poimirlem11 - Mathbox for Brendan Leahy

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Theorem poimirlem11 37963

Description: Lemma for poimir 37985 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	⊢ (𝜑 → 𝑁 ∈ ℕ)
poimirlem22.s	⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
poimirlem22.1	⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑_m (1...𝑁)))
poimirlem12.2	⊢ (𝜑 → 𝑇 ∈ 𝑆)
poimirlem11.3	⊢ (𝜑 → (2^nd ‘𝑇) = 0)
poimirlem11.4	⊢ (𝜑 → 𝑈 ∈ 𝑆)
poimirlem11.5	⊢ (𝜑 → (2^nd ‘𝑈) = 0)
poimirlem11.6	⊢ (𝜑 → 𝑀 ∈ (1...𝑁))

**Assertion**
Ref	Expression
poimirlem11	⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ⊆ ((2^nd ‘(1^st ‘𝑈)) “ (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**
Step	Hyp	Ref	Expression
1		eldif 3900	. . . . . . 7 ⊢ (𝑦 ∈ (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∖ ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀))) ↔ (𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀))))
2		imassrn 6028	. . . . . . . . . . . 12 ⊢ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ⊆ ran (2^nd ‘(1^st ‘𝑇))
3		poimirlem12.2	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
4		elrabi 3631	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑇 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
5		poimirlem22.s	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
6	4, 5	eleq2s 2855	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
7	3, 6	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
8		xp1st 7965	. . . . . . . . . . . . . . . . 17 ⊢ (𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1^st ‘𝑇) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
9	7, 8	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1^st ‘𝑇) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
10		xp2nd 7966	. . . . . . . . . . . . . . . 16 ⊢ ((1^st ‘𝑇) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2^nd ‘(1^st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
11	9, 10	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
12		fvex 6845	. . . . . . . . . . . . . . . 16 ⊢ (2^nd ‘(1^st ‘𝑇)) ∈ V
13		f1oeq1 6760	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (2^nd ‘(1^st ‘𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
14	12, 13	elab 3623	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘(1^st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
15	11, 14	sylib 218	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
16		f1of 6772	. . . . . . . . . . . . . 14 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)⟶(1...𝑁))
17	15, 16	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)⟶(1...𝑁))
18	17	frnd 6668	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (2^nd ‘(1^st ‘𝑇)) ⊆ (1...𝑁))
19	2, 18	sstrid 3934	. . . . . . . . . . 11 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ⊆ (1...𝑁))
20		poimirlem11.4	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
21		elrabi 3631	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑈 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
22	21, 5	eleq2s 2855	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
23	20, 22	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑈 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
24		xp1st 7965	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1^st ‘𝑈) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
25	23, 24	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1^st ‘𝑈) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
26		xp2nd 7966	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘𝑈) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2^nd ‘(1^st ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
27	25, 26	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
28		fvex 6845	. . . . . . . . . . . . . . 15 ⊢ (2^nd ‘(1^st ‘𝑈)) ∈ V
29		f1oeq1 6760	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (2^nd ‘(1^st ‘𝑈)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁)))
30	28, 29	elab 3623	. . . . . . . . . . . . . 14 ⊢ ((2^nd ‘(1^st ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))
31	27, 30	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))
32		f1ofo 6779	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑈)):(1...𝑁)–onto→(1...𝑁))
33	31, 32	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑈)):(1...𝑁)–onto→(1...𝑁))
34		foima 6749	. . . . . . . . . . . 12 ⊢ ((2^nd ‘(1^st ‘𝑈)):(1...𝑁)–onto→(1...𝑁) → ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑁)) = (1...𝑁))
35	33, 34	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑁)) = (1...𝑁))
