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Theorem poimirlem11 35561
Description: Lemma for poimir 35583 connecting walks that could yield from a given cube a given face opposite the first vertex of the walk. (Contributed by Brendan Leahy, 21-Aug-2020.)
Hypotheses
Ref Expression
poimir.0 (𝜑𝑁 ∈ ℕ)
poimirlem22.s 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
poimirlem22.1 (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
poimirlem12.2 (𝜑𝑇𝑆)
poimirlem11.3 (𝜑 → (2nd𝑇) = 0)
poimirlem11.4 (𝜑𝑈𝑆)
poimirlem11.5 (𝜑 → (2nd𝑈) = 0)
poimirlem11.6 (𝜑𝑀 ∈ (1...𝑁))
Assertion
Ref Expression
poimirlem11 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑀)) ⊆ ((2nd ‘(1st𝑈)) “ (1...𝑀)))
Distinct variable groups:   𝑓,𝑗,𝑡,𝑦   𝜑,𝑗,𝑦   𝑗,𝐹,𝑦   𝑗,𝑀,𝑦   𝑗,𝑁,𝑦   𝑇,𝑗,𝑦   𝑈,𝑗,𝑦   𝜑,𝑡   𝑓,𝐾,𝑗,𝑡   𝑓,𝑀,𝑡   𝑓,𝑁,𝑡   𝑇,𝑓   𝑈,𝑓   𝑓,𝐹,𝑡   𝑡,𝑇   𝑡,𝑈   𝑆,𝑗,𝑡,𝑦
Allowed substitution hints:   𝜑(𝑓)   𝑆(𝑓)   𝐾(𝑦)

Proof of Theorem poimirlem11
StepHypRef Expression
1 eldif 3893 . . . . . . 7 (𝑦 ∈ (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∖ ((2nd ‘(1st𝑈)) “ (1...𝑀))) ↔ (𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd ‘(1st𝑈)) “ (1...𝑀))))
2 imassrn 5957 . . . . . . . . . . . 12 ((2nd ‘(1st𝑇)) “ (1...𝑀)) ⊆ ran (2nd ‘(1st𝑇))
3 poimirlem12.2 . . . . . . . . . . . . . . . . . 18 (𝜑𝑇𝑆)
4 elrabi 3611 . . . . . . . . . . . . . . . . . . 19 (𝑇 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
5 poimirlem22.s . . . . . . . . . . . . . . . . . . 19 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
64, 5eleq2s 2858 . . . . . . . . . . . . . . . . . 18 (𝑇𝑆𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
73, 6syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
8 xp1st 7814 . . . . . . . . . . . . . . . . 17 (𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
97, 8syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (1st𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
10 xp2nd 7815 . . . . . . . . . . . . . . . 16 ((1st𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
119, 10syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
12 fvex 6751 . . . . . . . . . . . . . . . 16 (2nd ‘(1st𝑇)) ∈ V
13 f1oeq1 6670 . . . . . . . . . . . . . . . 16 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
1412, 13elab 3602 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
1511, 14sylib 221 . . . . . . . . . . . . . 14 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
16 f1of 6682 . . . . . . . . . . . . . 14 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)):(1...𝑁)⟶(1...𝑁))
1715, 16syl 17 . . . . . . . . . . . . 13 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)⟶(1...𝑁))
1817frnd 6574 . . . . . . . . . . . 12 (𝜑 → ran (2nd ‘(1st𝑇)) ⊆ (1...𝑁))
192, 18sstrid 3928 . . . . . . . . . . 11 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑀)) ⊆ (1...𝑁))
20 poimirlem11.4 . . . . . . . . . . . . . . . . 17 (𝜑𝑈𝑆)
21 elrabi 3611 . . . . . . . . . . . . . . . . . 18 (𝑈 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑈 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
2221, 5eleq2s 2858 . . . . . . . . . . . . . . . . 17 (𝑈𝑆𝑈 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
2320, 22syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑈 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
24 xp1st 7814 . . . . . . . . . . . . . . . 16 (𝑈 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st𝑈) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
2523, 24syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (1st𝑈) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
26 xp2nd 7815 . . . . . . . . . . . . . . 15 ((1st𝑈) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st𝑈)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
2725, 26syl 17 . . . . . . . . . . . . . 14 (𝜑 → (2nd ‘(1st𝑈)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
28 fvex 6751 . . . . . . . . . . . . . . 15 (2nd ‘(1st𝑈)) ∈ V
29 f1oeq1 6670 . . . . . . . . . . . . . . 15 (𝑓 = (2nd ‘(1st𝑈)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑈)):(1...𝑁)–1-1-onto→(1...𝑁)))
3028, 29elab 3602 . . . . . . . . . . . . . 14 ((2nd ‘(1st𝑈)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))
3127, 30sylib 221 . . . . . . . . . . . . 13 (𝜑 → (2nd ‘(1st𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))
32 f1ofo 6689 . . . . . . . . . . . . 13 ((2nd ‘(1st𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑈)):(1...𝑁)–onto→(1...𝑁))
3331, 32syl 17 . . . . . . . . . . . 12 (𝜑 → (2nd ‘(1st𝑈)):(1...𝑁)–onto→(1...𝑁))
34 foima 6659 . . . . . . . . . . . 12 ((2nd ‘(1st𝑈)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st𝑈)) “ (1...𝑁)) = (1...𝑁))
3533, 34syl 17 . . . . . . . . . . 11 (𝜑 → ((2nd ‘(1st𝑈)) “ (1...𝑁)) = (1...𝑁))
3619, 35sseqtrrd 3958 . . . . . . . . . 10 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑀)) ⊆ ((2nd ‘(1st𝑈)) “ (1...𝑁)))
