𝑉 + 1)))
172 peano2uz 12289 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑉 + 1) ∈ (ℤ‘(𝑉 + 1)) → ((𝑉 + 1) + 1) ∈ (ℤ‘(𝑉 + 1)))
173171, 172syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑉 + 1) + 1) ∈ (ℤ‘(𝑉 + 1)))
174 fzss1 12934 . . . . . . . . . . . . . . . . . . . . 21 (((𝑉 + 1) + 1) ∈ (ℤ‘(𝑉 + 1)) → (((𝑉 + 1) + 1)...𝑁) ⊆ ((𝑉 + 1)...𝑁))
175173, 174syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (((𝑉 + 1) + 1)...𝑁) ⊆ ((𝑉 + 1)...𝑁))
176 imass2 5958 . . . . . . . . . . . . . . . . . . . 20 ((((𝑉 + 1) + 1)...𝑁) ⊆ ((𝑉 + 1)...𝑁) → (𝑈 “ (((𝑉 + 1) + 1)...𝑁)) ⊆ (𝑈 “ ((𝑉 + 1)...𝑁)))
177175, 176syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑈 “ (((𝑉 + 1) + 1)...𝑁)) ⊆ (𝑈 “ ((𝑉 + 1)...𝑁)))
178177sselda 3964 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) → 𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁)))
179 fvun2 6748 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑈 “ (1...𝑉)) × {1}) Fn (𝑈 “ (1...𝑉)) ∧ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}) Fn (𝑈 “ ((𝑉 + 1)...𝑁)) ∧ (((𝑈 “ (1...𝑉)) ∩ (𝑈 “ ((𝑉 + 1)...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁)))) → ((((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})‘𝑛))
18072, 75, 179mp3an12 1442 . . . . . . . . . . . . . . . . . . . 20 ((((𝑈 “ (1...𝑉)) ∩ (𝑈 “ ((𝑉 + 1)...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) → ((((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})‘𝑛))
18184, 180sylan 580 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) → ((((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})‘𝑛))
18273fvconst2 6958 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁)) → (((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})‘𝑛) = 0)
183182adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) → (((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})‘𝑛) = 0)
184181, 183eqtrd 2853 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) → ((((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))‘𝑛) = 0)
185178, 184syldan 591 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) → ((((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))‘𝑛) = 0)
186185adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → ((((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))‘𝑛) = 0)
187166, 167, 168, 168, 112, 169, 186ofval 7407 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))‘𝑛) = ((𝑇𝑛) + 0))
188141adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) → (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁))
189 fvun2 6748 . . . . . . . . . . . . . . . . . . . 20 ((((𝑈 “ (1...(𝑉 + 1))) × {1}) Fn (𝑈 “ (1...(𝑉 + 1))) ∧ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑉 + 1) + 1)...𝑁)) ∧ (((𝑈 “ (1...(𝑉 + 1))) ∩ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁)))) → ((((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})‘𝑛))
190123, 125, 189mp3an12 1442 . . . . . . . . . . . . . . . . . . 19 ((((𝑈 “ (1...(𝑉 + 1))) ∩ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) → ((((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})‘𝑛))
191133, 190sylan 580 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) → ((((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})‘𝑛))
19273fvconst2 6958 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁)) → (((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})‘𝑛) = 0)
193192adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) → (((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})‘𝑛) = 0)
194191, 193eqtrd 2853 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) → ((((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}))‘𝑛) = 0)
195194adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → ((((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}))‘𝑛) = 0)
196166, 188, 168, 168, 112, 169, 195ofval 7407 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇f + (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})))‘𝑛) = ((𝑇𝑛) + 0))
197187, 196eqtr4d 2856 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))‘𝑛) = ((𝑇f + (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})))‘𝑛))
198165, 197mpdan 683 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) → ((𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))‘𝑛) = ((𝑇f + (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})))‘𝑛))
199162, 198jaodan 951 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (𝑈 “ (1...𝑉)) ∨ 𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁)))) → ((𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))‘𝑛) = ((𝑇f + (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})))‘𝑛))
200199adantlr 711 . . . . . . . . . . 11 (((𝜑𝑀 < 𝑉) ∧ (𝑛 ∈ (𝑈 “ (1...𝑉)) ∨ 𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁)))) → ((𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))‘𝑛) = ((𝑇f + (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})))‘𝑛))
201 poimirlem2.1 . . . . . . . . . . . . . . 15 (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗(𝑇f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))))
202201adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑀 < 𝑉) → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗(𝑇f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))))
203 vex 3495 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
204 ovex 7178 . . . . . . . . . . . . . . . . 17 (𝑦 + 1) ∈ V
205203, 204ifex 4511 . . . . . . . . . . . . . . . 16 if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) ∈ V
206205a1i 11 . . . . . . . . . . . . . . 15 (((𝜑𝑀 < 𝑉) ∧ 𝑦 = (𝑉 − 1)) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) ∈ V)
207 breq1 5060 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = (𝑉 − 1) → (𝑦 < 𝑀 ↔ (𝑉 − 1) < 𝑀))