36	19, 35	sseqtrrd 3960	. . . . . . . . . 10 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ⊆ ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑁)))
37	36	ssdifd 4086	. . . . . . . . 9 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∖ ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀))) ⊆ (((2^nd ‘(1^st ‘𝑈)) “ (1...𝑁)) ∖ ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀))))
38		dff1o3 6778	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2^nd ‘(1^st ‘𝑈)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ^◡(2^nd ‘(1^st ‘𝑈))))
39	38	simprbi 497	. . . . . . . . . . . 12 ⊢ ((2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^◡(2^nd ‘(1^st ‘𝑈)))
40	31, 39	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → Fun ^◡(2^nd ‘(1^st ‘𝑈)))
41		imadif 6574	. . . . . . . . . . 11 ⊢ (Fun ^◡(2^nd ‘(1^st ‘𝑈)) → ((2^nd ‘(1^st ‘𝑈)) “ ((1...𝑁) ∖ (1...𝑀))) = (((2^nd ‘(1^st ‘𝑈)) “ (1...𝑁)) ∖ ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀))))
42	40, 41	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) “ ((1...𝑁) ∖ (1...𝑀))) = (((2^nd ‘(1^st ‘𝑈)) “ (1...𝑁)) ∖ ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀))))
43		difun2 4422	. . . . . . . . . . . 12 ⊢ ((((𝑀 + 1)...𝑁) ∪ (1...𝑀)) ∖ (1...𝑀)) = (((𝑀 + 1)...𝑁) ∖ (1...𝑀))
44		poimirlem11.6	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ (1...𝑁))
45		fzsplit 13493	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ (1...𝑁) → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))
46	44, 45	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))
47		uncom 4099	. . . . . . . . . . . . . 14 ⊢ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) = (((𝑀 + 1)...𝑁) ∪ (1...𝑀))
48	46, 47	eqtrdi 2788	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1...𝑁) = (((𝑀 + 1)...𝑁) ∪ (1...𝑀)))
49	48	difeq1d 4066	. . . . . . . . . . . 12 ⊢ (𝜑 → ((1...𝑁) ∖ (1...𝑀)) = ((((𝑀 + 1)...𝑁) ∪ (1...𝑀)) ∖ (1...𝑀)))
50		incom 4150	. . . . . . . . . . . . . 14 ⊢ (((𝑀 + 1)...𝑁) ∩ (1...𝑀)) = ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))
51		elfznn 13496	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ (1...𝑁) → 𝑀 ∈ ℕ)
52	44, 51	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ ℕ)
53	52	nnred 12178	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ ℝ)
54	53	ltp1d 12075	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 < (𝑀 + 1))
55		fzdisj 13494	. . . . . . . . . . . . . . 15 ⊢ (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
56	54, 55	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
57	50, 56	eqtrid 2784	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝑀 + 1)...𝑁) ∩ (1...𝑀)) = ∅)
58		disj3 4395	. . . . . . . . . . . . 13 ⊢ ((((𝑀 + 1)...𝑁) ∩ (1...𝑀)) = ∅ ↔ ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...𝑁) ∖ (1...𝑀)))
59	57, 58	sylib 218	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...𝑁) ∖ (1...𝑀)))
60	43, 49, 59	3eqtr4a 2798	. . . . . . . . . . 11 ⊢ (𝜑 → ((1...𝑁) ∖ (1...𝑀)) = ((𝑀 + 1)...𝑁))
61	60	imaeq2d 6017	. . . . . . . . . 10 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) “ ((1...𝑁) ∖ (1...𝑀))) = ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)))
62	42, 61	eqtr3d 2774	. . . . . . . . 9 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑈)) “ (1...𝑁)) ∖ ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀))) = ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)))
63	37, 62	sseqtrd 3959	. . . . . . . 8 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∖ ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀))) ⊆ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)))
64	63	sselda 3922	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∖ ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)))) → 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)))
65	1, 64	sylan2br 596	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)))) → 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)))
66		fveq2 6832	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑈 → (2^nd ‘𝑡) = (2^nd ‘𝑈))
67	66	breq2d 5098	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑈 → (𝑦 < (2^nd ‘𝑡) ↔ 𝑦 < (2^nd ‘𝑈)))
68	67	ifbid 4491	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑈 → if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)))
69	68	csbeq1d 3842	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑈 → ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
70		2fveq3 6837	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑈 → (1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘𝑈)))
71		2fveq3 6837	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑈 → (2^nd ‘(1^st ‘𝑡)) = (2^nd ‘(1^st ‘𝑈)))
