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Theorem poimirlem2 33541
Description: Lemma for poimir 33572- 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 6181 . . . . . . . . . . . . . . . . 17 (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (𝑈:(1...𝑁)–onto→(1...𝑁) ∧ Fun 𝑈))
32simprbi 479 . . . . . . . . . . . . . . . 16 (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → Fun 𝑈)
41, 3syl 17 . . . . . . . . . . . . . . 15 (𝜑 → Fun 𝑈)
5 imadif 6011 . . . . . . . . . . . . . . 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 12432 . . . . . . . . . . . . . . . . . . . 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 11074 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑁 ∈ ℂ)
12 npcan1 10493 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
1311, 12syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
1413oveq2d 6706 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁))
159, 14eleqtrd 2732 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑉 + 1) ∈ (1...𝑁))
16 fzsplit 12405 . . . . . . . . . . . . . . . . . 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 3760 . . . . . . . . . . . . . . . 16 (𝜑 → ((1...𝑁) ∖ {(𝑉 + 1)}) = (((1...(𝑉 + 1)) ∪ (((𝑉 + 1) + 1)...𝑁)) ∖ {(𝑉 + 1)}))
19 difundir 3913 . . . . . . . . . . . . . . . . 17 (((1...(𝑉 + 1)) ∪ (((𝑉 + 1) + 1)...𝑁)) ∖ {(𝑉 + 1)}) = (((1...(𝑉 + 1)) ∖ {(𝑉 + 1)}) ∪ ((((𝑉 + 1) + 1)...𝑁) ∖ {(𝑉 + 1)}))
20 elfzuz 12376 . . . . . . . . . . . . . . . . . . . . . 22 (𝑉 ∈ (1...(𝑁 − 1)) → 𝑉 ∈ (ℤ‘1))
217, 20syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑉 ∈ (ℤ‘1))
22 fzsuc 12426 . . . . . . . . . . . . . . . . . . . . 21 (𝑉 ∈ (ℤ‘1) → (1...(𝑉 + 1)) = ((1...𝑉) ∪ {(𝑉 + 1)}))
2321, 22syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (1...(𝑉 + 1)) = ((1...𝑉) ∪ {(𝑉 + 1)}))
2423difeq1d 3760 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((1...(𝑉 + 1)) ∖ {(𝑉 + 1)}) = (((1...𝑉) ∪ {(𝑉 + 1)}) ∖ {(𝑉 + 1)}))
25 difun2 4081 . . . . . . . . . . . . . . . . . . . 20 (((1...𝑉) ∪ {(𝑉 + 1)}) ∖ {(𝑉 + 1)}) = ((1...𝑉) ∖ {(𝑉 + 1)})
26 elfzelz 12380 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑉 ∈ (1...(𝑁 − 1)) → 𝑉 ∈ ℤ)
277, 26syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑉 ∈ ℤ)
2827zred 11520 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑉 ∈ ℝ)
2928ltp1d 10992 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑉 < (𝑉 + 1))
3027peano2zd 11523 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝑉 + 1) ∈ ℤ)
3130zred 11520 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑉 + 1) ∈ ℝ)
3228, 31ltnled 10222 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑉 < (𝑉 + 1) ↔ ¬ (𝑉 + 1) ≤ 𝑉))
3329, 32mpbid 222 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ¬ (𝑉 + 1) ≤ 𝑉)
34 elfzle2 12383 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑉 + 1) ∈ (1...𝑉) → (𝑉 + 1) ≤ 𝑉)
3533, 34nsyl 135 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ¬ (𝑉 + 1) ∈ (1...𝑉))
36 difsn 4360 . . . . . . . . . . . . . . . . . . . . 21 (¬ (𝑉 + 1) ∈ (1...𝑉) → ((1...𝑉) ∖ {(𝑉 + 1)}) = (1...𝑉))
3735, 36syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((1...𝑉) ∖ {(𝑉 + 1)}) = (1...𝑉))
3825, 37syl5eq 2697 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (((1...𝑉) ∪ {(𝑉 + 1)}) ∖ {(𝑉 + 1)}) = (1...𝑉))
3924, 38eqtrd 2685 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1...(𝑉 + 1)) ∖ {(𝑉 + 1)}) = (1...𝑉))
4031ltp1d 10992 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑉 + 1) < ((𝑉 + 1) + 1))
41 peano2re 10247 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑉 + 1) ∈ ℝ → ((𝑉 + 1) + 1) ∈ ℝ)
4231, 41syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((𝑉 + 1) + 1) ∈ ℝ)
4331, 42ltnled 10222 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑉 + 1) < ((𝑉 + 1) + 1) ↔ ¬ ((𝑉 + 1) + 1) ≤ (𝑉 + 1)))
4440, 43mpbid 222 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ¬ ((𝑉 + 1) + 1) ≤ (𝑉 + 1))
45 elfzle1 12382 . . . . . . . . . . . . . . . . . . . 20 ((𝑉 + 1) ∈ (((𝑉 + 1) + 1)...𝑁) → ((𝑉 + 1) + 1) ≤ (𝑉 + 1))
4644, 45nsyl 135 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ¬ (𝑉 + 1) ∈ (((𝑉 + 1) + 1)...𝑁))
47 difsn 4360 . . . . . . . . . . . . . . . . . . 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 3801 . . . . . . . . . . . . . . . . 17 (𝜑 → (((1...(𝑉 + 1)) ∖ {(𝑉 + 1)}) ∪ ((((𝑉 + 1) + 1)...𝑁) ∖ {(𝑉 + 1)})) = ((1...𝑉) ∪ (((𝑉 + 1) + 1)...𝑁)))
5019, 49syl5eq 2697 . . . . . . . . . . . . . . . 16 (𝜑 → (((1...(𝑉 + 1)) ∪ (((𝑉 + 1) + 1)...𝑁)) ∖ {(𝑉 + 1)}) = ((1...𝑉) ∪ (((𝑉 + 1) + 1)...𝑁)))
5118, 50eqtrd 2685 . . . . . . . . . . . . . . 15 (𝜑 → ((1...𝑁) ∖ {(𝑉 + 1)}) = ((1...𝑉) ∪ (((𝑉 + 1) + 1)...𝑁)))
5251imaeq2d 5501 . . . . . . . . . . . . . 14 (𝜑 → (𝑈 “ ((1...𝑁) ∖ {(𝑉 + 1)})) = (𝑈 “ ((1...𝑉) ∪ (((𝑉 + 1) + 1)...𝑁))))
536, 52eqtr3d 2687 . . . . . . . . . . . . 13 (𝜑 → ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {(𝑉 + 1)})) = (𝑈 “ ((1...𝑉) ∪ (((𝑉 + 1) + 1)...𝑁))))
54 imaundi 5580 . . . . . . . . . . . . 13 (𝑈 “ ((1...𝑉) ∪ (((𝑉 + 1) + 1)...𝑁))) = ((𝑈 “ (1...𝑉)) ∪ (𝑈 “ (((𝑉 + 1) + 1)...𝑁)))
5553, 54syl6eq 2701 . . . . . . . . . . . 12 (𝜑 → ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {(𝑉 + 1)})) = ((𝑈 “ (1...𝑉)) ∪ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))))
5655eleq2d 2716 . . . . . . . . . . 11 (𝜑 → (𝑛 ∈ ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {(𝑉 + 1)})) ↔ 𝑛 ∈ ((𝑈 “ (1...𝑉)) ∪ (𝑈 “ (((𝑉 + 1) + 1)...𝑁)))))
57 eldif 3617 . . . . . . . . . . 11 (𝑛 ∈ ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {(𝑉 + 1)})) ↔ (𝑛 ∈ (𝑈 “ (1...𝑁)) ∧ ¬ 𝑛 ∈ (𝑈 “ {(𝑉 + 1)})))
58 elun 3786 . . . . . . . . . . 11 (𝑛 ∈ ((𝑈 “ (1...𝑉)) ∪ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) ↔ (𝑛 ∈ (𝑈 “ (1...𝑉)) ∨ 𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))))
5956, 57, 583bitr3g 302 . . . . . . . . . 10 (𝜑 → ((𝑛 ∈ (𝑈 “ (1...𝑁)) ∧ ¬ 𝑛 ∈ (𝑈 “ {(𝑉 + 1)})) ↔ (𝑛 ∈ (𝑈 “ (1...𝑉)) ∨ 𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁)))))
6059adantr 480 . . . . . . . . 9 ((𝜑𝑀 < 𝑉) → ((𝑛 ∈ (𝑈 “ (1...𝑁)) ∧ ¬ 𝑛 ∈ (𝑈 “ {(𝑉 + 1)})) ↔ (𝑛 ∈ (𝑈 “ (1...𝑉)) ∨ 𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁)))))
61 imassrn 5512 . . . . . . . . . . . . . . . 16 (𝑈 “ (1...𝑉)) ⊆ ran 𝑈
62 f1of 6175 . . . . . . . . . . . . . . . . . 18 (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈:(1...𝑁)⟶(1...𝑁))
631, 62syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝑈:(1...𝑁)⟶(1...𝑁))
64 frn 6091 . . . . . . . . . . . . . . . . 17 (𝑈:(1...𝑁)⟶(1...𝑁) → ran 𝑈 ⊆ (1...𝑁))
6563, 64syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ran 𝑈 ⊆ (1...𝑁))
6661, 65syl5ss 3647 . . . . . . . . . . . . . . 15 (𝜑 → (𝑈 “ (1...𝑉)) ⊆ (1...𝑁))
6766sselda 3636 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) → 𝑛 ∈ (1...𝑁))
68 poimirlem2.2 . . . . . . . . . . . . . . . . . 18 (𝜑𝑇:(1...𝑁)⟶ℤ)
69 ffn 6083 . . . . . . . . . . . . . . . . . 18 (𝑇:(1...𝑁)⟶ℤ → 𝑇 Fn (1...𝑁))
7068, 69syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝑇 Fn (1...𝑁))
7170adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) → 𝑇 Fn (1...𝑁))
72 1ex 10073 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ V
73 fnconstg 6131 . . . . . . . . . . . . . . . . . . . . 21 (1 ∈ V → ((𝑈 “ (1...𝑉)) × {1}) Fn (𝑈 “ (1...𝑉)))
7472, 73ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((𝑈 “ (1...𝑉)) × {1}) Fn (𝑈 “ (1...𝑉))
75 c0ex 10072 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ V