3736ssdifd 4071 . . . . . . . . 9 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∖ ((2nd ‘(1st𝑈)) “ (1...𝑀))) ⊆ (((2nd ‘(1st𝑈)) “ (1...𝑁)) ∖ ((2nd ‘(1st𝑈)) “ (1...𝑀))))
38 dff1o3 6688 . . . . . . . . . . . . 13 ((2nd ‘(1st𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st𝑈)):(1...𝑁)–onto→(1...𝑁) ∧ Fun (2nd ‘(1st𝑈))))
3938simprbi 500 . . . . . . . . . . . 12 ((2nd ‘(1st𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun (2nd ‘(1st𝑈)))
4031, 39syl 17 . . . . . . . . . . 11 (𝜑 → Fun (2nd ‘(1st𝑈)))
41 imadif 6484 . . . . . . . . . . 11 (Fun (2nd ‘(1st𝑈)) → ((2nd ‘(1st𝑈)) “ ((1...𝑁) ∖ (1...𝑀))) = (((2nd ‘(1st𝑈)) “ (1...𝑁)) ∖ ((2nd ‘(1st𝑈)) “ (1...𝑀))))
4240, 41syl 17 . . . . . . . . . 10 (𝜑 → ((2nd ‘(1st𝑈)) “ ((1...𝑁) ∖ (1...𝑀))) = (((2nd ‘(1st𝑈)) “ (1...𝑁)) ∖ ((2nd ‘(1st𝑈)) “ (1...𝑀))))
43 difun2 4411 . . . . . . . . . . . 12 ((((𝑀 + 1)...𝑁) ∪ (1...𝑀)) ∖ (1...𝑀)) = (((𝑀 + 1)...𝑁) ∖ (1...𝑀))
44 poimirlem11.6 . . . . . . . . . . . . . . 15 (𝜑𝑀 ∈ (1...𝑁))
45 fzsplit 13167 . . . . . . . . . . . . . . 15 (𝑀 ∈ (1...𝑁) → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))
4644, 45syl 17 . . . . . . . . . . . . . 14 (𝜑 → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))
47 uncom 4083 . . . . . . . . . . . . . 14 ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) = (((𝑀 + 1)...𝑁) ∪ (1...𝑀))
4846, 47eqtrdi 2796 . . . . . . . . . . . . 13 (𝜑 → (1...𝑁) = (((𝑀 + 1)...𝑁) ∪ (1...𝑀)))
4948difeq1d 4052 . . . . . . . . . . . 12 (𝜑 → ((1...𝑁) ∖ (1...𝑀)) = ((((𝑀 + 1)...𝑁) ∪ (1...𝑀)) ∖ (1...𝑀)))
50 incom 4131 . . . . . . . . . . . . . 14 (((𝑀 + 1)...𝑁) ∩ (1...𝑀)) = ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))
51 elfznn 13170 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ (1...𝑁) → 𝑀 ∈ ℕ)
5244, 51syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝑀 ∈ ℕ)
5352nnred 11874 . . . . . . . . . . . . . . . 16 (𝜑𝑀 ∈ ℝ)
5453ltp1d 11791 . . . . . . . . . . . . . . 15 (𝜑𝑀 < (𝑀 + 1))
55 fzdisj 13168 . . . . . . . . . . . . . . 15 (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
5654, 55syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
5750, 56syl5eq 2792 . . . . . . . . . . . . 13 (𝜑 → (((𝑀 + 1)...𝑁) ∩ (1...𝑀)) = ∅)
58 disj3 4384 . . . . . . . . . . . . 13 ((((𝑀 + 1)...𝑁) ∩ (1...𝑀)) = ∅ ↔ ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...𝑁) ∖ (1...𝑀)))
5957, 58sylib 221 . . . . . . . . . . . 12 (𝜑 → ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...𝑁) ∖ (1...𝑀)))
6043, 49, 593eqtr4a 2806 . . . . . . . . . . 11 (𝜑 → ((1...𝑁) ∖ (1...𝑀)) = ((𝑀 + 1)...𝑁))
6160imaeq2d 5946 . . . . . . . . . 10 (𝜑 → ((2nd ‘(1st𝑈)) “ ((1...𝑁) ∖ (1...𝑀))) = ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)))
6242, 61eqtr3d 2781 . . . . . . . . 9 (𝜑 → (((2nd ‘(1st𝑈)) “ (1...𝑁)) ∖ ((2nd ‘(1st𝑈)) “ (1...𝑀))) = ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)))
6337, 62sseqtrd 3957 . . . . . . . 8 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∖ ((2nd ‘(1st𝑈)) “ (1...𝑀))) ⊆ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)))
6463sselda 3917 . . . . . . 7 ((𝜑𝑦 ∈ (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∖ ((2nd ‘(1st𝑈)) “ (1...𝑀)))) → 𝑦 ∈ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)))
651, 64sylan2br 598 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd ‘(1st𝑈)) “ (1...𝑀)))) → 𝑦 ∈ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)))
66 fveq2 6738 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑈 → (2nd𝑡) = (2nd𝑈))
6766breq2d 5081 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑈 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑈)))
6867ifbid 4478 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑈 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)))
6968csbeq1d 3832 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑈if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
70 2fveq3 6743 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑈 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑈)))
71 2fveq3 6743 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑈 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑈)))
7271imaeq1d 5945 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑈 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑈)) “ (1...𝑗)))
7372xpeq1d 5597 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑈 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}))
7471imaeq1d 5945 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑈 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)))
7574xpeq1d 5597 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑈 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))
7673, 75uneq12d 4094 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑈 → ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))
7770, 76oveq12d 7252 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑈 → ((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑈)) ∘f + ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))