72	71	imaeq1d 6016	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑈 → ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)))
73	72	xpeq1d 5651	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑈 → (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}))
74	71	imaeq1d 6016	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑈 → ((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)))
75	74	xpeq1d 5651	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑈 → (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))
76	73, 75	uneq12d 4110	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑈 → ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))
77	70, 76	oveq12d 7376	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑈 → ((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))
78	77	csbeq2dv 3845	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑈 → ⦋if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))
79	69, 78	eqtrd 2772	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑈 → ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))
80	79	mpteq2dv 5180	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑈 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))))
81	80	eqeq2d 2748	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑈 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
82	81, 5	elrab2 3638	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ 𝑆 ↔ (𝑈 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
83	82	simprbi 497	. . . . . . . . . . 11 ⊢ (𝑈 ∈ 𝑆 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))))
84	20, 83	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})))))
85		poimirlem11.5	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2^nd ‘𝑈) = 0)
86		breq12 5091	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 = (𝑀 − 1) ∧ (2^nd ‘𝑈) = 0) → (𝑦 < (2^nd ‘𝑈) ↔ (𝑀 − 1) < 0))
87	85, 86	sylan2 594	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 = (𝑀 − 1) ∧ 𝜑) → (𝑦 < (2^nd ‘𝑈) ↔ (𝑀 − 1) < 0))
88	87	ancoms 458	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → (𝑦 < (2^nd ‘𝑈) ↔ (𝑀 − 1) < 0))
89		oveq1 7365	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑀 − 1) → (𝑦 + 1) = ((𝑀 − 1) + 1))
90	52	nncnd 12179	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ ℂ)
91		npcan1 11564	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℂ → ((𝑀 − 1) + 1) = 𝑀)
92	90, 91	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀)
93	89, 92	sylan9eqr 2794	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → (𝑦 + 1) = 𝑀)
94	88, 93	ifbieq2d 4494	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)) = if((𝑀 − 1) < 0, 𝑦, 𝑀))
95	52	nnzd 12539	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ ℤ)
96		poimir.0	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑁 ∈ ℕ)
97	96	nnzd 12539	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℤ)
98		elfzm1b 13545	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∈ (1...𝑁) ↔ (𝑀 − 1) ∈ (0...(𝑁 − 1))))
99	95, 97, 98	syl2anc 585	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 ∈ (1...𝑁) ↔ (𝑀 − 1) ∈ (0...(𝑁 − 1))))
100	44, 99	mpbid 232	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 − 1) ∈ (0...(𝑁 − 1)))
101		elfzle1 13470	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 − 1) ∈ (0...(𝑁 − 1)) → 0 ≤ (𝑀 − 1))
102	100, 101	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ≤ (𝑀 − 1))
103		0red 11136	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 ∈ ℝ)
104		nnm1nn0 12467	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ₀)
105	52, 104	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 − 1) ∈ ℕ₀)
106	105	nn0red 12488	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 − 1) ∈ ℝ)
107	103, 106	lenltd 11281	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0 ≤ (𝑀 − 1) ↔ ¬ (𝑀 − 1) < 0))
108	102, 107	mpbid 232	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ (𝑀 − 1) < 0)
109	108	iffalsed 4478	. . . . . . . . . . . . . 14 ⊢ (𝜑 → if((𝑀 − 1) < 0, 𝑦, 𝑀) = 𝑀)
110	109	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if((𝑀 − 1) < 0, 𝑦, 𝑀) = 𝑀)
111	94, 110	eqtrd 2772	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)) = 𝑀)
112	111	csbeq1d 3842	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑀 / 𝑗⦌((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))))
113		oveq2 7366	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑀 → (1...𝑗) = (1...𝑀))
114	113	imaeq2d 6017	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑀 → ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)))