76 fnconstg 6131 . . . . . . . . . . . . . . . . . . . . 21 (0 ∈ V → ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}) Fn (𝑈 “ ((𝑉 + 1)...𝑁)))
7775, 76ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}) Fn (𝑈 “ ((𝑉 + 1)...𝑁))
7874, 77pm3.2i 470 . . . . . . . . . . . . . . . . . . 19 (((𝑈 “ (1...𝑉)) × {1}) Fn (𝑈 “ (1...𝑉)) ∧ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}) Fn (𝑈 “ ((𝑉 + 1)...𝑁)))
79 imain 6012 . . . . . . . . . . . . . . . . . . . . 21 (Fun 𝑈 → (𝑈 “ ((1...𝑉) ∩ ((𝑉 + 1)...𝑁))) = ((𝑈 “ (1...𝑉)) ∩ (𝑈 “ ((𝑉 + 1)...𝑁))))
804, 79syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑈 “ ((1...𝑉) ∩ ((𝑉 + 1)...𝑁))) = ((𝑈 “ (1...𝑉)) ∩ (𝑈 “ ((𝑉 + 1)...𝑁))))
81 fzdisj 12406 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑉 < (𝑉 + 1) → ((1...𝑉) ∩ ((𝑉 + 1)...𝑁)) = ∅)
8229, 81syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1...𝑉) ∩ ((𝑉 + 1)...𝑁)) = ∅)
8382imaeq2d 5501 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑈 “ ((1...𝑉) ∩ ((𝑉 + 1)...𝑁))) = (𝑈 “ ∅))
84 ima0 5516 . . . . . . . . . . . . . . . . . . . . 21 (𝑈 “ ∅) = ∅
8583, 84syl6eq 2701 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑈 “ ((1...𝑉) ∩ ((𝑉 + 1)...𝑁))) = ∅)
8680, 85eqtr3d 2687 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑈 “ (1...𝑉)) ∩ (𝑈 “ ((𝑉 + 1)...𝑁))) = ∅)
87 fnun 6035 . . . . . . . . . . . . . . . . . . 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))))
8878, 86, 87sylancr 696 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})) Fn ((𝑈 “ (1...𝑉)) ∪ (𝑈 “ ((𝑉 + 1)...𝑁))))
89 imaundi 5580 . . . . . . . . . . . . . . . . . . . 20 (𝑈 “ ((1...𝑉) ∪ ((𝑉 + 1)...𝑁))) = ((𝑈 “ (1...𝑉)) ∪ (𝑈 “ ((𝑉 + 1)...𝑁)))
9010nnzd 11519 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝑁 ∈ ℤ)
91 peano2zm 11458 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
9290, 91syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (𝑁 − 1) ∈ ℤ)
93 uzid 11740 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
9492, 93syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
95 peano2uz 11779 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
9694, 95syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
9713, 96eqeltrrd 2731 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
98 fzss2 12419 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ (ℤ‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
9997, 98syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
10099, 7sseldd 3637 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑉 ∈ (1...𝑁))
101 fzsplit 12405 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑉 ∈ (1...𝑁) → (1...𝑁) = ((1...𝑉) ∪ ((𝑉 + 1)...𝑁)))
102100, 101syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (1...𝑁) = ((1...𝑉) ∪ ((𝑉 + 1)...𝑁)))
103102imaeq2d 5501 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑈 “ (1...𝑁)) = (𝑈 “ ((1...𝑉) ∪ ((𝑉 + 1)...𝑁))))
104 f1ofo 6182 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈:(1...𝑁)–onto→(1...𝑁))
1051, 104syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑈:(1...𝑁)–onto→(1...𝑁))
106 foima 6158 . . . . . . . . . . . . . . . . . . . . . 22 (𝑈:(1...𝑁)–onto→(1...𝑁) → (𝑈 “ (1...𝑁)) = (1...𝑁))
107105, 106syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑈 “ (1...𝑁)) = (1...𝑁))
108103, 107eqtr3d 2687 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑈 “ ((1...𝑉) ∪ ((𝑉 + 1)...𝑁))) = (1...𝑁))
10989, 108syl5eqr 2699 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑈 “ (1...𝑉)) ∪ (𝑈 “ ((𝑉 + 1)...𝑁))) = (1...𝑁))
110109fneq2d 6020 . . . . . . . . . . . . . . . . . 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)))
11188, 110mpbid 222 . . . . . . . . . . . . . . . . 17 (𝜑 → (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})) Fn (1...𝑁))
112111adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) → (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})) Fn (1...𝑁))
113 fzfid 12812 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) → (1...𝑁) ∈ Fin)
114 inidm 3855 . . . . . . . . . . . . . . . 16 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
115 eqidd 2652 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) ∧ 𝑛 ∈ (1...𝑁)) → (𝑇𝑛) = (𝑇𝑛))
116 fvun1 6308 . . . . . . . . . . . . . . . . . . . 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))
11774, 77, 116mp3an12 1454 . . . . . . . . . . . . . . . . . . 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))
11886, 117sylan 487 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) → ((((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (1...𝑉)) × {1})‘𝑛))
11972fvconst2 6510 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (𝑈 “ (1...𝑉)) → (((𝑈 “ (1...𝑉)) × {1})‘𝑛) = 1)
120119adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) → (((𝑈 “ (1...𝑉)) × {1})‘𝑛) = 1)
121118, 120eqtrd 2685 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) → ((((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))‘𝑛) = 1)
122121adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) ∧ 𝑛 ∈ (1...𝑁)) → ((((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))‘𝑛) = 1)
12371, 112, 113, 113, 114, 115, 122ofval 6948 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇𝑓 + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))‘𝑛) = ((𝑇𝑛) + 1))
124 fnconstg 6131 . . . . . . . . . . . . . . . . . . . . 21 (1 ∈ V → ((𝑈 “ (1...(𝑉 + 1))) × {1}) Fn (𝑈 “ (1...(𝑉 + 1))))
12572, 124ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((𝑈 “ (1...(𝑉 + 1))) × {1}) Fn (𝑈 “ (1...(𝑉 + 1)))
126 fnconstg 6131 . . . . . . . . . . . . . . . . . . . . 21 (0 ∈ V → ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑉 + 1) + 1)...𝑁)))
12775, 126ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑉 + 1) + 1)...𝑁))
128125, 127pm3.2i 470 . . . . . . . . . . . . . . . . . . 19 (((𝑈 “ (1...(𝑉 + 1))) × {1}) Fn (𝑈 “ (1...(𝑉 + 1))) ∧ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑉 + 1) + 1)...𝑁)))
129 imain 6012 . . . . . . . . . . . . . . . . . . . . 21 (Fun 𝑈 → (𝑈 “ ((1...(𝑉 + 1)) ∩ (((𝑉 + 1) + 1)...𝑁))) = ((𝑈 “ (1...(𝑉 + 1))) ∩ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))))
1304, 129syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑈 “ ((1...(𝑉 + 1)) ∩ (((𝑉 + 1) + 1)...𝑁))) = ((𝑈 “ (1...(𝑉 + 1))) ∩ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))))
131 fzdisj 12406 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑉 + 1) < ((𝑉 + 1) + 1) → ((1...(𝑉 + 1)) ∩ (((𝑉 + 1) + 1)...𝑁)) = ∅)
13240, 131syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1...(𝑉 + 1)) ∩ (((𝑉 + 1) + 1)...𝑁)) = ∅)
133132imaeq2d 5501 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑈 “ ((1...(𝑉 + 1)) ∩ (((𝑉 + 1) + 1)...𝑁))) = (𝑈 “ ∅))
134133, 84syl6eq 2701 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑈 “ ((1...(𝑉 + 1)) ∩ (((𝑉 + 1) + 1)...𝑁))) = ∅)
135130, 134eqtr3d 2687 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑈 “ (1...(𝑉 + 1))) ∩ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) = ∅)
136 fnun 6035 . . . . . . . . . . . . . . . . . . 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))))
137128, 135, 136sylancr 696 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑉 + 1))) ∪ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))))
138 imaundi 5580 . . . . . . . . . . . . . . . . . . . 20 (𝑈 “ ((1...(𝑉 + 1)) ∪ (((𝑉 + 1) + 1)...𝑁))) = ((𝑈 “ (1...(𝑉 + 1))) ∪ (𝑈 “ (((𝑉 + 1) + 1)...𝑁)))