7877csbeq2dv 3835 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑈if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑈)) ∘f + ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))
7969, 78eqtrd 2779 . . . . . . . . . . . . . . 15 (𝑡 = 𝑈if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑈)) ∘f + ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))
8079mpteq2dv 5167 . . . . . . . . . . . . . 14 (𝑡 = 𝑈 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑈)) ∘f + ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))))
8180eqeq2d 2750 . . . . . . . . . . . . 13 (𝑡 = 𝑈 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑈)) ∘f + ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
8281, 5elrab2 3620 . . . . . . . . . . . 12 (𝑈𝑆 ↔ (𝑈 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑈)) ∘f + ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
8382simprbi 500 . . . . . . . . . . 11 (𝑈𝑆𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑈)) ∘f + ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))))
8420, 83syl 17 . . . . . . . . . 10 (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑈)) ∘f + ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))))
85 poimirlem11.5 . . . . . . . . . . . . . . . 16 (𝜑 → (2nd𝑈) = 0)
86 breq12 5074 . . . . . . . . . . . . . . . 16 ((𝑦 = (𝑀 − 1) ∧ (2nd𝑈) = 0) → (𝑦 < (2nd𝑈) ↔ (𝑀 − 1) < 0))
8785, 86sylan2 596 . . . . . . . . . . . . . . 15 ((𝑦 = (𝑀 − 1) ∧ 𝜑) → (𝑦 < (2nd𝑈) ↔ (𝑀 − 1) < 0))
8887ancoms 462 . . . . . . . . . . . . . 14 ((𝜑𝑦 = (𝑀 − 1)) → (𝑦 < (2nd𝑈) ↔ (𝑀 − 1) < 0))
89 oveq1 7241 . . . . . . . . . . . . . . 15 (𝑦 = (𝑀 − 1) → (𝑦 + 1) = ((𝑀 − 1) + 1))
9052nncnd 11875 . . . . . . . . . . . . . . . 16 (𝜑𝑀 ∈ ℂ)
91 npcan1 11286 . . . . . . . . . . . . . . . 16 (𝑀 ∈ ℂ → ((𝑀 − 1) + 1) = 𝑀)
9290, 91syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑀 − 1) + 1) = 𝑀)
9389, 92sylan9eqr 2802 . . . . . . . . . . . . . 14 ((𝜑𝑦 = (𝑀 − 1)) → (𝑦 + 1) = 𝑀)
9488, 93ifbieq2d 4481 . . . . . . . . . . . . 13 ((𝜑𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)) = if((𝑀 − 1) < 0, 𝑦, 𝑀))
9552nnzd 12310 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑀 ∈ ℤ)
96 poimir.0 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑁 ∈ ℕ)
9796nnzd 12310 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ ℤ)
98 elfzm1b 13219 . . . . . . . . . . . . . . . . . . 19 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∈ (1...𝑁) ↔ (𝑀 − 1) ∈ (0...(𝑁 − 1))))
9995, 97, 98syl2anc 587 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑀 ∈ (1...𝑁) ↔ (𝑀 − 1) ∈ (0...(𝑁 − 1))))
10044, 99mpbid 235 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑀 − 1) ∈ (0...(𝑁 − 1)))
101 elfzle1 13144 . . . . . . . . . . . . . . . . 17 ((𝑀 − 1) ∈ (0...(𝑁 − 1)) → 0 ≤ (𝑀 − 1))
102100, 101syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → 0 ≤ (𝑀 − 1))
103 0red 10865 . . . . . . . . . . . . . . . . 17 (𝜑 → 0 ∈ ℝ)
104 nnm1nn0 12160 . . . . . . . . . . . . . . . . . . 19 (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0)
10552, 104syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑀 − 1) ∈ ℕ0)
106105nn0red 12180 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑀 − 1) ∈ ℝ)
107103, 106lenltd 11007 . . . . . . . . . . . . . . . 16 (𝜑 → (0 ≤ (𝑀 − 1) ↔ ¬ (𝑀 − 1) < 0))
108102, 107mpbid 235 . . . . . . . . . . . . . . 15 (𝜑 → ¬ (𝑀 − 1) < 0)
109108iffalsed 4466 . . . . . . . . . . . . . 14 (𝜑 → if((𝑀 − 1) < 0, 𝑦, 𝑀) = 𝑀)
110109adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑦 = (𝑀 − 1)) → if((𝑀 − 1) < 0, 𝑦, 𝑀) = 𝑀)
11194, 110eqtrd 2779 . . . . . . . . . . . 12 ((𝜑𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)) = 𝑀)
112111csbeq1d 3832 . . . . . . . . . . 11 ((𝜑𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑈)) ∘f + ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = 𝑀 / 𝑗((1st ‘(1st𝑈)) ∘f + ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))
113 oveq2 7242 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑀 → (1...𝑗) = (1...𝑀))
114113imaeq2d 5946 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑀 → ((2nd ‘(1st𝑈)) “ (1...𝑗)) = ((2nd ‘(1st𝑈)) “ (1...𝑀)))
115114xpeq1d 5597 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑀 → (((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}))
116 oveq1 7241 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑀 → (𝑗 + 1) = (𝑀 + 1))
117116oveq1d 7249 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑀 → ((𝑗 + 1)...𝑁) = ((𝑀 + 1)...𝑁))
118117imaeq2d 5946 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑀 → ((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)))
119118xpeq1d 5597 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑀 → (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))
120115, 119uneq12d 4094 . . . . . . . . . . . . . . 15 (𝑗 = 𝑀 → ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))
121120oveq2d 7250 . . . . . . . . . . . . . 14 (𝑗 = 𝑀 → ((1st ‘(1st𝑈)) ∘f + ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑈)) ∘f + ((((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))))