208207adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 = (𝑉 − 1)) → (𝑦 < 𝑀 ↔ (𝑉 − 1) < 𝑀))
209 simpr 485 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 = (𝑉 − 1)) → 𝑦 = (𝑉 − 1))
210 oveq1 7152 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = (𝑉 − 1) → (𝑦 + 1) = ((𝑉 − 1) + 1))
21127zcnd 12076 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑉 ∈ ℂ)
212 npcan1 11053 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑉 ∈ ℂ → ((𝑉 − 1) + 1) = 𝑉)
213211, 212syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((𝑉 − 1) + 1) = 𝑉)
214210, 213sylan9eqr 2875 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 = (𝑉 − 1)) → (𝑦 + 1) = 𝑉)
215208, 209, 214ifbieq12d 4490 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 = (𝑉 − 1)) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = if((𝑉 − 1) < 𝑀, (𝑉 − 1), 𝑉))
216215adantlr 711 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑀 < 𝑉) ∧ 𝑦 = (𝑉 − 1)) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = if((𝑉 − 1) < 𝑀, (𝑉 − 1), 𝑉))
217 poimirlem2.5 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝑀 ∈ ((0...𝑁) ∖ {𝑉}))
218217eldifad 3945 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑀 ∈ (0...𝑁))
219 elfzelz 12896 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℤ)
220218, 219syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑀 ∈ ℤ)
221 zltlem1 12023 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑀 ∈ ℤ ∧ 𝑉 ∈ ℤ) → (𝑀 < 𝑉𝑀 ≤ (𝑉 − 1)))
222220, 27, 221syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑀 < 𝑉𝑀 ≤ (𝑉 − 1)))
223220zred 12075 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑀 ∈ ℝ)
224 peano2zm 12013 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑉 ∈ ℤ → (𝑉 − 1) ∈ ℤ)
22527, 224syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝑉 − 1) ∈ ℤ)
226225zred 12075 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑉 − 1) ∈ ℝ)
227223, 226lenltd 10774 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑀 ≤ (𝑉 − 1) ↔ ¬ (𝑉 − 1) < 𝑀))
228222, 227bitrd 280 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑀 < 𝑉 ↔ ¬ (𝑉 − 1) < 𝑀))
229228biimpa 477 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑀 < 𝑉) → ¬ (𝑉 − 1) < 𝑀)
230229iffalsed 4474 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑀 < 𝑉) → if((𝑉 − 1) < 𝑀, (𝑉 − 1), 𝑉) = 𝑉)
231230adantr 481 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑀 < 𝑉) ∧ 𝑦 = (𝑉 − 1)) → if((𝑉 − 1) < 𝑀, (𝑉 − 1), 𝑉) = 𝑉)
232216, 231eqtrd 2853 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑀 < 𝑉) ∧ 𝑦 = (𝑉 − 1)) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = 𝑉)
233232eqeq2d 2829 . . . . . . . . . . . . . . . . 17 (((𝜑𝑀 < 𝑉) ∧ 𝑦 = (𝑉 − 1)) → (𝑗 = if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) ↔ 𝑗 = 𝑉))
234233biimpa 477 . . . . . . . . . . . . . . . 16 ((((𝜑𝑀 < 𝑉) ∧ 𝑦 = (𝑉 − 1)) ∧ 𝑗 = if(𝑦 < 𝑀, 𝑦, (𝑦 + 1))) → 𝑗 = 𝑉)
235 oveq2 7153 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑉 → (1...𝑗) = (1...𝑉))
236235imaeq2d 5922 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑉 → (𝑈 “ (1...𝑗)) = (𝑈 “ (1...𝑉)))
237236xpeq1d 5577 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑉 → ((𝑈 “ (1...𝑗)) × {1}) = ((𝑈 “ (1...𝑉)) × {1}))
238 oveq1 7152 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑉 → (𝑗 + 1) = (𝑉 + 1))
239238oveq1d 7160 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑉 → ((𝑗 + 1)...𝑁) = ((𝑉 + 1)...𝑁))
240239imaeq2d 5922 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑉 → (𝑈 “ ((𝑗 + 1)...𝑁)) = (𝑈 “ ((𝑉 + 1)...𝑁)))
241240xpeq1d 5577 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑉 → ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) = ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))
242237, 241uneq12d 4137 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑉 → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) = (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))
243242oveq2d 7161 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑉 → (𝑇f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))))
244234, 243syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑀 < 𝑉) ∧ 𝑦 = (𝑉 − 1)) ∧ 𝑗 = if(𝑦 < 𝑀, 𝑦, (𝑦 + 1))) → (𝑇f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))))
245206, 244csbied 3916 . . . . . . . . . . . . . 14 (((𝜑𝑀 < 𝑉) ∧ 𝑦 = (𝑉 − 1)) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗(𝑇f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))))
246 elfzm1b 12973 . . . . . . . . . . . . . . . . 17 ((𝑉 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑉 ∈ (1...𝑁) ↔ (𝑉 − 1) ∈ (0...(𝑁 − 1))))
24727, 88, 246syl2anc 584 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑉 ∈ (1...𝑁) ↔ (𝑉 − 1) ∈ (0...(𝑁 − 1))))
24898, 247mpbid 233 . . . . . . . . . . . . . . 15 (𝜑 → (𝑉 − 1) ∈ (0...(𝑁 − 1)))
249248adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑀 < 𝑉) → (𝑉 − 1) ∈ (0...(𝑁 − 1)))
250 ovexd 7180 . . . . . . . . . . . . . 14 ((𝜑𝑀 < 𝑉) → (𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))) ∈ V)
251202, 245, 249, 250fvmptd 6767 . . . . . . . . . . . . 13 ((𝜑𝑀 < 𝑉) → (𝐹‘(𝑉 − 1)) = (𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))))
252251fveq1d 6665 . . . . . . . . . . . 12 ((𝜑𝑀 < 𝑉) → ((𝐹‘(𝑉 − 1))‘𝑛) = ((𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))‘𝑛))
253252adantr 481 . . . . . . . . . . 11 (((𝜑𝑀 < 𝑉) ∧ (𝑛 ∈ (𝑈 “ (1...𝑉)) ∨ 𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁)))) → ((𝐹‘(𝑉 − 1))‘𝑛) = ((𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))‘𝑛))
254205a1i 11 . . . . . . . . . . . . . . 15 (((𝜑𝑀 < 𝑉) ∧ 𝑦 = 𝑉) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) ∈ V)