13917imaeq2d 5501 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑈 “ (1...𝑁)) = (𝑈 “ ((1...(𝑉 + 1)) ∪ (((𝑉 + 1) + 1)...𝑁))))
140139, 107eqtr3d 2687 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑈 “ ((1...(𝑉 + 1)) ∪ (((𝑉 + 1) + 1)...𝑁))) = (1...𝑁))
141138, 140syl5eqr 2699 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑈 “ (1...(𝑉 + 1))) ∪ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) = (1...𝑁))
142141fneq2d 6020 . . . . . . . . . . . . . . . . . 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)))
143137, 142mpbid 222 . . . . . . . . . . . . . . . . 17 (𝜑 → (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁))
144143adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) → (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁))
145 uzid 11740 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑉 ∈ ℤ → 𝑉 ∈ (ℤ𝑉))
14627, 145syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑉 ∈ (ℤ𝑉))
147 peano2uz 11779 . . . . . . . . . . . . . . . . . . . . . 22 (𝑉 ∈ (ℤ𝑉) → (𝑉 + 1) ∈ (ℤ𝑉))
148146, 147syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑉 + 1) ∈ (ℤ𝑉))
149 fzss2 12419 . . . . . . . . . . . . . . . . . . . . 21 ((𝑉 + 1) ∈ (ℤ𝑉) → (1...𝑉) ⊆ (1...(𝑉 + 1)))
150148, 149syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (1...𝑉) ⊆ (1...(𝑉 + 1)))
151 imass2 5536 . . . . . . . . . . . . . . . . . . . 20 ((1...𝑉) ⊆ (1...(𝑉 + 1)) → (𝑈 “ (1...𝑉)) ⊆ (𝑈 “ (1...(𝑉 + 1))))
152150, 151syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑈 “ (1...𝑉)) ⊆ (𝑈 “ (1...(𝑉 + 1))))
153152sselda 3636 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) → 𝑛 ∈ (𝑈 “ (1...(𝑉 + 1))))
154 fvun1 6308 . . . . . . . . . . . . . . . . . . . . 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))
155125, 127, 154mp3an12 1454 . . . . . . . . . . . . . . . . . . . 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))
156135, 155sylan 487 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 + 1)))) → ((((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (1...(𝑉 + 1))) × {1})‘𝑛))
15772fvconst2 6510 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (𝑈 “ (1...(𝑉 + 1))) → (((𝑈 “ (1...(𝑉 + 1))) × {1})‘𝑛) = 1)
158157adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 + 1)))) → (((𝑈 “ (1...(𝑉 + 1))) × {1})‘𝑛) = 1)
159156, 158eqtrd 2685 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 + 1)))) → ((((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}))‘𝑛) = 1)
160153, 159syldan 486 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) → ((((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}))‘𝑛) = 1)
161160adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) ∧ 𝑛 ∈ (1...𝑁)) → ((((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}))‘𝑛) = 1)
16271, 144, 113, 113, 114, 115, 161ofval 6948 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ (𝑈 “ (1...𝑉))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇𝑓 + (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})))‘𝑛) = ((𝑇𝑛) + 1))
163123, 162eqtr4d 2688 . . . . . . . . . . . . . 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))
16467, 163mpdan 703 . . . . . . . . . . . . 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))
165 imassrn 5512 . . . . . . . . . . . . . . . 16 (𝑈 “ (((𝑉 + 1) + 1)...𝑁)) ⊆ ran 𝑈
166165, 65syl5ss 3647 . . . . . . . . . . . . . . 15 (𝜑 → (𝑈 “ (((𝑉 + 1) + 1)...𝑁)) ⊆ (1...𝑁))
167166sselda 3636 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) → 𝑛 ∈ (1...𝑁))
16870adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) → 𝑇 Fn (1...𝑁))
169111adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) → (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})) Fn (1...𝑁))
170 fzfid 12812 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) → (1...𝑁) ∈ Fin)
171 eqidd 2652 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → (𝑇𝑛) = (𝑇𝑛))
172 uzid 11740 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑉 + 1) ∈ ℤ → (𝑉 + 1) ∈ (ℤ‘(𝑉 + 1)))
17330, 172syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑉 + 1) ∈ (ℤ‘(𝑉 + 1)))
174 peano2uz 11779 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑉 + 1) ∈ (ℤ‘(𝑉 + 1)) → ((𝑉 + 1) + 1) ∈ (ℤ‘(𝑉 + 1)))
175173, 174syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑉 + 1) + 1) ∈ (ℤ‘(𝑉 + 1)))
176 fzss1 12418 . . . . . . . . . . . . . . . . . . . . 21 (((𝑉 + 1) + 1) ∈ (ℤ‘(𝑉 + 1)) → (((𝑉 + 1) + 1)...𝑁) ⊆ ((𝑉 + 1)...𝑁))
177175, 176syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (((𝑉 + 1) + 1)...𝑁) ⊆ ((𝑉 + 1)...𝑁))
178 imass2 5536 . . . . . . . . . . . . . . . . . . . 20 ((((𝑉 + 1) + 1)...𝑁) ⊆ ((𝑉 + 1)...𝑁) → (𝑈 “ (((𝑉 + 1) + 1)...𝑁)) ⊆ (𝑈 “ ((𝑉 + 1)...𝑁)))
179177, 178syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑈 “ (((𝑉 + 1) + 1)...𝑁)) ⊆ (𝑈 “ ((𝑉 + 1)...𝑁)))
180179sselda 3636 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) → 𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁)))
181 fvun2 6309 . . . . . . . . . . . . . . . . . . . . 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))
18274, 77, 181mp3an12 1454 . . . . . . . . . . . . . . . . . . . 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))
18386, 182sylan 487 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) → ((((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})‘𝑛))
18475fvconst2 6510 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁)) → (((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})‘𝑛) = 0)
185184adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) → (((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})‘𝑛) = 0)
186183, 185eqtrd 2685 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) → ((((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))‘𝑛) = 0)
187180, 186syldan 486 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) → ((((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))‘𝑛) = 0)
188187adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → ((((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))‘𝑛) = 0)
189168, 169, 170, 170, 114, 171, 188ofval 6948 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇𝑓 + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))‘𝑛) = ((𝑇𝑛) + 0))
190143adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) → (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁))
191 fvun2 6309 . . . . . . . . . . . . . . . . . . . 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))
192125, 127, 191mp3an12 1454 . . . . . . . . . . . . . . . . . . 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))
193135, 192sylan 487 . . . . . . . . . . . . . . . . . 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))
19475fvconst2 6510 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁)) → (((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})‘𝑛) = 0)
195194adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) → (((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})‘𝑛) = 0)
196193, 195eqtrd 2685 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) → ((((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}))‘𝑛) = 0)
197196adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → ((((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}))‘𝑛) = 0)
198168, 190, 170, 170, 114, 171, 197ofval 6948 . . . . . . . . . . . . . . 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))
199189, 198eqtr4d 2688 . . . . . . . . . . . . . 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))