115	114	xpeq1d 5651	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑀 → (((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}))
116		oveq1 7365	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑀 → (𝑗 + 1) = (𝑀 + 1))
117	116	oveq1d 7373	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑀 → ((𝑗 + 1)...𝑁) = ((𝑀 + 1)...𝑁))
118	117	imaeq2d 6017	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑀 → ((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)))
119	118	xpeq1d 5651	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑀 → (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))
120	115, 119	uneq12d 4110	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑀 → ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))
121	120	oveq2d 7374	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑀 → ((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))))
122	121	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 = 𝑀) → ((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))))
123	44, 122	csbied 3874	. . . . . . . . . . . 12 ⊢ (𝜑 → ⦋𝑀 / 𝑗⦌((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))))
124	123	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋𝑀 / 𝑗⦌((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))))
125	112, 124	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋if(𝑦 < (2^nd ‘𝑈), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))))
126		ovexd 7393	. . . . . . . . . 10 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V)
127	84, 125, 100, 126	fvmptd 6947	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘(𝑀 − 1)) = ((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))))
128	127	fveq1d 6834	. . . . . . . 8 ⊢ (𝜑 → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦))
129	128	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦))
130		imassrn 6028	. . . . . . . . . 10 ⊢ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) ⊆ ran (2^nd ‘(1^st ‘𝑈))
131		f1of 6772	. . . . . . . . . . . 12 ⊢ ((2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑈)):(1...𝑁)⟶(1...𝑁))
132	31, 131	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑈)):(1...𝑁)⟶(1...𝑁))
133	132	frnd 6668	. . . . . . . . . 10 ⊢ (𝜑 → ran (2^nd ‘(1^st ‘𝑈)) ⊆ (1...𝑁))
134	130, 133	sstrid 3934	. . . . . . . . 9 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) ⊆ (1...𝑁))
135	134	sselda 3922	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → 𝑦 ∈ (1...𝑁))
136		xp1st 7965	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝑈) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1^st ‘(1^st ‘𝑈)) ∈ ((0..^𝐾) ↑_m (1...𝑁)))
137	25, 136	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑈)) ∈ ((0..^𝐾) ↑_m (1...𝑁)))
138		elmapfn 8803	. . . . . . . . . . 11 ⊢ ((1^st ‘(1^st ‘𝑈)) ∈ ((0..^𝐾) ↑_m (1...𝑁)) → (1^st ‘(1^st ‘𝑈)) Fn (1...𝑁))
139	137, 138	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑈)) Fn (1...𝑁))
140	139	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (1^st ‘(1^st ‘𝑈)) Fn (1...𝑁))
141		1ex 11129	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
142		fnconstg 6720	. . . . . . . . . . . . . 14 ⊢ (1 ∈ V → (((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)))
143	141, 142	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀))
144		c0ex 11127	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
145		fnconstg 6720	. . . . . . . . . . . . . 14 ⊢ (0 ∈ V → (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)))
146	144, 145	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))
147	143, 146	pm3.2i 470	. . . . . . . . . . . 12 ⊢ ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) ∧ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)))
148		imain 6575	. . . . . . . . . . . . . 14 ⊢ (Fun ^◡(2^nd ‘(1^st ‘𝑈)) → ((2^nd ‘(1^st ‘𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) ∩ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))))
149	40, 148	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) ∩ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))))
150	56	imaeq2d 6017	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ((2^nd ‘(1^st ‘𝑈)) “ ∅))
151		ima0 6034	. . . . . . . . . . . . . 14 ⊢ ((2^nd ‘(1^st ‘𝑈)) “ ∅) = ∅
152	150, 151	eqtrdi 2788	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ∅)
153	149, 152	eqtr3d 2774	. . . . . . . . . . . 12 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) ∩ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = ∅)