122121adantl 485 . . . . . . . . . . . . 13 ((𝜑𝑗 = 𝑀) → ((1st ‘(1st𝑈)) ∘f + ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑈)) ∘f + ((((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))))
12344, 122csbied 3866 . . . . . . . . . . . 12 (𝜑𝑀 / 𝑗((1st ‘(1st𝑈)) ∘f + ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑈)) ∘f + ((((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))))
124123adantr 484 . . . . . . . . . . 11 ((𝜑𝑦 = (𝑀 − 1)) → 𝑀 / 𝑗((1st ‘(1st𝑈)) ∘f + ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑈)) ∘f + ((((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))))
125112, 124eqtrd 2779 . . . . . . . . . 10 ((𝜑𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd𝑈), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑈)) ∘f + ((((2nd ‘(1st𝑈)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑈)) ∘f + ((((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))))
126 ovexd 7269 . . . . . . . . . 10 (𝜑 → ((1st ‘(1st𝑈)) ∘f + ((((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V)
12784, 125, 100, 126fvmptd 6846 . . . . . . . . 9 (𝜑 → (𝐹‘(𝑀 − 1)) = ((1st ‘(1st𝑈)) ∘f + ((((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))))
128127fveq1d 6740 . . . . . . . 8 (𝜑 → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1st ‘(1st𝑈)) ∘f + ((((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦))
129128adantr 484 . . . . . . 7 ((𝜑𝑦 ∈ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))) → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1st ‘(1st𝑈)) ∘f + ((((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦))
130 imassrn 5957 . . . . . . . . . 10 ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) ⊆ ran (2nd ‘(1st𝑈))
131 f1of 6682 . . . . . . . . . . . 12 ((2nd ‘(1st𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑈)):(1...𝑁)⟶(1...𝑁))
13231, 131syl 17 . . . . . . . . . . 11 (𝜑 → (2nd ‘(1st𝑈)):(1...𝑁)⟶(1...𝑁))
133132frnd 6574 . . . . . . . . . 10 (𝜑 → ran (2nd ‘(1st𝑈)) ⊆ (1...𝑁))
134130, 133sstrid 3928 . . . . . . . . 9 (𝜑 → ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) ⊆ (1...𝑁))
135134sselda 3917 . . . . . . . 8 ((𝜑𝑦 ∈ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))) → 𝑦 ∈ (1...𝑁))
136 xp1st 7814 . . . . . . . . . . . 12 ((1st𝑈) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑈)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
13725, 136syl 17 . . . . . . . . . . 11 (𝜑 → (1st ‘(1st𝑈)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
138 elmapfn 8569 . . . . . . . . . . 11 ((1st ‘(1st𝑈)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st𝑈)) Fn (1...𝑁))
139137, 138syl 17 . . . . . . . . . 10 (𝜑 → (1st ‘(1st𝑈)) Fn (1...𝑁))
140139adantr 484 . . . . . . . . 9 ((𝜑𝑦 ∈ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))) → (1st ‘(1st𝑈)) Fn (1...𝑁))
141 1ex 10858 . . . . . . . . . . . . . 14 1 ∈ V
142 fnconstg 6628 . . . . . . . . . . . . . 14 (1 ∈ V → (((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) Fn ((2nd ‘(1st𝑈)) “ (1...𝑀)))
143141, 142ax-mp 5 . . . . . . . . . . . . 13 (((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) Fn ((2nd ‘(1st𝑈)) “ (1...𝑀))
144 c0ex 10856 . . . . . . . . . . . . . 14 0 ∈ V
145 fnconstg 6628 . . . . . . . . . . . . . 14 (0 ∈ V → (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)))
146144, 145ax-mp 5 . . . . . . . . . . . . 13 (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))
147143, 146pm3.2i 474 . . . . . . . . . . . 12 ((((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) Fn ((2nd ‘(1st𝑈)) “ (1...𝑀)) ∧ (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)))
148 imain 6485 . . . . . . . . . . . . . 14 (Fun (2nd ‘(1st𝑈)) → ((2nd ‘(1st𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑈)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))))
14940, 148syl 17 . . . . . . . . . . . . 13 (𝜑 → ((2nd ‘(1st𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑈)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))))
15056imaeq2d 5946 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ((2nd ‘(1st𝑈)) “ ∅))
151 ima0 5962 . . . . . . . . . . . . . 14 ((2nd ‘(1st𝑈)) “ ∅) = ∅
152150, 151eqtrdi 2796 . . . . . . . . . . . . 13 (𝜑 → ((2nd ‘(1st𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ∅)
153149, 152eqtr3d 2781 . . . . . . . . . . . 12 (𝜑 → (((2nd ‘(1st𝑈)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))) = ∅)
154 fnun 6511 . . . . . . . . . . . 12 ((((((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) Fn ((2nd ‘(1st𝑈)) “ (1...𝑀)) ∧ (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))) ∧ (((2nd ‘(1st𝑈)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))) = ∅) → ((((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑈)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))))
155147, 153, 154sylancr 590 . . . . . . . . . . 11 (𝜑 → ((((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑈)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))))