255 breq1 5060 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑉 → (𝑦 < 𝑀𝑉 < 𝑀))
256 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑉𝑦 = 𝑉)
257 oveq1 7152 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑉 → (𝑦 + 1) = (𝑉 + 1))
258255, 256, 257ifbieq12d 4490 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑉 → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = if(𝑉 < 𝑀, 𝑉, (𝑉 + 1)))
259 ltnsym 10726 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑀 ∈ ℝ ∧ 𝑉 ∈ ℝ) → (𝑀 < 𝑉 → ¬ 𝑉 < 𝑀))
260223, 28, 259syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑀 < 𝑉 → ¬ 𝑉 < 𝑀))
261260imp 407 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑀 < 𝑉) → ¬ 𝑉 < 𝑀)
262261iffalsed 4474 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑀 < 𝑉) → if(𝑉 < 𝑀, 𝑉, (𝑉 + 1)) = (𝑉 + 1))
263258, 262sylan9eqr 2875 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑀 < 𝑉) ∧ 𝑦 = 𝑉) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = (𝑉 + 1))
264263eqeq2d 2829 . . . . . . . . . . . . . . . . 17 (((𝜑𝑀 < 𝑉) ∧ 𝑦 = 𝑉) → (𝑗 = if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) ↔ 𝑗 = (𝑉 + 1)))
265264biimpa 477 . . . . . . . . . . . . . . . 16 ((((𝜑𝑀 < 𝑉) ∧ 𝑦 = 𝑉) ∧ 𝑗 = if(𝑦 < 𝑀, 𝑦, (𝑦 + 1))) → 𝑗 = (𝑉 + 1))
266 oveq2 7153 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑉 + 1) → (1...𝑗) = (1...(𝑉 + 1)))
267266imaeq2d 5922 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑉 + 1) → (𝑈 “ (1...𝑗)) = (𝑈 “ (1...(𝑉 + 1))))
268267xpeq1d 5577 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑉 + 1) → ((𝑈 “ (1...𝑗)) × {1}) = ((𝑈 “ (1...(𝑉 + 1))) × {1}))
269 oveq1 7152 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (𝑉 + 1) → (𝑗 + 1) = ((𝑉 + 1) + 1))
270269oveq1d 7160 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑉 + 1) → ((𝑗 + 1)...𝑁) = (((𝑉 + 1) + 1)...𝑁))
271270imaeq2d 5922 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑉 + 1) → (𝑈 “ ((𝑗 + 1)...𝑁)) = (𝑈 “ (((𝑉 + 1) + 1)...𝑁)))
272271xpeq1d 5577 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑉 + 1) → ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) = ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}))
273268, 272uneq12d 4137 . . . . . . . . . . . . . . . . 17 (𝑗 = (𝑉 + 1) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) = (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})))
274273oveq2d 7161 . . . . . . . . . . . . . . . 16 (𝑗 = (𝑉 + 1) → (𝑇f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇f + (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}))))
275265, 274syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑀 < 𝑉) ∧ 𝑦 = 𝑉) ∧ 𝑗 = if(𝑦 < 𝑀, 𝑦, (𝑦 + 1))) → (𝑇f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇f + (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}))))
276254, 275csbied 3916 . . . . . . . . . . . . . 14 (((𝜑𝑀 < 𝑉) ∧ 𝑦 = 𝑉) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗(𝑇f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇f + (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}))))
277 fz1ssfz0 12991 . . . . . . . . . . . . . . . 16 (1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1))
278277, 7sseldi 3962 . . . . . . . . . . . . . . 15 (𝜑𝑉 ∈ (0...(𝑁 − 1)))
279278adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑀 < 𝑉) → 𝑉 ∈ (0...(𝑁 − 1)))
280 ovexd 7180 . . . . . . . . . . . . . 14 ((𝜑𝑀 < 𝑉) → (𝑇f + (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}))) ∈ V)
281202, 276, 279, 280fvmptd 6767 . . . . . . . . . . . . 13 ((𝜑𝑀 < 𝑉) → (𝐹𝑉) = (𝑇f + (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}))))
282281fveq1d 6665 . . . . . . . . . . . 12 ((𝜑𝑀 < 𝑉) → ((𝐹𝑉)‘𝑛) = ((𝑇f + (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})))‘𝑛))
283282adantr 481 . . . . . . . . . . 11 (((𝜑𝑀 < 𝑉) ∧ (𝑛 ∈ (𝑈 “ (1...𝑉)) ∨ 𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁)))) → ((𝐹𝑉)‘𝑛) = ((𝑇f + (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})))‘𝑛))
284200, 253, 2833eqtr4d 2863 . . . . . . . . . 10 (((𝜑𝑀 < 𝑉) ∧ (𝑛 ∈ (𝑈 “ (1...𝑉)) ∨ 𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁)))) → ((𝐹‘(𝑉 − 1))‘𝑛) = ((𝐹𝑉)‘𝑛))
285284ex 413 . . . . . . . . 9 ((𝜑𝑀 < 𝑉) → ((𝑛 ∈ (𝑈 “ (1...𝑉)) ∨ 𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) → ((𝐹‘(𝑉 − 1))‘𝑛) = ((𝐹𝑉)‘𝑛)))
28660, 285sylbid 241 . . . . . . . 8 ((𝜑𝑀 < 𝑉) → ((𝑛 ∈ (𝑈 “ (1...𝑁)) ∧ ¬ 𝑛 ∈ (𝑈 “ {(𝑉 + 1)})) → ((𝐹‘(𝑉 − 1))‘𝑛) = ((𝐹𝑉)‘𝑛)))
287286expdimp 453 . . . . . . 7 (((𝜑𝑀 < 𝑉) ∧ 𝑛 ∈ (𝑈 “ (1...𝑁))) → (¬ 𝑛 ∈ (𝑈 “ {(𝑉 + 1)}) → ((𝐹‘(𝑉 − 1))‘𝑛) = ((𝐹𝑉)‘𝑛)))
288287necon1ad 3030 . . . . . 6 (((𝜑𝑀 < 𝑉) ∧ 𝑛 ∈ (𝑈 “ (1...𝑁))) → (((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 ∈ (𝑈 “ {(𝑉 + 1)})))
289 elimasni 5949 . . . . . . . 8 (𝑛 ∈ (𝑈 “ {(𝑉 + 1)}) → (𝑉 + 1)𝑈𝑛)
290 eqcom 2825 . . . . . . . . 9 (𝑛 = (𝑈‘(𝑉 + 1)) ↔ (𝑈‘(𝑉 + 1)) = 𝑛)
291 f1ofn 6609 . . . . . . . . . . 11 (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈 Fn (1...𝑁))
2921, 291syl 17 . . . . . . . . . 10 (𝜑𝑈 Fn (1...𝑁))
293 fnbrfvb 6711 . . . . . . . . . 10 ((𝑈 Fn (1...𝑁) ∧ (𝑉 + 1) ∈ (1...𝑁)) → ((𝑈‘(𝑉 + 1)) = 𝑛 ↔ (𝑉 + 1)𝑈𝑛))
294292, 15, 293syl2anc 584 . . . . . . . . 9 (𝜑 → ((𝑈‘(𝑉 + 1)) = 𝑛 ↔ (𝑉 + 1)𝑈𝑛))
295290, 294syl5bb 284 . . . . . . . 8 (𝜑 → (𝑛 = (𝑈‘(𝑉 + 1)) ↔ (𝑉 + 1)𝑈𝑛))
296289, 295syl5ibr 247 . . . . . . 7 (𝜑 → (𝑛 ∈ (𝑈 “ {(𝑉 + 1)}) → 𝑛 = (𝑈‘(𝑉 + 1))))
297296ad2antrr 722 . . . . . 6 (((𝜑𝑀 < 𝑉) ∧ 𝑛 ∈ (𝑈 “ (1...𝑁))) → (𝑛 ∈ (𝑈 “ {(𝑉 + 1)}) → 𝑛 = (𝑈‘(𝑉 + 1))))
298288, 297syld 47 . . . . 5 (((𝜑𝑀 < 𝑉) ∧ 𝑛 ∈ (𝑈 “ (1...𝑁))) → (((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = (𝑈‘(𝑉 + 1))))
299298ralrimiva 3179 . . . 4 ((𝜑𝑀 < 𝑉) → ∀𝑛 ∈ (𝑈 “ (1...𝑁))(((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = (𝑈‘(𝑉 + 1))))