154		fnun 6604	. . . . . . . . . . . 12 ⊢ ((((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) ∧ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) ∧ (((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) ∩ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = ∅) → ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) ∪ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))))
155	147, 153, 154	sylancr 588	. . . . . . . . . . 11 ⊢ (𝜑 → ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) ∪ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))))
156		imaundi 6105	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘(1^st ‘𝑈)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) ∪ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)))
157	46	imaeq2d 6017	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑁)) = ((2^nd ‘(1^st ‘𝑈)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))))
158	157, 35	eqtr3d 2774	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (1...𝑁))
159	156, 158	eqtr3id 2786	. . . . . . . . . . . 12 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) ∪ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = (1...𝑁))
160	159	fneq2d 6584	. . . . . . . . . . 11 ⊢ (𝜑 → (((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) ∪ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) ↔ ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)))
161	155, 160	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))
162	161	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))
163		ovexd 7393	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (1...𝑁) ∈ V)
164		inidm 4168	. . . . . . . . 9 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
165		eqidd 2738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) ∧ 𝑦 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑈))‘𝑦) = ((1^st ‘(1^st ‘𝑈))‘𝑦))
166		fvun2 6924	. . . . . . . . . . . . 13 ⊢ (((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) Fn ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) ∧ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) ∧ ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) ∩ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)))) → (((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦))
167	143, 146, 166	mp3an12 1454	. . . . . . . . . . . 12 ⊢ (((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) ∩ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦))
168	153, 167	sylan 581	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦))
169	144	fvconst2 7150	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) → ((((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦) = 0)
170	169	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})‘𝑦) = 0)
171	168, 170	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = 0)
172	171	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) ∧ 𝑦 ∈ (1...𝑁)) → (((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = 0)
173	140, 162, 163, 163, 164, 165, 172	ofval 7633	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) ∧ 𝑦 ∈ (1...𝑁)) → (((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦) = (((1^st ‘(1^st ‘𝑈))‘𝑦) + 0))
174	135, 173	mpdan 688	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((1^st ‘(1^st ‘𝑈)) ∘_f + ((((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦) = (((1^st ‘(1^st ‘𝑈))‘𝑦) + 0))
175		elmapi 8787	. . . . . . . . . . . . 13 ⊢ ((1^st ‘(1^st ‘𝑈)) ∈ ((0..^𝐾) ↑_m (1...𝑁)) → (1^st ‘(1^st ‘𝑈)):(1...𝑁)⟶(0..^𝐾))
176	137, 175	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑈)):(1...𝑁)⟶(0..^𝐾))
177	176	ffvelcdmda 7028	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑈))‘𝑦) ∈ (0..^𝐾))
178		elfzonn0 13651	. . . . . . . . . . 11 ⊢ (((1^st ‘(1^st ‘𝑈))‘𝑦) ∈ (0..^𝐾) → ((1^st ‘(1^st ‘𝑈))‘𝑦) ∈ ℕ₀)
179	177, 178	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑈))‘𝑦) ∈ ℕ₀)
180	179	nn0cnd 12489	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑈))‘𝑦) ∈ ℂ)
181	135, 180	syldan 592	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((1^st ‘(1^st ‘𝑈))‘𝑦) ∈ ℂ)
182	181	addridd 11335	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → (((1^st ‘(1^st ‘𝑈))‘𝑦) + 0) = ((1^st ‘(1^st ‘𝑈))‘𝑦))
183	129, 174, 182	3eqtrd 2776	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ ((𝑀 + 1)...𝑁))) → ((𝐹‘(𝑀 − 1))‘𝑦) = ((1^st ‘(1^st ‘𝑈))‘𝑦))
184	65, 183	syldan 592	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)))) → ((𝐹‘(𝑀 − 1))‘𝑦) = ((1^st ‘(1^st ‘𝑈))‘𝑦))
185		fveq2 6832	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑇 → (2^nd ‘𝑡) = (2^nd ‘𝑇))
186	185	breq2d 5098	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑇 → (𝑦 < (2^nd ‘𝑡) ↔ 𝑦 < (2^nd ‘𝑇)))