200167, 199mpdan 703 . . . . . . . . . . . . 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))
201164, 200jaodan 843 . . . . . . . . . . . 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))
202201adantlr 751 . . . . . . . . . . 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))
203 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})))))
204203adantr 480 . . . . . . . . . . . . . 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})))))
205 vex 3234 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
206 ovex 6718 . . . . . . . . . . . . . . . . 17 (𝑦 + 1) ∈ V
207205, 206ifex 4189 . . . . . . . . . . . . . . . 16 if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) ∈ V
208207a1i 11 . . . . . . . . . . . . . . 15 (((𝜑𝑀 < 𝑉) ∧ 𝑦 = (𝑉 − 1)) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) ∈ V)
209 breq1 4688 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = (𝑉 − 1) → (𝑦 < 𝑀 ↔ (𝑉 − 1) < 𝑀))
210209adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 = (𝑉 − 1)) → (𝑦 < 𝑀 ↔ (𝑉 − 1) < 𝑀))
211 simpr 476 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 = (𝑉 − 1)) → 𝑦 = (𝑉 − 1))
212 oveq1 6697 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = (𝑉 − 1) → (𝑦 + 1) = ((𝑉 − 1) + 1))
21327zcnd 11521 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑉 ∈ ℂ)
214 npcan1 10493 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑉 ∈ ℂ → ((𝑉 − 1) + 1) = 𝑉)
215213, 214syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((𝑉 − 1) + 1) = 𝑉)
216212, 215sylan9eqr 2707 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 = (𝑉 − 1)) → (𝑦 + 1) = 𝑉)
217210, 211, 216ifbieq12d 4146 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 = (𝑉 − 1)) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = if((𝑉 − 1) < 𝑀, (𝑉 − 1), 𝑉))
218217adantlr 751 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑀 < 𝑉) ∧ 𝑦 = (𝑉 − 1)) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = if((𝑉 − 1) < 𝑀, (𝑉 − 1), 𝑉))
219 poimirlem2.5 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝑀 ∈ ((0...𝑁) ∖ {𝑉}))
220219eldifad 3619 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑀 ∈ (0...𝑁))
221 elfzelz 12380 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℤ)
222220, 221syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑀 ∈ ℤ)
223 zltlem1 11468 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑀 ∈ ℤ ∧ 𝑉 ∈ ℤ) → (𝑀 < 𝑉𝑀 ≤ (𝑉 − 1)))
224222, 27, 223syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑀 < 𝑉𝑀 ≤ (𝑉 − 1)))
225222zred 11520 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑀 ∈ ℝ)
226 peano2zm 11458 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑉 ∈ ℤ → (𝑉 − 1) ∈ ℤ)
22727, 226syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝑉 − 1) ∈ ℤ)
228227zred 11520 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑉 − 1) ∈ ℝ)
229225, 228lenltd 10221 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑀 ≤ (𝑉 − 1) ↔ ¬ (𝑉 − 1) < 𝑀))
230224, 229bitrd 268 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑀 < 𝑉 ↔ ¬ (𝑉 − 1) < 𝑀))
231230biimpa 500 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑀 < 𝑉) → ¬ (𝑉 − 1) < 𝑀)
232231iffalsed 4130 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑀 < 𝑉) → if((𝑉 − 1) < 𝑀, (𝑉 − 1), 𝑉) = 𝑉)
233232adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑀 < 𝑉) ∧ 𝑦 = (𝑉 − 1)) → if((𝑉 − 1) < 𝑀, (𝑉 − 1), 𝑉) = 𝑉)
234218, 233eqtrd 2685 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑀 < 𝑉) ∧ 𝑦 = (𝑉 − 1)) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = 𝑉)
235234eqeq2d 2661 . . . . . . . . . . . . . . . . 17 (((𝜑𝑀 < 𝑉) ∧ 𝑦 = (𝑉 − 1)) → (𝑗 = if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) ↔ 𝑗 = 𝑉))
236235biimpa 500 . . . . . . . . . . . . . . . 16 ((((𝜑𝑀 < 𝑉) ∧ 𝑦 = (𝑉 − 1)) ∧ 𝑗 = if(𝑦 < 𝑀, 𝑦, (𝑦 + 1))) → 𝑗 = 𝑉)
237 oveq2 6698 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑉 → (1...𝑗) = (1...𝑉))
238237imaeq2d 5501 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑉 → (𝑈 “ (1...𝑗)) = (𝑈 “ (1...𝑉)))
239238xpeq1d 5172 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑉 → ((𝑈 “ (1...𝑗)) × {1}) = ((𝑈 “ (1...𝑉)) × {1}))
240 oveq1 6697 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑉 → (𝑗 + 1) = (𝑉 + 1))
241240oveq1d 6705 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑉 → ((𝑗 + 1)...𝑁) = ((𝑉 + 1)...𝑁))
242241imaeq2d 5501 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑉 → (𝑈 “ ((𝑗 + 1)...𝑁)) = (𝑈 “ ((𝑉 + 1)...𝑁)))
243242xpeq1d 5172 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑉 → ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) = ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))
244239, 243uneq12d 3801 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑉 → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) = (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))
245244oveq2d 6706 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑉 → (𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇𝑓 + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))))
246236, 245syl 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}))))
247208, 246csbied 3593 . . . . . . . . . . . . . 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}))))
248 elfzm1b 12456 . . . . . . . . . . . . . . . . 17 ((𝑉 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑉 ∈ (1...𝑁) ↔ (𝑉 − 1) ∈ (0...(𝑁 − 1))))
24927, 90, 248syl2anc 694 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑉 ∈ (1...𝑁) ↔ (𝑉 − 1) ∈ (0...(𝑁 − 1))))
250100, 249mpbid 222 . . . . . . . . . . . . . . 15 (𝜑 → (𝑉 − 1) ∈ (0...(𝑁 − 1)))
251250adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑀 < 𝑉) → (𝑉 − 1) ∈ (0...(𝑁 − 1)))
252 ovexd 6720 . . . . . . . . . . . . . 14 ((𝜑𝑀 < 𝑉) → (𝑇𝑓 + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))) ∈ V)
253204, 247, 251, 252fvmptd 6327 . . . . . . . . . . . . 13 ((𝜑𝑀 < 𝑉) → (𝐹‘(𝑉 − 1)) = (𝑇𝑓 + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))))
254253fveq1d 6231 . . . . . . . . . . . 12 ((𝜑𝑀 < 𝑉) → ((𝐹‘(𝑉 − 1))‘𝑛) = ((𝑇𝑓 + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))‘𝑛))
255254adantr 480 . . . . . . . . . . 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))
256207a1i 11 . . . . . . . . . . . . . . 15 (((𝜑𝑀 < 𝑉) ∧ 𝑦 = 𝑉) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) ∈ V)
257 breq1 4688 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑉 → (𝑦 < 𝑀𝑉 < 𝑀))
258 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑉𝑦 = 𝑉)
259 oveq1 6697 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑉 → (𝑦 + 1) = (𝑉 + 1))
260257, 258, 259ifbieq12d 4146 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑉 → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = if(𝑉 < 𝑀, 𝑉, (𝑉 + 1)))
261 ltnsym 10173 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑀 ∈ ℝ ∧ 𝑉 ∈ ℝ) → (𝑀 < 𝑉 → ¬ 𝑉 < 𝑀))
262225, 28, 261syl2anc 694 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑀 < 𝑉 → ¬ 𝑉 < 𝑀))
263262imp 444 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑀 < 𝑉) → ¬ 𝑉 < 𝑀)
264263iffalsed 4130 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑀 < 𝑉) → if(𝑉 < 𝑀, 𝑉, (𝑉 + 1)) = (𝑉 + 1))
265260, 264sylan9eqr 2707 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑀 < 𝑉) ∧ 𝑦 = 𝑉) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = (𝑉 + 1))