300 fvex 6676 . . . . 5 (𝑈‘(𝑉 + 1)) ∈ V
301 eqeq2 2830 . . . . . . 7 (𝑚 = (𝑈‘(𝑉 + 1)) → (𝑛 = 𝑚𝑛 = (𝑈‘(𝑉 + 1))))
302301imbi2d 342 . . . . . 6 (𝑚 = (𝑈‘(𝑉 + 1)) → ((((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = 𝑚) ↔ (((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = (𝑈‘(𝑉 + 1)))))
303302ralbidv 3194 . . . . 5 (𝑚 = (𝑈‘(𝑉 + 1)) → (∀𝑛 ∈ (𝑈 “ (1...𝑁))(((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = 𝑚) ↔ ∀𝑛 ∈ (𝑈 “ (1...𝑁))(((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = (𝑈‘(𝑉 + 1)))))
304300, 303spcev 3604 . . . 4 (∀𝑛 ∈ (𝑈 “ (1...𝑁))(((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = (𝑈‘(𝑉 + 1))) → ∃𝑚𝑛 ∈ (𝑈 “ (1...𝑁))(((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = 𝑚))
305299, 304syl 17 . . 3 ((𝜑𝑀 < 𝑉) → ∃𝑚𝑛 ∈ (𝑈 “ (1...𝑁))(((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = 𝑚))
306 imadif 6431 . . . . . . . . . . . . . . 15 (Fun 𝑈 → (𝑈 “ ((1...𝑁) ∖ {𝑉})) = ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {𝑉})))
3074, 306syl 17 . . . . . . . . . . . . . 14 (𝜑 → (𝑈 “ ((1...𝑁) ∖ {𝑉})) = ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {𝑉})))
308100difeq1d 4095 . . . . . . . . . . . . . . . 16 (𝜑 → ((1...𝑁) ∖ {𝑉}) = (((1...𝑉) ∪ ((𝑉 + 1)...𝑁)) ∖ {𝑉}))
309 difundir 4254 . . . . . . . . . . . . . . . . 17 (((1...𝑉) ∪ ((𝑉 + 1)...𝑁)) ∖ {𝑉}) = (((1...𝑉) ∖ {𝑉}) ∪ (((𝑉 + 1)...𝑁) ∖ {𝑉}))
310213, 21eqeltrd 2910 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((𝑉 − 1) + 1) ∈ (ℤ‘1))
311 uzid 12246 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑉 − 1) ∈ ℤ → (𝑉 − 1) ∈ (ℤ‘(𝑉 − 1)))
312225, 311syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑉 − 1) ∈ (ℤ‘(𝑉 − 1)))
313 peano2uz 12289 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑉 − 1) ∈ (ℤ‘(𝑉 − 1)) → ((𝑉 − 1) + 1) ∈ (ℤ‘(𝑉 − 1)))
314312, 313syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑉 − 1) + 1) ∈ (ℤ‘(𝑉 − 1)))
315213, 314eqeltrrd 2911 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑉 ∈ (ℤ‘(𝑉 − 1)))
316 fzsplit2 12920 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑉 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑉 ∈ (ℤ‘(𝑉 − 1))) → (1...𝑉) = ((1...(𝑉 − 1)) ∪ (((𝑉 − 1) + 1)...𝑉)))
317310, 315, 316syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (1...𝑉) = ((1...(𝑉 − 1)) ∪ (((𝑉 − 1) + 1)...𝑉)))
318213oveq1d 7160 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (((𝑉 − 1) + 1)...𝑉) = (𝑉...𝑉))
319 fzsn 12937 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑉 ∈ ℤ → (𝑉...𝑉) = {𝑉})
32027, 319syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑉...𝑉) = {𝑉})
321318, 320eqtrd 2853 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (((𝑉 − 1) + 1)...𝑉) = {𝑉})
322321uneq2d 4136 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((1...(𝑉 − 1)) ∪ (((𝑉 − 1) + 1)...𝑉)) = ((1...(𝑉 − 1)) ∪ {𝑉}))
323317, 322eqtrd 2853 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (1...𝑉) = ((1...(𝑉 − 1)) ∪ {𝑉}))
324323difeq1d 4095 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((1...𝑉) ∖ {𝑉}) = (((1...(𝑉 − 1)) ∪ {𝑉}) ∖ {𝑉}))
325 difun2 4425 . . . . . . . . . . . . . . . . . . . 20 (((1...(𝑉 − 1)) ∪ {𝑉}) ∖ {𝑉}) = ((1...(𝑉 − 1)) ∖ {𝑉})
32628ltm1d 11560 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑉 − 1) < 𝑉)
327226, 28ltnled 10775 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑉 − 1) < 𝑉 ↔ ¬ 𝑉 ≤ (𝑉 − 1)))
328326, 327mpbid 233 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ¬ 𝑉 ≤ (𝑉 − 1))
329 elfzle2 12899 . . . . . . . . . . . . . . . . . . . . . 22 (𝑉 ∈ (1...(𝑉 − 1)) → 𝑉 ≤ (𝑉 − 1))
330328, 329nsyl 142 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ¬ 𝑉 ∈ (1...(𝑉 − 1)))
331 difsn 4723 . . . . . . . . . . . . . . . . . . . . 21 𝑉 ∈ (1...(𝑉 − 1)) → ((1...(𝑉 − 1)) ∖ {𝑉}) = (1...(𝑉 − 1)))
332330, 331syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((1...(𝑉 − 1)) ∖ {𝑉}) = (1...(𝑉 − 1)))
333325, 332syl5eq 2865 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (((1...(𝑉 − 1)) ∪ {𝑉}) ∖ {𝑉}) = (1...(𝑉 − 1)))
334324, 333eqtrd 2853 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1...𝑉) ∖ {𝑉}) = (1...(𝑉 − 1)))
335 elfzle1 12898 . . . . . . . . . . . . . . . . . . . 20 (𝑉 ∈ ((𝑉 + 1)...𝑁) → (𝑉 + 1) ≤ 𝑉)
33633, 335nsyl 142 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ¬ 𝑉 ∈ ((𝑉 + 1)...𝑁))
337 difsn 4723 . . . . . . . . . . . . . . . . . . 19 𝑉 ∈ ((𝑉 + 1)...𝑁) → (((𝑉 + 1)...𝑁) ∖ {𝑉}) = ((𝑉 + 1)...𝑁))
338336, 337syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((𝑉 + 1)...𝑁) ∖ {𝑉}) = ((𝑉 + 1)...𝑁))
339334, 338uneq12d 4137 . . . . . . . . . . . . . . . . 17 (𝜑 → (((1...𝑉) ∖ {𝑉}) ∪ (((𝑉 + 1)...𝑁) ∖ {𝑉})) = ((1...(𝑉 − 1)) ∪ ((𝑉 + 1)...𝑁)))
340309, 339syl5eq 2865 . . . . . . . . . . . . . . . 16 (𝜑 → (((1...𝑉) ∪ ((𝑉 + 1)...𝑁)) ∖ {𝑉}) = ((1...(𝑉 − 1)) ∪ ((𝑉 + 1)...𝑁)))
341308, 340eqtrd 2853 . . . . . . . . . . . . . . 15 (𝜑 → ((1...𝑁) ∖ {𝑉}) = ((1...(𝑉 − 1)) ∪ ((𝑉 + 1)...𝑁)))
342341imaeq2d 5922 . . . . . . . . . . . . . 14 (𝜑 → (𝑈 “ ((1...𝑁) ∖ {𝑉})) = (𝑈 “ ((1...(𝑉 − 1)) ∪ ((𝑉 + 1)...𝑁))))
343307, 342eqtr3d 2855 . . . . . . . . . . . . 13 (𝜑 → ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {𝑉})) = (𝑈 “ ((1...(𝑉 − 1)) ∪ ((𝑉 + 1)...𝑁))))
344 imaundi 6001 . . . . . . . . . . . . 13 (𝑈 “ ((1...(𝑉 − 1)) ∪ ((𝑉 + 1)...𝑁))) = ((𝑈 “ (1...(𝑉 − 1))) ∪ (𝑈 “ ((𝑉 + 1)...𝑁)))
345343, 344syl6eq 2869 . . . . . . . . . . . 12 (𝜑 → ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {𝑉})) = ((𝑈 “ (1...(𝑉 − 1))) ∪ (𝑈 “ ((𝑉 + 1)...𝑁))))
346345eleq2d 2895 . . . . . . . . . . 11 (𝜑 → (𝑛 ∈ ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {𝑉})) ↔ 𝑛 ∈ ((𝑈 “ (1...(𝑉 − 1))) ∪ (𝑈 “ ((𝑉 + 1)...𝑁)))))
347 eldif 3943 . . . . . . . . . . 11 (𝑛 ∈ ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {𝑉})) ↔ (𝑛 ∈ (𝑈 “ (1...𝑁)) ∧ ¬ 𝑛 ∈ (𝑈 “ {𝑉})))