156 imaundi 6030 . . . . . . . . . . . . 13 ((2nd ‘(1st𝑈)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑈)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)))
15746imaeq2d 5946 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑈)) “ (1...𝑁)) = ((2nd ‘(1st𝑈)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))))
158157, 35eqtr3d 2781 . . . . . . . . . . . . 13 (𝜑 → ((2nd ‘(1st𝑈)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (1...𝑁))
159156, 158eqtr3id 2794 . . . . . . . . . . . 12 (𝜑 → (((2nd ‘(1st𝑈)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))) = (1...𝑁))
160159fneq2d 6493 . . . . . . . . . . 11 (𝜑 → (((((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑈)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))) ↔ ((((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)))
161155, 160mpbid 235 . . . . . . . . . 10 (𝜑 → ((((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))
162161adantr 484 . . . . . . . . 9 ((𝜑𝑦 ∈ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))) → ((((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))
163 ovexd 7269 . . . . . . . . 9 ((𝜑𝑦 ∈ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))) → (1...𝑁) ∈ V)
164 inidm 4149 . . . . . . . . 9 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
165 eqidd 2740 . . . . . . . . 9 (((𝜑𝑦 ∈ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))) ∧ 𝑦 ∈ (1...𝑁)) → ((1st ‘(1st𝑈))‘𝑦) = ((1st ‘(1st𝑈))‘𝑦))
166 fvun2 6824 . . . . . . . . . . . . 13 (((((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) Fn ((2nd ‘(1st𝑈)) “ (1...𝑀)) ∧ (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) ∧ ((((2nd ‘(1st𝑈)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ 𝑦 ∈ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)))) → (((((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦))
167143, 146, 166mp3an12 1453 . . . . . . . . . . . 12 (((((2nd ‘(1st𝑈)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ 𝑦 ∈ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))) → (((((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦))
168153, 167sylan 583 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))) → (((((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦))
169144fvconst2 7040 . . . . . . . . . . . 12 (𝑦 ∈ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) → ((((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦) = 0)
170169adantl 485 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))) → ((((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦) = 0)
171168, 170eqtrd 2779 . . . . . . . . . 10 ((𝜑𝑦 ∈ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))) → (((((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = 0)
172171adantr 484 . . . . . . . . 9 (((𝜑𝑦 ∈ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))) ∧ 𝑦 ∈ (1...𝑁)) → (((((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = 0)
173140, 162, 163, 163, 164, 165, 172ofval 7500 . . . . . . . 8 (((𝜑𝑦 ∈ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))) ∧ 𝑦 ∈ (1...𝑁)) → (((1st ‘(1st𝑈)) ∘f + ((((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦) = (((1st ‘(1st𝑈))‘𝑦) + 0))
174135, 173mpdan 687 . . . . . . 7 ((𝜑𝑦 ∈ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))) → (((1st ‘(1st𝑈)) ∘f + ((((2nd ‘(1st𝑈)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦) = (((1st ‘(1st𝑈))‘𝑦) + 0))
175 elmapi 8553 . . . . . . . . . . . . 13 ((1st ‘(1st𝑈)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st𝑈)):(1...𝑁)⟶(0..^𝐾))
176137, 175syl 17 . . . . . . . . . . . 12 (𝜑 → (1st ‘(1st𝑈)):(1...𝑁)⟶(0..^𝐾))
177176ffvelrnda 6925 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (1...𝑁)) → ((1st ‘(1st𝑈))‘𝑦) ∈ (0..^𝐾))
178 elfzonn0 13316 . . . . . . . . . . 11 (((1st ‘(1st𝑈))‘𝑦) ∈ (0..^𝐾) → ((1st ‘(1st𝑈))‘𝑦) ∈ ℕ0)
179177, 178syl 17 . . . . . . . . . 10 ((𝜑𝑦 ∈ (1...𝑁)) → ((1st ‘(1st𝑈))‘𝑦) ∈ ℕ0)
180179nn0cnd 12181 . . . . . . . . 9 ((𝜑𝑦 ∈ (1...𝑁)) → ((1st ‘(1st𝑈))‘𝑦) ∈ ℂ)
181135, 180syldan 594 . . . . . . . 8 ((𝜑𝑦 ∈ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))) → ((1st ‘(1st𝑈))‘𝑦) ∈ ℂ)
182181addid1d 11061 . . . . . . 7 ((𝜑𝑦 ∈ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))) → (((1st ‘(1st𝑈))‘𝑦) + 0) = ((1st ‘(1st𝑈))‘𝑦))
183129, 174, 1823eqtrd 2783 . . . . . 6 ((𝜑𝑦 ∈ ((2nd ‘(1st𝑈)) “ ((𝑀 + 1)...𝑁))) → ((𝐹‘(𝑀 − 1))‘𝑦) = ((1st ‘(1st𝑈))‘𝑦))
18465, 183syldan 594 . . . . 5 ((𝜑 ∧ (𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd ‘(1st𝑈)) “ (1...𝑀)))) → ((𝐹‘(𝑀 − 1))‘𝑦) = ((1st ‘(1st𝑈))‘𝑦))
185 fveq2 6738 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
186185breq2d 5081 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
187186ifbid 4478 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
188187csbeq1d 3832 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑇if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
189 2fveq3 6743 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