187	186	ifbid 4491	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)))
188	187	csbeq1d 3842	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
189		2fveq3 6837	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑇 → (1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘𝑇)))
190		2fveq3 6837	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑇 → (2^nd ‘(1^st ‘𝑡)) = (2^nd ‘(1^st ‘𝑇)))
191	190	imaeq1d 6016	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)))
192	191	xpeq1d 5651	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑇 → (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}))
193	190	imaeq1d 6016	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
194	193	xpeq1d 5651	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑇 → (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
195	192, 194	uneq12d 4110	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑇 → ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))
196	189, 195	oveq12d 7376	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑇 → ((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
197	196	csbeq2dv 3845	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
198	188, 197	eqtrd 2772	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
199	198	mpteq2dv 5180	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑇 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
200	199	eqeq2d 2748	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑇 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
201	200, 5	elrab2 3638	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ 𝑆 ↔ (𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
202	201	simprbi 497	. . . . . . . . . . 11 ⊢ (𝑇 ∈ 𝑆 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
203	3, 202	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
204		poimirlem11.3	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2^nd ‘𝑇) = 0)
205		breq12 5091	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 = (𝑀 − 1) ∧ (2^nd ‘𝑇) = 0) → (𝑦 < (2^nd ‘𝑇) ↔ (𝑀 − 1) < 0))
206	204, 205	sylan2 594	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 = (𝑀 − 1) ∧ 𝜑) → (𝑦 < (2^nd ‘𝑇) ↔ (𝑀 − 1) < 0))
207	206	ancoms 458	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → (𝑦 < (2^nd ‘𝑇) ↔ (𝑀 − 1) < 0))
208	207, 93	ifbieq2d 4494	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) = if((𝑀 − 1) < 0, 𝑦, 𝑀))
209	208, 110	eqtrd 2772	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) = 𝑀)
210	209	csbeq1d 3842	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑀 / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
211	113	imaeq2d 6017	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑀 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)))
212	211	xpeq1d 5651	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑀 → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}))
213	117	imaeq2d 6017	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑀 → ((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))
214	213	xpeq1d 5651	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑀 → (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))
215	212, 214	uneq12d 4110	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑀 → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))
216	215	oveq2d 7374	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑀 → ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
217	216	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 = 𝑀) → ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
218	44, 217	csbied 3874	. . . . . . . . . . . 12 ⊢ (𝜑 → ⦋𝑀 / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
219	218	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋𝑀 / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
220	210, 219	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
221		ovexd 7393	. . . . . . . . . 10 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))) ∈ V)
222	203, 220, 100, 221	fvmptd 6947	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘(𝑀 − 1)) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))))
223	222	fveq1d 6834	. . . . . . . 8 ⊢ (𝜑 → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦))
224	223	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀))) → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦))
225	19	sselda 3922	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀))) → 𝑦 ∈ (1...𝑁))
226		xp1st 7965	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝑇) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_m (1...𝑁)))