348 elun 4122 . . . . . . . . . . 11 (𝑛 ∈ ((𝑈 “ (1...(𝑉 − 1))) ∪ (𝑈 “ ((𝑉 + 1)...𝑁))) ↔ (𝑛 ∈ (𝑈 “ (1...(𝑉 − 1))) ∨ 𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))))
349346, 347, 3483bitr3g 314 . . . . . . . . . 10 (𝜑 → ((𝑛 ∈ (𝑈 “ (1...𝑁)) ∧ ¬ 𝑛 ∈ (𝑈 “ {𝑉})) ↔ (𝑛 ∈ (𝑈 “ (1...(𝑉 − 1))) ∨ 𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁)))))
350349adantr 481 . . . . . . . . 9 ((𝜑𝑉 < 𝑀) → ((𝑛 ∈ (𝑈 “ (1...𝑁)) ∧ ¬ 𝑛 ∈ (𝑈 “ {𝑉})) ↔ (𝑛 ∈ (𝑈 “ (1...(𝑉 − 1))) ∨ 𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁)))))
351 imassrn 5933 . . . . . . . . . . . . . . . 16 (𝑈 “ (1...(𝑉 − 1))) ⊆ ran 𝑈
352351, 64sstrid 3975 . . . . . . . . . . . . . . 15 (𝜑 → (𝑈 “ (1...(𝑉 − 1))) ⊆ (1...𝑁))
353352sselda 3964 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) → 𝑛 ∈ (1...𝑁))
35468adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) → 𝑇 Fn (1...𝑁))
355 fnconstg 6560 . . . . . . . . . . . . . . . . . . . . 21 (1 ∈ V → ((𝑈 “ (1...(𝑉 − 1))) × {1}) Fn (𝑈 “ (1...(𝑉 − 1))))
35670, 355ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((𝑈 “ (1...(𝑉 − 1))) × {1}) Fn (𝑈 “ (1...(𝑉 − 1)))
357 fnconstg 6560 . . . . . . . . . . . . . . . . . . . . 21 (0 ∈ V → ((𝑈 “ (𝑉...𝑁)) × {0}) Fn (𝑈 “ (𝑉...𝑁)))
35873, 357ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((𝑈 “ (𝑉...𝑁)) × {0}) Fn (𝑈 “ (𝑉...𝑁))
359356, 358pm3.2i 471 . . . . . . . . . . . . . . . . . . 19 (((𝑈 “ (1...(𝑉 − 1))) × {1}) Fn (𝑈 “ (1...(𝑉 − 1))) ∧ ((𝑈 “ (𝑉...𝑁)) × {0}) Fn (𝑈 “ (𝑉...𝑁)))
360 imain 6432 . . . . . . . . . . . . . . . . . . . . 21 (Fun 𝑈 → (𝑈 “ ((1...(𝑉 − 1)) ∩ (𝑉...𝑁))) = ((𝑈 “ (1...(𝑉 − 1))) ∩ (𝑈 “ (𝑉...𝑁))))
3614, 360syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑈 “ ((1...(𝑉 − 1)) ∩ (𝑉...𝑁))) = ((𝑈 “ (1...(𝑉 − 1))) ∩ (𝑈 “ (𝑉...𝑁))))
362 fzdisj 12922 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑉 − 1) < 𝑉 → ((1...(𝑉 − 1)) ∩ (𝑉...𝑁)) = ∅)
363326, 362syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1...(𝑉 − 1)) ∩ (𝑉...𝑁)) = ∅)
364363imaeq2d 5922 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑈 “ ((1...(𝑉 − 1)) ∩ (𝑉...𝑁))) = (𝑈 “ ∅))
365364, 82syl6eq 2869 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑈 “ ((1...(𝑉 − 1)) ∩ (𝑉...𝑁))) = ∅)
366361, 365eqtr3d 2855 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑈 “ (1...(𝑉 − 1))) ∩ (𝑈 “ (𝑉...𝑁))) = ∅)
367 fnun 6456 . . . . . . . . . . . . . . . . . . 19 (((((𝑈 “ (1...(𝑉 − 1))) × {1}) Fn (𝑈 “ (1...(𝑉 − 1))) ∧ ((𝑈 “ (𝑉...𝑁)) × {0}) Fn (𝑈 “ (𝑉...𝑁))) ∧ ((𝑈 “ (1...(𝑉 − 1))) ∩ (𝑈 “ (𝑉...𝑁))) = ∅) → (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑉 − 1))) ∪ (𝑈 “ (𝑉...𝑁))))
368359, 366, 367sylancr 587 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑉 − 1))) ∪ (𝑈 “ (𝑉...𝑁))))
369 imaundi 6001 . . . . . . . . . . . . . . . . . . . 20 (𝑈 “ ((1...(𝑉 − 1)) ∪ (𝑉...𝑁))) = ((𝑈 “ (1...(𝑉 − 1))) ∪ (𝑈 “ (𝑉...𝑁)))
370 uzss 12253 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑉 ∈ (ℤ‘(𝑉 − 1)) → (ℤ𝑉) ⊆ (ℤ‘(𝑉 − 1)))
371315, 370syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (ℤ𝑉) ⊆ (ℤ‘(𝑉 − 1)))
372 elfzuz3 12893 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑉 ∈ (1...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ𝑉))
3737, 372syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (𝑁 − 1) ∈ (ℤ𝑉))
374371, 373sseldd 3965 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝑁 − 1) ∈ (ℤ‘(𝑉 − 1)))
375 peano2uz 12289 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑁 − 1) ∈ (ℤ‘(𝑉 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑉 − 1)))
376374, 375syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑉 − 1)))
37713, 376eqeltrrd 2911 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑁 ∈ (ℤ‘(𝑉 − 1)))
378 fzsplit2 12920 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑉 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑉 − 1))) → (1...𝑁) = ((1...(𝑉 − 1)) ∪ (((𝑉 − 1) + 1)...𝑁)))
379310, 377, 378syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (1...𝑁) = ((1...(𝑉 − 1)) ∪ (((𝑉 − 1) + 1)...𝑁)))
380213oveq1d 7160 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (((𝑉 − 1) + 1)...𝑁) = (𝑉...𝑁))
381380uneq2d 4136 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((1...(𝑉 − 1)) ∪ (((𝑉 − 1) + 1)...𝑁)) = ((1...(𝑉 − 1)) ∪ (𝑉...𝑁)))
382379, 381eqtrd 2853 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (1...𝑁) = ((1...(𝑉 − 1)) ∪ (𝑉...𝑁)))
383382imaeq2d 5922 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑈 “ (1...𝑁)) = (𝑈 “ ((1...(𝑉 − 1)) ∪ (𝑉...𝑁))))
384383, 105eqtr3d 2855 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑈 “ ((1...(𝑉 − 1)) ∪ (𝑉...𝑁))) = (1...𝑁))
385369, 384syl5eqr 2867 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑈 “ (1...(𝑉 − 1))) ∪ (𝑈 “ (𝑉...𝑁))) = (1...𝑁))
386385fneq2d 6440 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑉 − 1))) ∪ (𝑈 “ (𝑉...𝑁))) ↔ (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0})) Fn (1...𝑁)))
387368, 386mpbid 233 . . . . . . . . . . . . . . . . 17 (𝜑 → (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0})) Fn (1...𝑁))
388387adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) → (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0})) Fn (1...𝑁))
389 fzfid 13329 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) → (1...𝑁) ∈ Fin)
390 eqidd 2819 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) ∧ 𝑛 ∈ (1...𝑁)) → (𝑇𝑛) = (𝑇𝑛))
391 fvun1 6747 . . . . . . . . . . . . . . . . . . . 20 ((((𝑈 “ (1...(𝑉 − 1))) × {1}) Fn (𝑈 “ (1...(𝑉 − 1))) ∧ ((𝑈 “ (𝑉...𝑁)) × {0}) Fn (𝑈 “ (𝑉...𝑁)) ∧ (((𝑈 “ (1...(𝑉 − 1))) ∩ (𝑈 “ (𝑉...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ (1...(𝑉 − 1))))) → ((((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (1...(𝑉 − 1))) × {1})‘𝑛))