190 2fveq3 6743 . . . . . . . . . . . . . . . . . . . . 21 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
191190imaeq1d 5945 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
192191xpeq1d 5597 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
193190imaeq1d 5945 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
194193xpeq1d 5597 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
195192, 194uneq12d 4094 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑇 → ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))
196189, 195oveq12d 7252 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑇 → ((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
197196csbeq2dv 3835 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑇if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
198188, 197eqtrd 2779 . . . . . . . . . . . . . . 15 (𝑡 = 𝑇if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
199198mpteq2dv 5167 . . . . . . . . . . . . . 14 (𝑡 = 𝑇 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
200199eqeq2d 2750 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
201200, 5elrab2 3620 . . . . . . . . . . . 12 (𝑇𝑆 ↔ (𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
202201simprbi 500 . . . . . . . . . . 11 (𝑇𝑆𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
2033, 202syl 17 . . . . . . . . . 10 (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
204 poimirlem11.3 . . . . . . . . . . . . . . . 16 (𝜑 → (2nd𝑇) = 0)
205 breq12 5074 . . . . . . . . . . . . . . . 16 ((𝑦 = (𝑀 − 1) ∧ (2nd𝑇) = 0) → (𝑦 < (2nd𝑇) ↔ (𝑀 − 1) < 0))
206204, 205sylan2 596 . . . . . . . . . . . . . . 15 ((𝑦 = (𝑀 − 1) ∧ 𝜑) → (𝑦 < (2nd𝑇) ↔ (𝑀 − 1) < 0))
207206ancoms 462 . . . . . . . . . . . . . 14 ((𝜑𝑦 = (𝑀 − 1)) → (𝑦 < (2nd𝑇) ↔ (𝑀 − 1) < 0))
208207, 93ifbieq2d 4481 . . . . . . . . . . . . 13 ((𝜑𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = if((𝑀 − 1) < 0, 𝑦, 𝑀))
209208, 110eqtrd 2779 . . . . . . . . . . . 12 ((𝜑𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = 𝑀)
210209csbeq1d 3832 . . . . . . . . . . 11 ((𝜑𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = 𝑀 / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
211113imaeq2d 5946 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑀 → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑀)))
212211xpeq1d 5597 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑀 → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}))
213117imaeq2d 5946 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑀 → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
214213xpeq1d 5597 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑀 → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))
215212, 214uneq12d 4094 . . . . . . . . . . . . . . 15 (𝑗 = 𝑀 → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))
216215oveq2d 7250 . . . . . . . . . . . . . 14 (𝑗 = 𝑀 → ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
217216adantl 485 . . . . . . . . . . . . 13 ((𝜑𝑗 = 𝑀) → ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
21844, 217csbied 3866 . . . . . . . . . . . 12 (𝜑𝑀 / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
219218adantr 484 . . . . . . . . . . 11 ((𝜑𝑦 = (𝑀 − 1)) → 𝑀 / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
220210, 219eqtrd 2779 . . . . . . . . . 10 ((𝜑𝑦 = (𝑀 − 1)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
221 ovexd 7269 . . . . . . . . . 10 (𝜑 → ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V)
222203, 220, 100, 221fvmptd 6846 . . . . . . . . 9 (𝜑 → (𝐹‘(𝑀 − 1)) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
223222fveq1d 6740 . . . . . . . 8 (𝜑 → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦))
224223adantr 484 . . . . . . 7 ((𝜑𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀))) → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦))
22519sselda 3917 . . . . . . . 8 ((𝜑𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀))) → 𝑦 ∈ (1...𝑁))
226 xp1st 7814 . . . . . . . . . . . 12 ((1st𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
2279, 226syl 17 . . . . . . . . . . 11 (𝜑 → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
228 elmapfn 8569 . . . . . . . . . . 11 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st𝑇)) Fn (1...𝑁))
229227, 228syl 17 . . . . . . . . . 10 (𝜑 → (1st ‘(1st𝑇)) Fn (1...𝑁))
230229adantr 484 . . . . . . . . 9 ((𝜑𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀))) → (1st ‘(1st𝑇)) Fn (1...𝑁))
231 fnconstg 6628 . . . . . . . . . . . . . 14 (1 ∈ V → (((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑀)))
232141, 231ax-mp 5 . . . . . . . . . . . . 13 (((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑀))
233 fnconstg 6628 . . . . . . . . . . . . . 14 (0 ∈ V → (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
234144, 233ax-mp 5 . . . . . . . . . . . . 13 (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))