227	9, 226	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_m (1...𝑁)))
228		elmapfn 8803	. . . . . . . . . . 11 ⊢ ((1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_m (1...𝑁)) → (1^st ‘(1^st ‘𝑇)) Fn (1...𝑁))
229	227, 228	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)) Fn (1...𝑁))
230	229	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀))) → (1^st ‘(1^st ‘𝑇)) Fn (1...𝑁))
231		fnconstg 6720	. . . . . . . . . . . . . 14 ⊢ (1 ∈ V → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)))
232	141, 231	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀))
233		fnconstg 6720	. . . . . . . . . . . . . 14 ⊢ (0 ∈ V → (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))
234	144, 233	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))
235	232, 234	pm3.2i 470	. . . . . . . . . . . 12 ⊢ ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∧ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))
236		dff1o3 6778	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ^◡(2^nd ‘(1^st ‘𝑇))))
237	236	simprbi 497	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^◡(2^nd ‘(1^st ‘𝑇)))
238	15, 237	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Fun ^◡(2^nd ‘(1^st ‘𝑇)))
239		imain 6575	. . . . . . . . . . . . . 14 ⊢ (Fun ^◡(2^nd ‘(1^st ‘𝑇)) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∩ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))))
240	238, 239	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∩ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))))
241	56	imaeq2d 6017	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ((2^nd ‘(1^st ‘𝑇)) “ ∅))
242		ima0 6034	. . . . . . . . . . . . . 14 ⊢ ((2^nd ‘(1^st ‘𝑇)) “ ∅) = ∅
243	241, 242	eqtrdi 2788	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑀) ∩ ((𝑀 + 1)...𝑁))) = ∅)
244	240, 243	eqtr3d 2774	. . . . . . . . . . . 12 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∩ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅)
245		fnun 6604	. . . . . . . . . . . 12 ⊢ ((((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∧ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) ∧ (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∩ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))))
246	235, 244, 245	sylancr 588	. . . . . . . . . . 11 ⊢ (𝜑 → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))))
247		imaundi 6105	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)))
248	46	imaeq2d 6017	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))))
249		f1ofo 6779	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–onto→(1...𝑁))
250	15, 249	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–onto→(1...𝑁))
251		foima 6749	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
252	250, 251	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
253	248, 252	eqtr3d 2774	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) = (1...𝑁))
254	247, 253	eqtr3id 2786	. . . . . . . . . . . 12 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = (1...𝑁))
255	254	fneq2d 6584	. . . . . . . . . . 11 ⊢ (𝜑 → (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∪ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) ↔ ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁)))
256	246, 255	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))
257	256	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀))) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})) Fn (1...𝑁))
258		ovexd 7393	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀))) → (1...𝑁) ∈ V)
259		eqidd 2738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀))) ∧ 𝑦 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘𝑦) = ((1^st ‘(1^st ‘𝑇))‘𝑦))
260		fvun1 6923	. . . . . . . . . . . . 13 ⊢ (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∧ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) ∧ ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∩ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)))) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1})‘𝑦))
261	232, 234, 260	mp3an12 1454	. . . . . . . . . . . 12 ⊢ (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∩ ((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁))) = ∅ ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀))) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1})‘𝑦))
262	244, 261	sylan 581	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀))) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1})‘𝑦))
263	141	fvconst2 7150	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1})‘𝑦) = 1)