392356, 358, 391mp3an12 1442 . . . . . . . . . . . . . . . . . . 19 ((((𝑈 “ (1...(𝑉 − 1))) ∩ (𝑈 “ (𝑉...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) → ((((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (1...(𝑉 − 1))) × {1})‘𝑛))
393366, 392sylan 580 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) → ((((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (1...(𝑉 − 1))) × {1})‘𝑛))
39470fvconst2 6958 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (𝑈 “ (1...(𝑉 − 1))) → (((𝑈 “ (1...(𝑉 − 1))) × {1})‘𝑛) = 1)
395394adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) → (((𝑈 “ (1...(𝑉 − 1))) × {1})‘𝑛) = 1)
396393, 395eqtrd 2853 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) → ((((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0}))‘𝑛) = 1)
397396adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) ∧ 𝑛 ∈ (1...𝑁)) → ((((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0}))‘𝑛) = 1)
398354, 388, 389, 389, 112, 390, 397ofval 7407 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇f + (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0})))‘𝑛) = ((𝑇𝑛) + 1))
399109adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) → (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})) Fn (1...𝑁))
400 fzss2 12935 . . . . . . . . . . . . . . . . . . . . 21 (𝑉 ∈ (ℤ‘(𝑉 − 1)) → (1...(𝑉 − 1)) ⊆ (1...𝑉))
401315, 400syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (1...(𝑉 − 1)) ⊆ (1...𝑉))
402 imass2 5958 . . . . . . . . . . . . . . . . . . . 20 ((1...(𝑉 − 1)) ⊆ (1...𝑉) → (𝑈 “ (1...(𝑉 − 1))) ⊆ (𝑈 “ (1...𝑉)))
403401, 402syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑈 “ (1...(𝑉 − 1))) ⊆ (𝑈 “ (1...𝑉)))
404403sselda 3964 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) → 𝑛 ∈ (𝑈 “ (1...𝑉)))
405404, 119syldan 591 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) → ((((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))‘𝑛) = 1)
406405adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) ∧ 𝑛 ∈ (1...𝑁)) → ((((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))‘𝑛) = 1)
407354, 399, 389, 389, 112, 390, 406ofval 7407 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))‘𝑛) = ((𝑇𝑛) + 1))
408398, 407eqtr4d 2856 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇f + (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0})))‘𝑛) = ((𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))‘𝑛))
409353, 408mpdan 683 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) → ((𝑇f + (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0})))‘𝑛) = ((𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))‘𝑛))
410 imassrn 5933 . . . . . . . . . . . . . . . 16 (𝑈 “ ((𝑉 + 1)...𝑁)) ⊆ ran 𝑈
411410, 64sstrid 3975 . . . . . . . . . . . . . . 15 (𝜑 → (𝑈 “ ((𝑉 + 1)...𝑁)) ⊆ (1...𝑁))
412411sselda 3964 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) → 𝑛 ∈ (1...𝑁))
41368adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) → 𝑇 Fn (1...𝑁))
414387adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) → (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0})) Fn (1...𝑁))
415 fzfid 13329 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) → (1...𝑁) ∈ Fin)
416 eqidd 2819 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → (𝑇𝑛) = (𝑇𝑛))
417 fzss1 12934 . . . . . . . . . . . . . . . . . . . . 21 ((𝑉 + 1) ∈ (ℤ𝑉) → ((𝑉 + 1)...𝑁) ⊆ (𝑉...𝑁))
418146, 417syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑉 + 1)...𝑁) ⊆ (𝑉...𝑁))
419 imass2 5958 . . . . . . . . . . . . . . . . . . . 20 (((𝑉 + 1)...𝑁) ⊆ (𝑉...𝑁) → (𝑈 “ ((𝑉 + 1)...𝑁)) ⊆ (𝑈 “ (𝑉...𝑁)))
420418, 419syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑈 “ ((𝑉 + 1)...𝑁)) ⊆ (𝑈 “ (𝑉...𝑁)))
421420sselda 3964 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) → 𝑛 ∈ (𝑈 “ (𝑉...𝑁)))
422 fvun2 6748 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑈 “ (1...(𝑉 − 1))) × {1}) Fn (𝑈 “ (1...(𝑉 − 1))) ∧ ((𝑈 “ (𝑉...𝑁)) × {0}) Fn (𝑈 “ (𝑉...𝑁)) ∧ (((𝑈 “ (1...(𝑉 − 1))) ∩ (𝑈 “ (𝑉...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ (𝑉...𝑁)))) → ((((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (𝑉...𝑁)) × {0})‘𝑛))
423356, 358, 422mp3an12 1442 . . . . . . . . . . . . . . . . . . . 20 ((((𝑈 “ (1...(𝑉 − 1))) ∩ (𝑈 “ (𝑉...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ (𝑉...𝑁))) → ((((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (𝑉...𝑁)) × {0})‘𝑛))
424366, 423sylan 580 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ (𝑈 “ (𝑉...𝑁))) → ((((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (𝑉...𝑁)) × {0})‘𝑛))
42573fvconst2 6958 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (𝑈 “ (𝑉...𝑁)) → (((𝑈 “ (𝑉...𝑁)) × {0})‘𝑛) = 0)
426425adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ (𝑈 “ (𝑉...𝑁))) → (((𝑈 “ (𝑉...𝑁)) × {0})‘𝑛) = 0)
427424, 426eqtrd 2853 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (𝑈 “ (𝑉...𝑁))) → ((((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0}))‘𝑛) = 0)
428421, 427syldan 591 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) → ((((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0}))‘𝑛) = 0)
429428adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → ((((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0}))‘𝑛) = 0)
430413, 414, 415, 415, 112, 416, 429ofval 7407 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇f + (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0})))‘𝑛) = ((𝑇𝑛) + 0))
431109adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) → (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})) Fn (1...𝑁))
432184adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → ((((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))‘𝑛) = 0)