266265eqeq2d 2661 . . . . . . . . . . . . . . . . 17 (((𝜑𝑀 < 𝑉) ∧ 𝑦 = 𝑉) → (𝑗 = if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) ↔ 𝑗 = (𝑉 + 1)))
267266biimpa 500 . . . . . . . . . . . . . . . 16 ((((𝜑𝑀 < 𝑉) ∧ 𝑦 = 𝑉) ∧ 𝑗 = if(𝑦 < 𝑀, 𝑦, (𝑦 + 1))) → 𝑗 = (𝑉 + 1))
268 oveq2 6698 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑉 + 1) → (1...𝑗) = (1...(𝑉 + 1)))
269268imaeq2d 5501 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑉 + 1) → (𝑈 “ (1...𝑗)) = (𝑈 “ (1...(𝑉 + 1))))
270269xpeq1d 5172 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑉 + 1) → ((𝑈 “ (1...𝑗)) × {1}) = ((𝑈 “ (1...(𝑉 + 1))) × {1}))
271 oveq1 6697 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (𝑉 + 1) → (𝑗 + 1) = ((𝑉 + 1) + 1))
272271oveq1d 6705 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑉 + 1) → ((𝑗 + 1)...𝑁) = (((𝑉 + 1) + 1)...𝑁))
273272imaeq2d 5501 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑉 + 1) → (𝑈 “ ((𝑗 + 1)...𝑁)) = (𝑈 “ (((𝑉 + 1) + 1)...𝑁)))
274273xpeq1d 5172 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑉 + 1) → ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) = ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}))
275270, 274uneq12d 3801 . . . . . . . . . . . . . . . . 17 (𝑗 = (𝑉 + 1) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) = (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})))
276275oveq2d 6706 . . . . . . . . . . . . . . . 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}))))
277267, 276syl 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}))))
278256, 277csbied 3593 . . . . . . . . . . . . . 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}))))
279 fz1ssfz0 12474 . . . . . . . . . . . . . . . 16 (1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1))
280279, 7sseldi 3634 . . . . . . . . . . . . . . 15 (𝜑𝑉 ∈ (0...(𝑁 − 1)))
281280adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑀 < 𝑉) → 𝑉 ∈ (0...(𝑁 − 1)))
282 ovexd 6720 . . . . . . . . . . . . . 14 ((𝜑𝑀 < 𝑉) → (𝑇𝑓 + (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}))) ∈ V)
283204, 278, 281, 282fvmptd 6327 . . . . . . . . . . . . 13 ((𝜑𝑀 < 𝑉) → (𝐹𝑉) = (𝑇𝑓 + (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0}))))
284283fveq1d 6231 . . . . . . . . . . . 12 ((𝜑𝑀 < 𝑉) → ((𝐹𝑉)‘𝑛) = ((𝑇𝑓 + (((𝑈 “ (1...(𝑉 + 1))) × {1}) ∪ ((𝑈 “ (((𝑉 + 1) + 1)...𝑁)) × {0})))‘𝑛))
285284adantr 480 . . . . . . . . . . 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))
286202, 255, 2853eqtr4d 2695 . . . . . . . . . 10 (((𝜑𝑀 < 𝑉) ∧ (𝑛 ∈ (𝑈 “ (1...𝑉)) ∨ 𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁)))) → ((𝐹‘(𝑉 − 1))‘𝑛) = ((𝐹𝑉)‘𝑛))
287286ex 449 . . . . . . . . 9 ((𝜑𝑀 < 𝑉) → ((𝑛 ∈ (𝑈 “ (1...𝑉)) ∨ 𝑛 ∈ (𝑈 “ (((𝑉 + 1) + 1)...𝑁))) → ((𝐹‘(𝑉 − 1))‘𝑛) = ((𝐹𝑉)‘𝑛)))
28860, 287sylbid 230 . . . . . . . 8 ((𝜑𝑀 < 𝑉) → ((𝑛 ∈ (𝑈 “ (1...𝑁)) ∧ ¬ 𝑛 ∈ (𝑈 “ {(𝑉 + 1)})) → ((𝐹‘(𝑉 − 1))‘𝑛) = ((𝐹𝑉)‘𝑛)))
289288expdimp 452 . . . . . . 7 (((𝜑𝑀 < 𝑉) ∧ 𝑛 ∈ (𝑈 “ (1...𝑁))) → (¬ 𝑛 ∈ (𝑈 “ {(𝑉 + 1)}) → ((𝐹‘(𝑉 − 1))‘𝑛) = ((𝐹𝑉)‘𝑛)))
290289necon1ad 2840 . . . . . 6 (((𝜑𝑀 < 𝑉) ∧ 𝑛 ∈ (𝑈 “ (1...𝑁))) → (((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 ∈ (𝑈 “ {(𝑉 + 1)})))
291 elimasni 5527 . . . . . . . 8 (𝑛 ∈ (𝑈 “ {(𝑉 + 1)}) → (𝑉 + 1)𝑈𝑛)
292 eqcom 2658 . . . . . . . . 9 (𝑛 = (𝑈‘(𝑉 + 1)) ↔ (𝑈‘(𝑉 + 1)) = 𝑛)
293 f1ofn 6176 . . . . . . . . . . 11 (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈 Fn (1...𝑁))
2941, 293syl 17 . . . . . . . . . 10 (𝜑𝑈 Fn (1...𝑁))
295 fnbrfvb 6274 . . . . . . . . . 10 ((𝑈 Fn (1...𝑁) ∧ (𝑉 + 1) ∈ (1...𝑁)) → ((𝑈‘(𝑉 + 1)) = 𝑛 ↔ (𝑉 + 1)𝑈𝑛))
296294, 15, 295syl2anc 694 . . . . . . . . 9 (𝜑 → ((𝑈‘(𝑉 + 1)) = 𝑛 ↔ (𝑉 + 1)𝑈𝑛))
297292, 296syl5bb 272 . . . . . . . 8 (𝜑 → (𝑛 = (𝑈‘(𝑉 + 1)) ↔ (𝑉 + 1)𝑈𝑛))
298291, 297syl5ibr 236 . . . . . . 7 (𝜑 → (𝑛 ∈ (𝑈 “ {(𝑉 + 1)}) → 𝑛 = (𝑈‘(𝑉 + 1))))
299298ad2antrr 762 . . . . . 6 (((𝜑𝑀 < 𝑉) ∧ 𝑛 ∈ (𝑈 “ (1...𝑁))) → (𝑛 ∈ (𝑈 “ {(𝑉 + 1)}) → 𝑛 = (𝑈‘(𝑉 + 1))))
300290, 299syld 47 . . . . 5 (((𝜑𝑀 < 𝑉) ∧ 𝑛 ∈ (𝑈 “ (1...𝑁))) → (((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = (𝑈‘(𝑉 + 1))))
301300ralrimiva 2995 . . . 4 ((𝜑𝑀 < 𝑉) → ∀𝑛 ∈ (𝑈 “ (1...𝑁))(((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = (𝑈‘(𝑉 + 1))))
302 fvex 6239 . . . . 5 (𝑈‘(𝑉 + 1)) ∈ V
303 eqeq2 2662 . . . . . . 7 (𝑚 = (𝑈‘(𝑉 + 1)) → (𝑛 = 𝑚𝑛 = (𝑈‘(𝑉 + 1))))
304303imbi2d 329 . . . . . 6 (𝑚 = (𝑈‘(𝑉 + 1)) → ((((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = 𝑚) ↔ (((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = (𝑈‘(𝑉 + 1)))))
305304ralbidv 3015 . . . . 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)))))
306302, 305spcev 3331 . . . 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))
307301, 306syl 17 . . 3 ((𝜑𝑀 < 𝑉) → ∃𝑚𝑛 ∈ (𝑈 “ (1...𝑁))(((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = 𝑚))
308 imadif 6011 . . . . . . . . . . . . . . 15 (Fun 𝑈 → (𝑈 “ ((1...𝑁) ∖ {𝑉})) = ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {𝑉})))
3094, 308syl 17 . . . . . . . . . . . . . 14 (𝜑 → (𝑈 “ ((1...𝑁) ∖ {𝑉})) = ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {𝑉})))
310102difeq1d 3760 . . . . . . . . . . . . . . . 16 (𝜑 → ((1...𝑁) ∖ {𝑉}) = (((1...𝑉) ∪ ((𝑉 + 1)...𝑁)) ∖ {𝑉}))
311 difundir 3913 . . . . . . . . . . . . . . . . 17 (((1...𝑉) ∪ ((𝑉 + 1)...𝑁)) ∖ {𝑉}) = (((1...𝑉) ∖ {𝑉}) ∪ (((𝑉 + 1)...𝑁) ∖ {𝑉}))
312215, 21eqeltrd 2730 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((𝑉 − 1) + 1) ∈ (ℤ‘1))
313 uzid 11740 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑉 − 1) ∈ ℤ → (𝑉 − 1) ∈ (ℤ‘(𝑉 − 1)))
314227, 313syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑉 − 1) ∈ (ℤ‘(𝑉 − 1)))
315 peano2uz 11779 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑉 − 1) ∈ (ℤ‘(𝑉 − 1)) → ((𝑉 − 1) + 1) ∈ (ℤ‘(𝑉 − 1)))
316314, 315syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑉 − 1) + 1) ∈ (ℤ‘(𝑉 − 1)))
317215, 316eqeltrrd 2731 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑉 ∈ (ℤ‘(𝑉 − 1)))
318 fzsplit2 12404 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑉 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑉 ∈ (ℤ‘(𝑉 − 1))) → (1...𝑉) = ((1...(𝑉 − 1)) ∪ (((𝑉 − 1) + 1)...𝑉)))
319312, 317, 318syl2anc 694 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (1...𝑉) = ((1...(𝑉 − 1)) ∪ (((𝑉 − 1) + 1)...𝑉)))
320215oveq1d 6705 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (((𝑉 − 1) + 1)...𝑉) = (𝑉...𝑉))
321 fzsn 12421 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑉 ∈ ℤ → (𝑉...𝑉) = {𝑉})
32227, 321syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑉...𝑉) = {𝑉})
323320, 322eqtrd 2685 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (((𝑉 − 1) + 1)...𝑉) = {𝑉})
324323uneq2d 3800 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((1...(𝑉 − 1)) ∪ (((𝑉 − 1) + 1)...𝑉)) = ((1...(𝑉 − 1)) ∪ {𝑉}))
325319, 324eqtrd 2685 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (1...𝑉) = ((1...(𝑉 − 1)) ∪ {𝑉}))
326325difeq1d 3760 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((1...𝑉) ∖ {𝑉}) = (((1...(𝑉 − 1)) ∪ {𝑉}) ∖ {𝑉}))
327 difun2 4081 . . . . . . . . . . . . . . . . . . . 20 (((1...(𝑉 − 1)) ∪ {𝑉}) ∖ {𝑉}) = ((1...(𝑉 − 1)) ∖ {𝑉})
32828ltm1d 10994 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑉 − 1) < 𝑉)
329228, 28ltnled 10222 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑉 − 1) < 𝑉 ↔ ¬ 𝑉 ≤ (𝑉 − 1)))
330328, 329mpbid 222 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ¬ 𝑉 ≤ (𝑉 − 1))
331 elfzle2 12383 . . . . . . . . . . . . . . . . . . . . . 22 (𝑉 ∈ (1...(𝑉 − 1)) → 𝑉 ≤ (𝑉 − 1))