235232, 234pm3.2i 474 . . . . . . . . . . . 12 ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑀)) ∧ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
236 dff1o3 6688 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun (2nd ‘(1st𝑇))))
237236simprbi 500 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun (2nd ‘(1st𝑇)))
23815, 237syl 17 . . . . . . . . . . . . . 14 (𝜑 → Fun (2nd ‘(1st𝑇)))
239 imain 6485 . . . . . . . . . . . . . 14 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
240238, 239syl 17 . . . . . . . . . . . . 13 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
24156imaeq2d 5946 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ((2nd ‘(1st𝑇)) “ ∅))
242 ima0 5962 . . . . . . . . . . . . . 14 ((2nd ‘(1st𝑇)) “ ∅) = ∅
243241, 242eqtrdi 2796 . . . . . . . . . . . . 13 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ∅)
244240, 243eqtr3d 2781 . . . . . . . . . . . 12 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅)
245 fnun 6511 . . . . . . . . . . . 12 ((((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑀)) ∧ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) ∧ (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
246235, 244, 245sylancr 590 . . . . . . . . . . 11 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))))
247 imaundi 6030 . . . . . . . . . . . . 13 ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)))
24846imaeq2d 5946 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))))
249 f1ofo 6689 . . . . . . . . . . . . . . . 16 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁))
25015, 249syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁))
251 foima 6659 . . . . . . . . . . . . . . 15 ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
252250, 251syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
253248, 252eqtr3d 2781 . . . . . . . . . . . . 13 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (1...𝑁))
254247, 253eqtr3id 2794 . . . . . . . . . . . 12 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = (1...𝑁))
255254fneq2d 6493 . . . . . . . . . . 11 (𝜑 → (((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...𝑀)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) ↔ ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)))
256246, 255mpbid 235 . . . . . . . . . 10 (𝜑 → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))
257256adantr 484 . . . . . . . . 9 ((𝜑𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀))) → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))
258 ovexd 7269 . . . . . . . . 9 ((𝜑𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀))) → (1...𝑁) ∈ V)
259 eqidd 2740 . . . . . . . . 9 (((𝜑𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀))) ∧ 𝑦 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑦) = ((1st ‘(1st𝑇))‘𝑦))
260 fvun1 6823 . . . . . . . . . . . . 13 (((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑀)) ∧ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) ∧ ((((2nd ‘(1st𝑇)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ 𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀)))) → (((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1})‘𝑦))
261232, 234, 260mp3an12 1453 . . . . . . . . . . . 12 (((((2nd ‘(1st𝑇)) “ (1...𝑀)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ 𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀))) → (((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1})‘𝑦))
262244, 261sylan 583 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀))) → (((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1})‘𝑦))
263141fvconst2 7040 . . . . . . . . . . . 12 (𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀)) → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1})‘𝑦) = 1)
264263adantl 485 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀))) → ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1})‘𝑦) = 1)
265262, 264eqtrd 2779 . . . . . . . . . 10 ((𝜑𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀))) → (((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = 1)
266265adantr 484 . . . . . . . . 9 (((𝜑𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀))) ∧ 𝑦 ∈ (1...𝑁)) → (((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = 1)
267230, 257, 258, 258, 164, 259, 266ofval 7500 . . . . . . . 8 (((𝜑𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀))) ∧ 𝑦 ∈ (1...𝑁)) → (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦) = (((1st ‘(1st𝑇))‘𝑦) + 1))
268225, 267mpdan 687 . . . . . . 7 ((𝜑𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀))) → (((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑀)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦) = (((1st ‘(1st𝑇))‘𝑦) + 1))
269224, 268eqtrd 2779 . . . . . 6 ((𝜑𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀))) → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1st ‘(1st𝑇))‘𝑦) + 1))