433413, 431, 415, 415, 112, 416, 432ofval 7407 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))‘𝑛) = ((𝑇𝑛) + 0))
434430, 433eqtr4d 2856 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇f + (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0})))‘𝑛) = ((𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))‘𝑛))
435412, 434mpdan 683 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) → ((𝑇f + (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0})))‘𝑛) = ((𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))‘𝑛))
436409, 435jaodan 951 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (𝑈 “ (1...(𝑉 − 1))) ∨ 𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁)))) → ((𝑇f + (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0})))‘𝑛) = ((𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))‘𝑛))
437436adantlr 711 . . . . . . . . . . 11 (((𝜑𝑉 < 𝑀) ∧ (𝑛 ∈ (𝑈 “ (1...(𝑉 − 1))) ∨ 𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁)))) → ((𝑇f + (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0})))‘𝑛) = ((𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))‘𝑛))
438201adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑉 < 𝑀) → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗(𝑇f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))))
439205a1i 11 . . . . . . . . . . . . . . 15 (((𝜑𝑉 < 𝑀) ∧ 𝑦 = (𝑉 − 1)) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) ∈ V)
440215adantlr 711 . . . . . . . . . . . . . . . 16 (((𝜑𝑉 < 𝑀) ∧ 𝑦 = (𝑉 − 1)) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = if((𝑉 − 1) < 𝑀, (𝑉 − 1), 𝑉))
441 lttr 10705 . . . . . . . . . . . . . . . . . . . . 21 (((𝑉 − 1) ∈ ℝ ∧ 𝑉 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (((𝑉 − 1) < 𝑉𝑉 < 𝑀) → (𝑉 − 1) < 𝑀))
442226, 28, 223, 441syl3anc 1363 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (((𝑉 − 1) < 𝑉𝑉 < 𝑀) → (𝑉 − 1) < 𝑀))
443326, 442mpand 691 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑉 < 𝑀 → (𝑉 − 1) < 𝑀))
444443imp 407 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑉 < 𝑀) → (𝑉 − 1) < 𝑀)
445444iftrued 4471 . . . . . . . . . . . . . . . . 17 ((𝜑𝑉 < 𝑀) → if((𝑉 − 1) < 𝑀, (𝑉 − 1), 𝑉) = (𝑉 − 1))
446445adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑉 < 𝑀) ∧ 𝑦 = (𝑉 − 1)) → if((𝑉 − 1) < 𝑀, (𝑉 − 1), 𝑉) = (𝑉 − 1))
447440, 446eqtrd 2853 . . . . . . . . . . . . . . 15 (((𝜑𝑉 < 𝑀) ∧ 𝑦 = (𝑉 − 1)) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = (𝑉 − 1))
448 simpll 763 . . . . . . . . . . . . . . . 16 (((𝜑𝑉 < 𝑀) ∧ 𝑦 = (𝑉 − 1)) → 𝜑)
449 oveq2 7153 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (𝑉 − 1) → (1...𝑗) = (1...(𝑉 − 1)))
450449imaeq2d 5922 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑉 − 1) → (𝑈 “ (1...𝑗)) = (𝑈 “ (1...(𝑉 − 1))))
451450xpeq1d 5577 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑉 − 1) → ((𝑈 “ (1...𝑗)) × {1}) = ((𝑈 “ (1...(𝑉 − 1))) × {1}))
452451adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 = (𝑉 − 1)) → ((𝑈 “ (1...𝑗)) × {1}) = ((𝑈 “ (1...(𝑉 − 1))) × {1}))
453 oveq1 7152 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = (𝑉 − 1) → (𝑗 + 1) = ((𝑉 − 1) + 1))
454453, 213sylan9eqr 2875 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 = (𝑉 − 1)) → (𝑗 + 1) = 𝑉)
455454oveq1d 7160 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 = (𝑉 − 1)) → ((𝑗 + 1)...𝑁) = (𝑉...𝑁))
456455imaeq2d 5922 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 = (𝑉 − 1)) → (𝑈 “ ((𝑗 + 1)...𝑁)) = (𝑈 “ (𝑉...𝑁)))
457456xpeq1d 5577 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 = (𝑉 − 1)) → ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) = ((𝑈 “ (𝑉...𝑁)) × {0}))
458452, 457uneq12d 4137 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 = (𝑉 − 1)) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) = (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0})))
459458oveq2d 7161 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 = (𝑉 − 1)) → (𝑇f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇f + (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0}))))
460448, 459sylan 580 . . . . . . . . . . . . . . 15 ((((𝜑𝑉 < 𝑀) ∧ 𝑦 = (𝑉 − 1)) ∧ 𝑗 = (𝑉 − 1)) → (𝑇f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇f + (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0}))))
461439, 447, 460csbied2 3917 . . . . . . . . . . . . . 14 (((𝜑𝑉 < 𝑀) ∧ 𝑦 = (𝑉 − 1)) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗(𝑇f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇f + (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0}))))
462248adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑉 < 𝑀) → (𝑉 − 1) ∈ (0...(𝑁 − 1)))
463 ovexd 7180 . . . . . . . . . . . . . 14 ((𝜑𝑉 < 𝑀) → (𝑇f + (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0}))) ∈ V)
464438, 461, 462, 463fvmptd 6767 . . . . . . . . . . . . 13 ((𝜑𝑉 < 𝑀) → (𝐹‘(𝑉 − 1)) = (𝑇f + (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0}))))
465464fveq1d 6665 . . . . . . . . . . . 12 ((𝜑𝑉 < 𝑀) → ((𝐹‘(𝑉 − 1))‘𝑛) = ((𝑇f + (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0})))‘𝑛))
466465adantr 481 . . . . . . . . . . 11 (((𝜑𝑉 < 𝑀) ∧ (𝑛 ∈ (𝑈 “ (1...(𝑉 − 1))) ∨ 𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁)))) → ((𝐹‘(𝑉 − 1))‘𝑛) = ((𝑇f + (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0})))‘𝑛))
467205a1i 11 . . . . . . . . . . . . . . . 16 ((𝑉 < 𝑀𝑦 = 𝑉) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) ∈ V)
468 iftrue 4469 . . . . . . . . . . . . . . . . . . . 20 (𝑉 < 𝑀 → if(𝑉 < 𝑀, 𝑉, (𝑉 + 1)) = 𝑉)
469258, 468sylan9eqr 2875 . . . . . . . . . . . . . . . . . . 19 ((𝑉 < 𝑀𝑦 = 𝑉) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = 𝑉)