332330, 331nsyl 135 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ¬ 𝑉 ∈ (1...(𝑉 − 1)))
333 difsn 4360 . . . . . . . . . . . . . . . . . . . . 21 𝑉 ∈ (1...(𝑉 − 1)) → ((1...(𝑉 − 1)) ∖ {𝑉}) = (1...(𝑉 − 1)))
334332, 333syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((1...(𝑉 − 1)) ∖ {𝑉}) = (1...(𝑉 − 1)))
335327, 334syl5eq 2697 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (((1...(𝑉 − 1)) ∪ {𝑉}) ∖ {𝑉}) = (1...(𝑉 − 1)))
336326, 335eqtrd 2685 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1...𝑉) ∖ {𝑉}) = (1...(𝑉 − 1)))
337 elfzle1 12382 . . . . . . . . . . . . . . . . . . . 20 (𝑉 ∈ ((𝑉 + 1)...𝑁) → (𝑉 + 1) ≤ 𝑉)
33833, 337nsyl 135 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ¬ 𝑉 ∈ ((𝑉 + 1)...𝑁))
339 difsn 4360 . . . . . . . . . . . . . . . . . . 19 𝑉 ∈ ((𝑉 + 1)...𝑁) → (((𝑉 + 1)...𝑁) ∖ {𝑉}) = ((𝑉 + 1)...𝑁))
340338, 339syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((𝑉 + 1)...𝑁) ∖ {𝑉}) = ((𝑉 + 1)...𝑁))
341336, 340uneq12d 3801 . . . . . . . . . . . . . . . . 17 (𝜑 → (((1...𝑉) ∖ {𝑉}) ∪ (((𝑉 + 1)...𝑁) ∖ {𝑉})) = ((1...(𝑉 − 1)) ∪ ((𝑉 + 1)...𝑁)))
342311, 341syl5eq 2697 . . . . . . . . . . . . . . . 16 (𝜑 → (((1...𝑉) ∪ ((𝑉 + 1)...𝑁)) ∖ {𝑉}) = ((1...(𝑉 − 1)) ∪ ((𝑉 + 1)...𝑁)))
343310, 342eqtrd 2685 . . . . . . . . . . . . . . 15 (𝜑 → ((1...𝑁) ∖ {𝑉}) = ((1...(𝑉 − 1)) ∪ ((𝑉 + 1)...𝑁)))
344343imaeq2d 5501 . . . . . . . . . . . . . 14 (𝜑 → (𝑈 “ ((1...𝑁) ∖ {𝑉})) = (𝑈 “ ((1...(𝑉 − 1)) ∪ ((𝑉 + 1)...𝑁))))
345309, 344eqtr3d 2687 . . . . . . . . . . . . 13 (𝜑 → ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {𝑉})) = (𝑈 “ ((1...(𝑉 − 1)) ∪ ((𝑉 + 1)...𝑁))))
346 imaundi 5580 . . . . . . . . . . . . 13 (𝑈 “ ((1...(𝑉 − 1)) ∪ ((𝑉 + 1)...𝑁))) = ((𝑈 “ (1...(𝑉 − 1))) ∪ (𝑈 “ ((𝑉 + 1)...𝑁)))
347345, 346syl6eq 2701 . . . . . . . . . . . 12 (𝜑 → ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {𝑉})) = ((𝑈 “ (1...(𝑉 − 1))) ∪ (𝑈 “ ((𝑉 + 1)...𝑁))))
348347eleq2d 2716 . . . . . . . . . . 11 (𝜑 → (𝑛 ∈ ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {𝑉})) ↔ 𝑛 ∈ ((𝑈 “ (1...(𝑉 − 1))) ∪ (𝑈 “ ((𝑉 + 1)...𝑁)))))
349 eldif 3617 . . . . . . . . . . 11 (𝑛 ∈ ((𝑈 “ (1...𝑁)) ∖ (𝑈 “ {𝑉})) ↔ (𝑛 ∈ (𝑈 “ (1...𝑁)) ∧ ¬ 𝑛 ∈ (𝑈 “ {𝑉})))
350 elun 3786 . . . . . . . . . . 11 (𝑛 ∈ ((𝑈 “ (1...(𝑉 − 1))) ∪ (𝑈 “ ((𝑉 + 1)...𝑁))) ↔ (𝑛 ∈ (𝑈 “ (1...(𝑉 − 1))) ∨ 𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))))
351348, 349, 3503bitr3g 302 . . . . . . . . . 10 (𝜑 → ((𝑛 ∈ (𝑈 “ (1...𝑁)) ∧ ¬ 𝑛 ∈ (𝑈 “ {𝑉})) ↔ (𝑛 ∈ (𝑈 “ (1...(𝑉 − 1))) ∨ 𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁)))))
352351adantr 480 . . . . . . . . 9 ((𝜑𝑉 < 𝑀) → ((𝑛 ∈ (𝑈 “ (1...𝑁)) ∧ ¬ 𝑛 ∈ (𝑈 “ {𝑉})) ↔ (𝑛 ∈ (𝑈 “ (1...(𝑉 − 1))) ∨ 𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁)))))
353 imassrn 5512 . . . . . . . . . . . . . . . 16 (𝑈 “ (1...(𝑉 − 1))) ⊆ ran 𝑈
354353, 65syl5ss 3647 . . . . . . . . . . . . . . 15 (𝜑 → (𝑈 “ (1...(𝑉 − 1))) ⊆ (1...𝑁))
355354sselda 3636 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) → 𝑛 ∈ (1...𝑁))
35670adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) → 𝑇 Fn (1...𝑁))
357 fnconstg 6131 . . . . . . . . . . . . . . . . . . . . 21 (1 ∈ V → ((𝑈 “ (1...(𝑉 − 1))) × {1}) Fn (𝑈 “ (1...(𝑉 − 1))))
35872, 357ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((𝑈 “ (1...(𝑉 − 1))) × {1}) Fn (𝑈 “ (1...(𝑉 − 1)))
359 fnconstg 6131 . . . . . . . . . . . . . . . . . . . . 21 (0 ∈ V → ((𝑈 “ (𝑉...𝑁)) × {0}) Fn (𝑈 “ (𝑉...𝑁)))
36075, 359ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((𝑈 “ (𝑉...𝑁)) × {0}) Fn (𝑈 “ (𝑉...𝑁))
361358, 360pm3.2i 470 . . . . . . . . . . . . . . . . . . 19 (((𝑈 “ (1...(𝑉 − 1))) × {1}) Fn (𝑈 “ (1...(𝑉 − 1))) ∧ ((𝑈 “ (𝑉...𝑁)) × {0}) Fn (𝑈 “ (𝑉...𝑁)))
362 imain 6012 . . . . . . . . . . . . . . . . . . . . 21 (Fun 𝑈 → (𝑈 “ ((1...(𝑉 − 1)) ∩ (𝑉...𝑁))) = ((𝑈 “ (1...(𝑉 − 1))) ∩ (𝑈 “ (𝑉...𝑁))))
3634, 362syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑈 “ ((1...(𝑉 − 1)) ∩ (𝑉...𝑁))) = ((𝑈 “ (1...(𝑉 − 1))) ∩ (𝑈 “ (𝑉...𝑁))))
364 fzdisj 12406 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑉 − 1) < 𝑉 → ((1...(𝑉 − 1)) ∩ (𝑉...𝑁)) = ∅)
365328, 364syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1...(𝑉 − 1)) ∩ (𝑉...𝑁)) = ∅)
366365imaeq2d 5501 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑈 “ ((1...(𝑉 − 1)) ∩ (𝑉...𝑁))) = (𝑈 “ ∅))
367366, 84syl6eq 2701 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑈 “ ((1...(𝑉 − 1)) ∩ (𝑉...𝑁))) = ∅)
368363, 367eqtr3d 2687 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑈 “ (1...(𝑉 − 1))) ∩ (𝑈 “ (𝑉...𝑁))) = ∅)
369 fnun 6035 . . . . . . . . . . . . . . . . . . 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))))
370361, 368, 369sylancr 696 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑉 − 1))) ∪ (𝑈 “ (𝑉...𝑁))))
371 imaundi 5580 . . . . . . . . . . . . . . . . . . . 20 (𝑈 “ ((1...(𝑉 − 1)) ∪ (𝑉...𝑁))) = ((𝑈 “ (1...(𝑉 − 1))) ∪ (𝑈 “ (𝑉...𝑁)))
372 uzss 11746 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑉 ∈ (ℤ‘(𝑉 − 1)) → (ℤ𝑉) ⊆ (ℤ‘(𝑉 − 1)))
373317, 372syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (ℤ𝑉) ⊆ (ℤ‘(𝑉 − 1)))
374 elfzuz3 12377 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑉 ∈ (1...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ𝑉))
3757, 374syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (𝑁 − 1) ∈ (ℤ𝑉))
376373, 375sseldd 3637 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝑁 − 1) ∈ (ℤ‘(𝑉 − 1)))
377 peano2uz 11779 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑁 − 1) ∈ (ℤ‘(𝑉 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑉 − 1)))
378376, 377syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑉 − 1)))
37913, 378eqeltrrd 2731 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑁 ∈ (ℤ‘(𝑉 − 1)))
380 fzsplit2 12404 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑉 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑉 − 1))) → (1...𝑁) = ((1...(𝑉 − 1)) ∪ (((𝑉 − 1) + 1)...𝑁)))
381312, 379, 380syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (1...𝑁) = ((1...(𝑉 − 1)) ∪ (((𝑉 − 1) + 1)...𝑁)))
382215oveq1d 6705 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (((𝑉 − 1) + 1)...𝑁) = (𝑉...𝑁))
383382uneq2d 3800 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((1...(𝑉 − 1)) ∪ (((𝑉 − 1) + 1)...𝑁)) = ((1...(𝑉 − 1)) ∪ (𝑉...𝑁)))
384381, 383eqtrd 2685 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (1...𝑁) = ((1...(𝑉 − 1)) ∪ (𝑉...𝑁)))
385384imaeq2d 5501 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑈 “ (1...𝑁)) = (𝑈 “ ((1...(𝑉 − 1)) ∪ (𝑉...𝑁))))
386385, 107eqtr3d 2687 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑈 “ ((1...(𝑉 − 1)) ∪ (𝑉...𝑁))) = (1...𝑁))
387371, 386syl5eqr 2699 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑈 “ (1...(𝑉 − 1))) ∪ (𝑈 “ (𝑉...𝑁))) = (1...𝑁))
388387fneq2d 6020 . . . . . . . . . . . . . . . . . 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)))
389370, 388mpbid 222 . . . . . . . . . . . . . . . . 17 (𝜑 → (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0})) Fn (1...𝑁))
390389adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) → (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0})) Fn (1...𝑁))
391 fzfid 12812 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) → (1...𝑁) ∈ Fin)
392 eqidd 2652 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) ∧ 𝑛 ∈ (1...𝑁)) → (𝑇𝑛) = (𝑇𝑛))