270269adantrr 717 . . . . 5 ((𝜑 ∧ (𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd ‘(1st𝑈)) “ (1...𝑀)))) → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1st ‘(1st𝑇))‘𝑦) + 1))
271 poimirlem22.1 . . . . . . . . 9 (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
27296, 5, 271, 20, 85poimirlem10 35560 . . . . . . . 8 (𝜑 → ((𝐹‘(𝑁 − 1)) ∘f − ((1...𝑁) × {1})) = (1st ‘(1st𝑈)))
27396, 5, 271, 3, 204poimirlem10 35560 . . . . . . . 8 (𝜑 → ((𝐹‘(𝑁 − 1)) ∘f − ((1...𝑁) × {1})) = (1st ‘(1st𝑇)))
274272, 273eqtr3d 2781 . . . . . . 7 (𝜑 → (1st ‘(1st𝑈)) = (1st ‘(1st𝑇)))
275274fveq1d 6740 . . . . . 6 (𝜑 → ((1st ‘(1st𝑈))‘𝑦) = ((1st ‘(1st𝑇))‘𝑦))
276275adantr 484 . . . . 5 ((𝜑 ∧ (𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd ‘(1st𝑈)) “ (1...𝑀)))) → ((1st ‘(1st𝑈))‘𝑦) = ((1st ‘(1st𝑇))‘𝑦))
277184, 270, 2763eqtr3d 2787 . . . 4 ((𝜑 ∧ (𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd ‘(1st𝑈)) “ (1...𝑀)))) → (((1st ‘(1st𝑇))‘𝑦) + 1) = ((1st ‘(1st𝑇))‘𝑦))
278 elmapi 8553 . . . . . . . . . . . 12 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
279227, 278syl 17 . . . . . . . . . . 11 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
280279ffvelrnda 6925 . . . . . . . . . 10 ((𝜑𝑦 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑦) ∈ (0..^𝐾))
281 elfzonn0 13316 . . . . . . . . . 10 (((1st ‘(1st𝑇))‘𝑦) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘𝑦) ∈ ℕ0)
282280, 281syl 17 . . . . . . . . 9 ((𝜑𝑦 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑦) ∈ ℕ0)
283282nn0red 12180 . . . . . . . 8 ((𝜑𝑦 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑦) ∈ ℝ)
284283ltp1d 11791 . . . . . . . 8 ((𝜑𝑦 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑦) < (((1st ‘(1st𝑇))‘𝑦) + 1))
285283, 284gtned 10996 . . . . . . 7 ((𝜑𝑦 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑦) + 1) ≠ ((1st ‘(1st𝑇))‘𝑦))
286225, 285syldan 594 . . . . . 6 ((𝜑𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀))) → (((1st ‘(1st𝑇))‘𝑦) + 1) ≠ ((1st ‘(1st𝑇))‘𝑦))
287286neneqd 2948 . . . . 5 ((𝜑𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀))) → ¬ (((1st ‘(1st𝑇))‘𝑦) + 1) = ((1st ‘(1st𝑇))‘𝑦))
288287adantrr 717 . . . 4 ((𝜑 ∧ (𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd ‘(1st𝑈)) “ (1...𝑀)))) → ¬ (((1st ‘(1st𝑇))‘𝑦) + 1) = ((1st ‘(1st𝑇))‘𝑦))
289277, 288pm2.65da 817 . . 3 (𝜑 → ¬ (𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd ‘(1st𝑈)) “ (1...𝑀))))
290 iman 405 . . 3 ((𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀)) → 𝑦 ∈ ((2nd ‘(1st𝑈)) “ (1...𝑀))) ↔ ¬ (𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2nd ‘(1st𝑈)) “ (1...𝑀))))
291289, 290sylibr 237 . 2 (𝜑 → (𝑦 ∈ ((2nd ‘(1st𝑇)) “ (1...𝑀)) → 𝑦 ∈ ((2nd ‘(1st𝑈)) “ (1...𝑀))))
292291ssrdv 3923 1 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑀)) ⊆ ((2nd ‘(1st𝑈)) “ (1...𝑀)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1543  wcel 2112  {cab 2716  wne 2943  {crab 3068  Vcvv 3423  csb 3828  cdif 3880  cun 3881  cin 3882  wss 3883  c0 4253  ifcif 4455  {csn 4557   class class class wbr 5069  cmpt 5151   × cxp 5566  ccnv 5567  ran crn 5569  cima 5571  Fun wfun 6394   Fn wfn 6395  wf 6396  ontowfo 6398  1-1-ontowf1o 6399  cfv 6400  (class class class)co 7234  f cof 7488  1st c1st 7780  2nd c2nd 7781  m cmap 8531  cc 10756  0cc0 10758  1c1 10759   + caddc 10761   < clt 10896  cle 10897  cmin 11091  cn 11859  0cn0 12119  cz 12205  ...cfz 13124  ..^cfzo 13267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-rep 5195  ax-sep 5208  ax-nul 5215  ax-pow 5274  ax-pr 5338  ax-un 7544  ax-cnex 10814  ax-resscn 10815  ax-1cn 10816  ax-icn 10817  ax-addcl 10818  ax-addrcl 10819  ax-mulcl 10820  ax-mulrcl 10821  ax-mulcom 10822  ax-addass 10823  ax-mulass 10824  ax-distr 10825  ax-i2m1 10826  ax-1ne0 10827  ax-1rid 10828  ax-rnegex 10829  ax-rrecex 10830  ax-cnre 10831  ax-pre-lttri 10832  ax-pre-lttrn 10833  ax-pre-ltadd 10834  ax-pre-mulgt0 10835
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3425  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4456  df-pw 4531  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4836  df-iun 4922  df-br 5070  df-opab 5132  df-mpt 5152  df-tr 5178  df-id 5471  df-eprel 5477  df-po 5485  df-so 5486  df-fr 5526  df-we 5528  df-xp 5574  df-rel 5575  df-cnv 5576  df-co 5577  df-dm 5578  df-rn 5579  df-res 5580  df-ima 5581  df-pred 6178  df-ord 6236  df-on 6237  df-lim 6238  df-suc 6239  df-iota 6358  df-fun 6402  df-fn 6403  df-f 6404  df-f1 6405  df-fo 6406  df-f1o 6407  df-fv 6408  df-riota 7191  df-ov 7237  df-oprab 7238  df-mpo 7239  df-of 7490  df-om 7666  df-1st 7782  df-2nd 7783  df-wrecs 8070  df-recs 8131  df-rdg 8169  df-er 8414  df-map 8533  df-en 8650  df-dom 8651  df-sdom 8652  df-pnf 10898  df-mnf 10899  df-xr 10900  df-ltxr 10901  df-le 10902  df-sub 11093  df-neg 11094  df-nn 11860  df-n0 12120  df-z 12206  df-uz 12468  df-fz 13125  df-fzo 13268
This theorem is referenced by:  poimirlem13  35563
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