470469eqeq2d 2829 . . . . . . . . . . . . . . . . . 18 ((𝑉 < 𝑀𝑦 = 𝑉) → (𝑗 = if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) ↔ 𝑗 = 𝑉))
471470biimpa 477 . . . . . . . . . . . . . . . . 17 (((𝑉 < 𝑀𝑦 = 𝑉) ∧ 𝑗 = if(𝑦 < 𝑀, 𝑦, (𝑦 + 1))) → 𝑗 = 𝑉)
472471, 243syl 17 . . . . . . . . . . . . . . . 16 (((𝑉 < 𝑀𝑦 = 𝑉) ∧ 𝑗 = if(𝑦 < 𝑀, 𝑦, (𝑦 + 1))) → (𝑇f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))))
473467, 472csbied 3916 . . . . . . . . . . . . . . 15 ((𝑉 < 𝑀𝑦 = 𝑉) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗(𝑇f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))))
474473adantll 710 . . . . . . . . . . . . . 14 (((𝜑𝑉 < 𝑀) ∧ 𝑦 = 𝑉) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗(𝑇f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))))
475278adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑉 < 𝑀) → 𝑉 ∈ (0...(𝑁 − 1)))
476 ovexd 7180 . . . . . . . . . . . . . 14 ((𝜑𝑉 < 𝑀) → (𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))) ∈ V)
477438, 474, 475, 476fvmptd 6767 . . . . . . . . . . . . 13 ((𝜑𝑉 < 𝑀) → (𝐹𝑉) = (𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))))
478477fveq1d 6665 . . . . . . . . . . . 12 ((𝜑𝑉 < 𝑀) → ((𝐹𝑉)‘𝑛) = ((𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))‘𝑛))
479478adantr 481 . . . . . . . . . . 11 (((𝜑𝑉 < 𝑀) ∧ (𝑛 ∈ (𝑈 “ (1...(𝑉 − 1))) ∨ 𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁)))) → ((𝐹𝑉)‘𝑛) = ((𝑇f + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))‘𝑛))
480437, 466, 4793eqtr4d 2863 . . . . . . . . . 10 (((𝜑𝑉 < 𝑀) ∧ (𝑛 ∈ (𝑈 “ (1...(𝑉 − 1))) ∨ 𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁)))) → ((𝐹‘(𝑉 − 1))‘𝑛) = ((𝐹𝑉)‘𝑛))
481480ex 413 . . . . . . . . 9 ((𝜑𝑉 < 𝑀) → ((𝑛 ∈ (𝑈 “ (1...(𝑉 − 1))) ∨ 𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) → ((𝐹‘(𝑉 − 1))‘𝑛) = ((𝐹𝑉)‘𝑛)))
482350, 481sylbid 241 . . . . . . . 8 ((𝜑𝑉 < 𝑀) → ((𝑛 ∈ (𝑈 “ (1...𝑁)) ∧ ¬ 𝑛 ∈ (𝑈 “ {𝑉})) → ((𝐹‘(𝑉 − 1))‘𝑛) = ((𝐹𝑉)‘𝑛)))
483482expdimp 453 . . . . . . 7 (((𝜑𝑉 < 𝑀) ∧ 𝑛 ∈ (𝑈 “ (1...𝑁))) → (¬ 𝑛 ∈ (𝑈 “ {𝑉}) → ((𝐹‘(𝑉 − 1))‘𝑛) = ((𝐹𝑉)‘𝑛)))
484483necon1ad 3030 . . . . . 6 (((𝜑𝑉 < 𝑀) ∧ 𝑛 ∈ (𝑈 “ (1...𝑁))) → (((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 ∈ (𝑈 “ {𝑉})))
485 elimasni 5949 . . . . . . . 8 (𝑛 ∈ (𝑈 “ {𝑉}) → 𝑉𝑈𝑛)
486 eqcom 2825 . . . . . . . . 9 (𝑛 = (𝑈𝑉) ↔ (𝑈𝑉) = 𝑛)
487 fnbrfvb 6711 . . . . . . . . . 10 ((𝑈 Fn (1...𝑁) ∧ 𝑉 ∈ (1...𝑁)) → ((𝑈𝑉) = 𝑛𝑉𝑈𝑛))
488292, 98, 487syl2anc 584 . . . . . . . . 9 (𝜑 → ((𝑈𝑉) = 𝑛𝑉𝑈𝑛))
489486, 488syl5bb 284 . . . . . . . 8 (𝜑 → (𝑛 = (𝑈𝑉) ↔ 𝑉𝑈𝑛))
490485, 489syl5ibr 247 . . . . . . 7 (𝜑 → (𝑛 ∈ (𝑈 “ {𝑉}) → 𝑛 = (𝑈𝑉)))
491490ad2antrr 722 . . . . . 6 (((𝜑𝑉 < 𝑀) ∧ 𝑛 ∈ (𝑈 “ (1...𝑁))) → (𝑛 ∈ (𝑈 “ {𝑉}) → 𝑛 = (𝑈𝑉)))
492484, 491syld 47 . . . . 5 (((𝜑𝑉 < 𝑀) ∧ 𝑛 ∈ (𝑈 “ (1...𝑁))) → (((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = (𝑈𝑉)))
493492ralrimiva 3179 . . . 4 ((𝜑𝑉 < 𝑀) → ∀𝑛 ∈ (𝑈 “ (1...𝑁))(((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = (𝑈𝑉)))
494 fvex 6676 . . . . 5 (𝑈𝑉) ∈ V
495 eqeq2 2830 . . . . . . 7 (𝑚 = (𝑈𝑉) → (𝑛 = 𝑚𝑛 = (𝑈𝑉)))
496495imbi2d 342 . . . . . 6 (𝑚 = (𝑈𝑉) → ((((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = 𝑚) ↔ (((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = (𝑈𝑉))))
497496ralbidv 3194 . . . . 5 (𝑚 = (𝑈𝑉) → (∀𝑛 ∈ (𝑈 “ (1...𝑁))(((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = 𝑚) ↔ ∀𝑛 ∈ (𝑈 “ (1...𝑁))(((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = (𝑈𝑉))))
498494, 497spcev 3604 . . . 4 (∀𝑛 ∈ (𝑈 “ (1...𝑁))(((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = (𝑈𝑉)) → ∃𝑚𝑛 ∈ (𝑈 “ (1...𝑁))(((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = 𝑚))
499493, 498syl 17 . . 3 ((𝜑𝑉 < 𝑀) → ∃𝑚𝑛 ∈ (𝑈 “ (1...𝑁))(((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = 𝑚))
500 eldifsni 4714 . . . . 5 (𝑀 ∈ ((0...𝑁) ∖ {𝑉}) → 𝑀𝑉)
501217, 500syl 17 . . . 4 (𝜑𝑀𝑉)
502223, 28lttri2d 10767 . . . 4 (𝜑 → (𝑀𝑉 ↔ (𝑀 < 𝑉𝑉 < 𝑀)))
503501, 502mpbid 233 . . 3 (𝜑 → (𝑀 < 𝑉𝑉 < 𝑀))
504305, 499, 503mpjaodan 952 . 2 (𝜑 → ∃𝑚𝑛 ∈ (𝑈 “ (1...𝑁))(((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = 𝑚))
505 nfv 1906 . . . 4 𝑚((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛)
506505rmo2 3867 . . 3 (∃*𝑛 ∈ (𝑈 “ (1...𝑁))((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) ↔ ∃𝑚𝑛 ∈ (𝑈 “ (1...𝑁))(((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = 𝑚))
507 rmoeq1 3406 . . . 4 ((𝑈 “ (1...𝑁)) = (1...𝑁) → (∃*𝑛 ∈ (𝑈 “ (1...𝑁))((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) ↔ ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛)))
508105, 507syl 17 . . 3 (𝜑 → (∃*𝑛 ∈ (𝑈 “ (1...𝑁))((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) ↔ ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛)))
509506, 508syl5bbr 286 . 2 (𝜑 → (∃𝑚𝑛 ∈ (𝑈 “ (1...𝑁))(((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = 𝑚) ↔ ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛)))
510504, 509mpbid 233 1 (𝜑 → ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wex 1771  wcel 2105  wne 3013  wral 3135  ∃*wrmo 3138  Vcvv 3492  csb 3880  cdif 3930  cun 3931  cin 3932  wss 3933  c0 4288  ifcif 4463  {csn 4557   class class class wbr 5057  cmpt 5137   × cxp 5546  ccnv 5547  ran crn 5549  cima 5551  Fun wfun 6342   Fn wfn 6343  wf 6344  ontowfo 6346  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  f cof 7396  Fincfn 8497  cc 10523  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   < clt 10663  cle 10664  cmin 10858  cn 11626  cz 11969  cuz 12231  ...cfz 12880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881
This theorem is referenced by:  poimirlem8  34781  poimirlem18  34791  poimirlem21  34794  poimirlem22  34795
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