393 fvun1 6308 . . . . . . . . . . . . . . . . . . . 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))
394358, 360, 393mp3an12 1454 . . . . . . . . . . . . . . . . . . 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))
395368, 394sylan 487 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) → ((((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (1...(𝑉 − 1))) × {1})‘𝑛))
39672fvconst2 6510 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (𝑈 “ (1...(𝑉 − 1))) → (((𝑈 “ (1...(𝑉 − 1))) × {1})‘𝑛) = 1)
397396adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) → (((𝑈 “ (1...(𝑉 − 1))) × {1})‘𝑛) = 1)
398395, 397eqtrd 2685 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) → ((((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0}))‘𝑛) = 1)
399398adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) ∧ 𝑛 ∈ (1...𝑁)) → ((((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0}))‘𝑛) = 1)
400356, 390, 391, 391, 114, 392, 399ofval 6948 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇𝑓 + (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0})))‘𝑛) = ((𝑇𝑛) + 1))
401111adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) → (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})) Fn (1...𝑁))
402 fzss2 12419 . . . . . . . . . . . . . . . . . . . . 21 (𝑉 ∈ (ℤ‘(𝑉 − 1)) → (1...(𝑉 − 1)) ⊆ (1...𝑉))
403317, 402syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (1...(𝑉 − 1)) ⊆ (1...𝑉))
404 imass2 5536 . . . . . . . . . . . . . . . . . . . 20 ((1...(𝑉 − 1)) ⊆ (1...𝑉) → (𝑈 “ (1...(𝑉 − 1))) ⊆ (𝑈 “ (1...𝑉)))
405403, 404syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑈 “ (1...(𝑉 − 1))) ⊆ (𝑈 “ (1...𝑉)))
406405sselda 3636 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) → 𝑛 ∈ (𝑈 “ (1...𝑉)))
407406, 121syldan 486 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) → ((((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))‘𝑛) = 1)
408407adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) ∧ 𝑛 ∈ (1...𝑁)) → ((((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))‘𝑛) = 1)
409356, 401, 391, 391, 114, 392, 408ofval 6948 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ (𝑈 “ (1...(𝑉 − 1)))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇𝑓 + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))‘𝑛) = ((𝑇𝑛) + 1))
410400, 409eqtr4d 2688 . . . . . . . . . . . . . 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))
411355, 410mpdan 703 . . . . . . . . . . . . 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))
412 imassrn 5512 . . . . . . . . . . . . . . . 16 (𝑈 “ ((𝑉 + 1)...𝑁)) ⊆ ran 𝑈
413412, 65syl5ss 3647 . . . . . . . . . . . . . . 15 (𝜑 → (𝑈 “ ((𝑉 + 1)...𝑁)) ⊆ (1...𝑁))
414413sselda 3636 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) → 𝑛 ∈ (1...𝑁))
41570adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) → 𝑇 Fn (1...𝑁))
416389adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) → (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0})) Fn (1...𝑁))
417 fzfid 12812 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) → (1...𝑁) ∈ Fin)
418 eqidd 2652 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → (𝑇𝑛) = (𝑇𝑛))
419 fzss1 12418 . . . . . . . . . . . . . . . . . . . . 21 ((𝑉 + 1) ∈ (ℤ𝑉) → ((𝑉 + 1)...𝑁) ⊆ (𝑉...𝑁))
420148, 419syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑉 + 1)...𝑁) ⊆ (𝑉...𝑁))
421 imass2 5536 . . . . . . . . . . . . . . . . . . . 20 (((𝑉 + 1)...𝑁) ⊆ (𝑉...𝑁) → (𝑈 “ ((𝑉 + 1)...𝑁)) ⊆ (𝑈 “ (𝑉...𝑁)))
422420, 421syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑈 “ ((𝑉 + 1)...𝑁)) ⊆ (𝑈 “ (𝑉...𝑁)))
423422sselda 3636 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) → 𝑛 ∈ (𝑈 “ (𝑉...𝑁)))
424 fvun2 6309 . . . . . . . . . . . . . . . . . . . . 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))
425358, 360, 424mp3an12 1454 . . . . . . . . . . . . . . . . . . . 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))
426368, 425sylan 487 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ (𝑈 “ (𝑉...𝑁))) → ((((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (𝑉...𝑁)) × {0})‘𝑛))
42775fvconst2 6510 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (𝑈 “ (𝑉...𝑁)) → (((𝑈 “ (𝑉...𝑁)) × {0})‘𝑛) = 0)
428427adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ (𝑈 “ (𝑉...𝑁))) → (((𝑈 “ (𝑉...𝑁)) × {0})‘𝑛) = 0)
429426, 428eqtrd 2685 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ (𝑈 “ (𝑉...𝑁))) → ((((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0}))‘𝑛) = 0)
430423, 429syldan 486 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) → ((((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0}))‘𝑛) = 0)
431430adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → ((((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0}))‘𝑛) = 0)
432415, 416, 417, 417, 114, 418, 431ofval 6948 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇𝑓 + (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0})))‘𝑛) = ((𝑇𝑛) + 0))
433111adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) → (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})) Fn (1...𝑁))
434186adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → ((((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))‘𝑛) = 0)
435415, 433, 417, 417, 114, 418, 434ofval 6948 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇𝑓 + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))‘𝑛) = ((𝑇𝑛) + 0))
436432, 435eqtr4d 2688 . . . . . . . . . . . . . 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))
437414, 436mpdan 703 . . . . . . . . . . . . 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))
438411, 437jaodan 843 . . . . . . . . . . . 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))
439438adantlr 751 . . . . . . . . . . 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))
440203adantr 480 . . . . . . . . . . . . . 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})))))
441207a1i 11 . . . . . . . . . . . . . . 15 (((𝜑𝑉 < 𝑀) ∧ 𝑦 = (𝑉 − 1)) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) ∈ V)
442217adantlr 751 . . . . . . . . . . . . . . . 16 (((𝜑𝑉 < 𝑀) ∧ 𝑦 = (𝑉 − 1)) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = if((𝑉 − 1) < 𝑀, (𝑉 − 1), 𝑉))
443 lttr 10152 . . . . . . . . . . . . . . . . . . . . 21 (((𝑉 − 1) ∈ ℝ ∧ 𝑉 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (((𝑉 − 1) < 𝑉𝑉 < 𝑀) → (𝑉 − 1) < 𝑀))
444228, 28, 225, 443syl3anc 1366 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (((𝑉 − 1) < 𝑉𝑉 < 𝑀) → (𝑉 − 1) < 𝑀))
445328, 444mpand 711 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑉 < 𝑀 → (𝑉 − 1) < 𝑀))
446445imp 444 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑉 < 𝑀) → (𝑉 − 1) < 𝑀)
447446iftrued 4127 . . . . . . . . . . . . . . . . 17 ((𝜑𝑉 < 𝑀) → if((𝑉 − 1) < 𝑀, (𝑉 − 1), 𝑉) = (𝑉 − 1))
448447adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑉 < 𝑀) ∧ 𝑦 = (𝑉 − 1)) → if((𝑉 − 1) < 𝑀, (𝑉 − 1), 𝑉) = (𝑉 − 1))
449442, 448eqtrd 2685 . . . . . . . . . . . . . . 15 (((𝜑𝑉 < 𝑀) ∧ 𝑦 = (𝑉 − 1)) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = (𝑉 − 1))
450 simpll 805 . . . . . . . . . . . . . . . 16 (((𝜑𝑉 < 𝑀) ∧ 𝑦 = (𝑉 − 1)) → 𝜑)
451 oveq2 6698 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = (𝑉 − 1) → (1...𝑗) = (1...(𝑉 − 1)))
452451imaeq2d 5501 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑉 − 1) → (𝑈 “ (1...𝑗)) = (𝑈 “ (1...(𝑉 − 1))))
453452xpeq1d 5172 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑉 − 1) → ((𝑈 “ (1...𝑗)) × {1}) = ((𝑈 “ (1...(𝑉 − 1))) × {1}))
454453adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 = (𝑉 − 1)) → ((𝑈 “ (1...𝑗)) × {1}) = ((𝑈 “ (1...(𝑉 − 1))) × {1}))
455 oveq1 6697 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = (𝑉 − 1) → (𝑗 + 1) = ((𝑉 − 1) + 1))
456455, 215sylan9eqr 2707 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 = (𝑉 − 1)) → (𝑗 + 1) = 𝑉)
457456oveq1d 6705 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 = (𝑉 − 1)) → ((𝑗 + 1)...𝑁) = (𝑉...𝑁))
458457imaeq2d 5501 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 = (𝑉 − 1)) → (𝑈 “ ((𝑗 + 1)...𝑁)) = (𝑈 “ (𝑉...𝑁)))
459458xpeq1d 5172 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 = (𝑉 − 1)) → ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) = ((𝑈 “ (𝑉...𝑁)) × {0}))
460454, 459uneq12d 3801 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 = (𝑉 − 1)) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) = (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0})))
461460oveq2d 6706 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 = (𝑉 − 1)) → (𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇𝑓 + (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0}))))
462450, 461sylan 487 . . . . . . . . . . . . . . 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}))))
463441, 449, 462csbied2 3594 . . . . . . . . . . . . . 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}))))
464250adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑉 < 𝑀) → (𝑉 − 1) ∈ (0...(𝑁 − 1)))
465 ovexd 6720 . . . . . . . . . . . . . 14 ((𝜑𝑉 < 𝑀) → (𝑇𝑓 + (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0}))) ∈ V)
466440, 463, 464, 465fvmptd 6327 . . . . . . . . . . . . 13 ((𝜑𝑉 < 𝑀) → (𝐹‘(𝑉 − 1)) = (𝑇𝑓 + (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0}))))
467466fveq1d 6231 . . . . . . . . . . . 12 ((𝜑𝑉 < 𝑀) → ((𝐹‘(𝑉 − 1))‘𝑛) = ((𝑇𝑓 + (((𝑈 “ (1...(𝑉 − 1))) × {1}) ∪ ((𝑈 “ (𝑉...𝑁)) × {0})))‘𝑛))
468467adantr 480 . . . . . . . . . . 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))
469207a1i 11 . . . . . . . . . . . . . . . 16 ((𝑉 < 𝑀𝑦 = 𝑉) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) ∈ V)
470 iftrue 4125 . . . . . . . . . . . . . . . . . . . 20 (𝑉 < 𝑀 → if(𝑉 < 𝑀, 𝑉, (𝑉 + 1)) = 𝑉)
471260, 470sylan9eqr 2707 . . . . . . . . . . . . . . . . . . 19 ((𝑉 < 𝑀𝑦 = 𝑉) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = 𝑉)
472471eqeq2d 2661 . . . . . . . . . . . . . . . . . 18 ((𝑉 < 𝑀𝑦 = 𝑉) → (𝑗 = if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) ↔ 𝑗 = 𝑉))
473472biimpa 500 . . . . . . . . . . . . . . . . 17 (((𝑉 < 𝑀𝑦 = 𝑉) ∧ 𝑗 = if(𝑦 < 𝑀, 𝑦, (𝑦 + 1))) → 𝑗 = 𝑉)
474473, 245syl 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}))))
475469, 474csbied 3593 . . . . . . . . . . . . . . 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}))))
476475adantll 750 . . . . . . . . . . . . . 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}))))
477280adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑉 < 𝑀) → 𝑉 ∈ (0...(𝑁 − 1)))
478 ovexd 6720 . . . . . . . . . . . . . 14 ((𝜑𝑉 < 𝑀) → (𝑇𝑓 + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))) ∈ V)
479440, 476, 477, 478fvmptd 6327 . . . . . . . . . . . . 13 ((𝜑𝑉 < 𝑀) → (𝐹𝑉) = (𝑇𝑓 + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0}))))
480479fveq1d 6231 . . . . . . . . . . . 12 ((𝜑𝑉 < 𝑀) → ((𝐹𝑉)‘𝑛) = ((𝑇𝑓 + (((𝑈 “ (1...𝑉)) × {1}) ∪ ((𝑈 “ ((𝑉 + 1)...𝑁)) × {0})))‘𝑛))
481480adantr 480 . . . . . . . . . . 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))
482439, 468, 4813eqtr4d 2695 . . . . . . . . . 10 (((𝜑𝑉 < 𝑀) ∧ (𝑛 ∈ (𝑈 “ (1...(𝑉 − 1))) ∨ 𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁)))) → ((𝐹‘(𝑉 − 1))‘𝑛) = ((𝐹𝑉)‘𝑛))
483482ex 449 . . . . . . . . 9 ((𝜑𝑉 < 𝑀) → ((𝑛 ∈ (𝑈 “ (1...(𝑉 − 1))) ∨ 𝑛 ∈ (𝑈 “ ((𝑉 + 1)...𝑁))) → ((𝐹‘(𝑉 − 1))‘𝑛) = ((𝐹𝑉)‘𝑛)))
484352, 483sylbid 230 . . . . . . . 8 ((𝜑𝑉 < 𝑀) → ((𝑛 ∈ (𝑈 “ (1...𝑁)) ∧ ¬ 𝑛 ∈ (𝑈 “ {𝑉})) → ((𝐹‘(𝑉 − 1))‘𝑛) = ((𝐹𝑉)‘𝑛)))
485484expdimp 452 . . . . . . 7 (((𝜑𝑉 < 𝑀) ∧ 𝑛 ∈ (𝑈 “ (1...𝑁))) → (¬ 𝑛 ∈ (𝑈 “ {𝑉}) → ((𝐹‘(𝑉 − 1))‘𝑛) = ((𝐹𝑉)‘𝑛)))
486485necon1ad 2840 . . . . . 6 (((𝜑𝑉 < 𝑀) ∧ 𝑛 ∈ (𝑈 “ (1...𝑁))) → (((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 ∈ (𝑈 “ {𝑉})))
487 elimasni 5527 . . . . . . . 8 (𝑛 ∈ (𝑈 “ {𝑉}) → 𝑉𝑈𝑛)
488 eqcom 2658 . . . . . . . . 9 (𝑛 = (𝑈𝑉) ↔ (𝑈𝑉) = 𝑛)
489 fnbrfvb 6274 . . . . . . . . . 10 ((𝑈 Fn (1...𝑁) ∧ 𝑉 ∈ (1...𝑁)) → ((𝑈𝑉) = 𝑛𝑉𝑈𝑛))
490294, 100, 489syl2anc 694 . . . . . . . . 9 (𝜑 → ((𝑈𝑉) = 𝑛𝑉𝑈𝑛))
491488, 490syl5bb 272 . . . . . . . 8 (𝜑 → (𝑛 = (𝑈𝑉) ↔ 𝑉𝑈𝑛))
492487, 491syl5ibr 236 . . . . . . 7 (𝜑 → (𝑛 ∈ (𝑈 “ {𝑉}) → 𝑛 = (𝑈𝑉)))
493492ad2antrr 762 . . . . . 6 (((𝜑𝑉 < 𝑀) ∧ 𝑛 ∈ (𝑈 “ (1...𝑁))) → (𝑛 ∈ (𝑈 “ {𝑉}) → 𝑛 = (𝑈𝑉)))
494486, 493syld 47 . . . . 5 (((𝜑𝑉 < 𝑀) ∧ 𝑛 ∈ (𝑈 “ (1...𝑁))) → (((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = (𝑈𝑉)))
495494ralrimiva 2995 . . . 4 ((𝜑𝑉 < 𝑀) → ∀𝑛 ∈ (𝑈 “ (1...𝑁))(((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = (𝑈𝑉)))
496 fvex 6239 . . . . 5 (𝑈𝑉) ∈ V
497 eqeq2 2662 . . . . . . 7 (𝑚 = (𝑈𝑉) → (𝑛 = 𝑚𝑛 = (𝑈𝑉)))
498497imbi2d 329 . . . . . 6 (𝑚 = (𝑈𝑉) → ((((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = 𝑚) ↔ (((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = (𝑈𝑉))))
499498ralbidv 3015 . . . . 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))))
500496, 499spcev 3331 . . . 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))
501495, 500syl 17 . . 3 ((𝜑𝑉 < 𝑀) → ∃𝑚𝑛 ∈ (𝑈 “ (1...𝑁))(((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = 𝑚))
502 eldifsni 4353 . . . . 5 (𝑀 ∈ ((0...𝑁) ∖ {𝑉}) → 𝑀𝑉)
503219, 502syl 17 . . . 4 (𝜑𝑀𝑉)
504225, 28lttri2d 10214 . . . 4 (𝜑 → (𝑀𝑉 ↔ (𝑀 < 𝑉𝑉 < 𝑀)))
505503, 504mpbid 222 . . 3 (𝜑 → (𝑀 < 𝑉𝑉 < 𝑀))
506307, 501, 505mpjaodan 844 . 2 (𝜑 → ∃𝑚𝑛 ∈ (𝑈 “ (1...𝑁))(((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = 𝑚))
507 nfv 1883 . . . 4 𝑚((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛)
508507rmo2 3559 . . 3 (∃*𝑛 ∈ (𝑈 “ (1...𝑁))((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) ↔ ∃𝑚𝑛 ∈ (𝑈 “ (1...𝑁))(((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = 𝑚))
509 rmoeq1 3171 . . . 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)))
510107, 509syl 17 . . 3 (𝜑 → (∃*𝑛 ∈ (𝑈 “ (1...𝑁))((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) ↔ ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛)))
511508, 510syl5bbr 274 . 2 (𝜑 → (∃𝑚𝑛 ∈ (𝑈 “ (1...𝑁))(((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛) → 𝑛 = 𝑚) ↔ ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛)))
512506, 511mpbid 222 1 (𝜑 → ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹𝑉)‘𝑛))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wex 1744  wcel 2030  wne 2823  wral 2941  ∃*wrmo 2944  Vcvv 3231  csb 3566  cdif 3604  cun 3605  cin 3606  wss 3607  c0 3948  ifcif 4119  {csn 4210   class class class wbr 4685  cmpt 4762   × cxp 5141  ccnv 5142  ran crn 5144  cima 5146  Fun wfun 5920   Fn wfn 5921  wf 5922  ontowfo 5924  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  𝑓 cof 6937  Fincfn 7997  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112  cle 10113  cmin 10304  cn 11058  cz 11415  cuz 11725  ...cfz 12364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365
This theorem is referenced by:  poimirlem8  33547  poimirlem18  33557  poimirlem21  33560  poimirlem22  33561
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