264	263	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀))) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1})‘𝑦) = 1)
265	262, 264	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀))) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = 1)
266	265	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀))) ∧ 𝑦 ∈ (1...𝑁)) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0}))‘𝑦) = 1)
267	230, 257, 258, 258, 164, 259, 266	ofval 7633	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀))) ∧ 𝑦 ∈ (1...𝑁)) → (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦) = (((1^st ‘(1^st ‘𝑇))‘𝑦) + 1))
268	225, 267	mpdan 688	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀))) → (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑀 + 1)...𝑁)) × {0})))‘𝑦) = (((1^st ‘(1^st ‘𝑇))‘𝑦) + 1))
269	224, 268	eqtrd 2772	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀))) → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1^st ‘(1^st ‘𝑇))‘𝑦) + 1))
270	269	adantrr 718	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)))) → ((𝐹‘(𝑀 − 1))‘𝑦) = (((1^st ‘(1^st ‘𝑇))‘𝑦) + 1))
271		poimirlem22.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑_m (1...𝑁)))
272	96, 5, 271, 20, 85	poimirlem10 37962	. . . . . . . 8 ⊢ (𝜑 → ((𝐹‘(𝑁 − 1)) ∘_f − ((1...𝑁) × {1})) = (1^st ‘(1^st ‘𝑈)))
273	96, 5, 271, 3, 204	poimirlem10 37962	. . . . . . . 8 ⊢ (𝜑 → ((𝐹‘(𝑁 − 1)) ∘_f − ((1...𝑁) × {1})) = (1^st ‘(1^st ‘𝑇)))
274	272, 273	eqtr3d 2774	. . . . . . 7 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑈)) = (1^st ‘(1^st ‘𝑇)))
275	274	fveq1d 6834	. . . . . 6 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑈))‘𝑦) = ((1^st ‘(1^st ‘𝑇))‘𝑦))
276	275	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)))) → ((1^st ‘(1^st ‘𝑈))‘𝑦) = ((1^st ‘(1^st ‘𝑇))‘𝑦))
277	184, 270, 276	3eqtr3d 2780	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)))) → (((1^st ‘(1^st ‘𝑇))‘𝑦) + 1) = ((1^st ‘(1^st ‘𝑇))‘𝑦))
278		elmapi 8787	. . . . . . . . . . . 12 ⊢ ((1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_m (1...𝑁)) → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
279	227, 278	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
280	279	ffvelcdmda 7028	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘𝑦) ∈ (0..^𝐾))
281		elfzonn0 13651	. . . . . . . . . 10 ⊢ (((1^st ‘(1^st ‘𝑇))‘𝑦) ∈ (0..^𝐾) → ((1^st ‘(1^st ‘𝑇))‘𝑦) ∈ ℕ₀)
282	280, 281	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘𝑦) ∈ ℕ₀)
283	282	nn0red 12488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘𝑦) ∈ ℝ)
284	283	ltp1d 12075	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘𝑦) < (((1^st ‘(1^st ‘𝑇))‘𝑦) + 1))
285	283, 284	gtned 11270	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...𝑁)) → (((1^st ‘(1^st ‘𝑇))‘𝑦) + 1) ≠ ((1^st ‘(1^st ‘𝑇))‘𝑦))
286	225, 285	syldan 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀))) → (((1^st ‘(1^st ‘𝑇))‘𝑦) + 1) ≠ ((1^st ‘(1^st ‘𝑇))‘𝑦))
287	286	neneqd 2938	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀))) → ¬ (((1^st ‘(1^st ‘𝑇))‘𝑦) + 1) = ((1^st ‘(1^st ‘𝑇))‘𝑦))
288	287	adantrr 718	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)))) → ¬ (((1^st ‘(1^st ‘𝑇))‘𝑦) + 1) = ((1^st ‘(1^st ‘𝑇))‘𝑦))
289	277, 288	pm2.65da 817	. . 3 ⊢ (𝜑 → ¬ (𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀))))
290		iman 401	. . 3 ⊢ ((𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) → 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀))) ↔ ¬ (𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ∧ ¬ 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀))))
291	289, 290	sylibr 234	. 2 ⊢ (𝜑 → (𝑦 ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) → 𝑦 ∈ ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀))))
292	291	ssrdv 3928	1 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑀)) ⊆ ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑀)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 {cab 2715 ≠ wne 2933 {crab 3390 Vcvv 3430 ⦋csb 3838 ∖ cdif 3887 ∪ cun 3888 ∩ cin 3889 ⊆ wss 3890 ∅c0 4274 ifcif 4467 {csn 4568 class class class wbr 5086 ↦ cmpt 5167 × cxp 5620 ^◡ccnv 5621 ran crn 5623 “ cima 5625 Fun wfun 6484 Fn wfn 6485 ⟶wf 6486 –onto→wfo 6488 –1-1-onto→wf1o 6489 ‘cfv 6490 (class class class)co 7358 ∘_f cof 7620 1^st c1st 7931 2^nd c2nd 7932 ↑_m cmap 8764 ℂcc 11025 0cc0 11027 1c1 11028 + caddc 11030 < clt 11168 ≤ cle 11169 − cmin 11366 ℕcn 12163 ℕ₀cn0 12426 ℤcz 12513 ...cfz 13450 ..^cfzo 13597

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-er 8634 df-map 8766 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12164 df-n0 12427 df-z 12514 df-uz 12778 df-fz 13451 df-fzo 13598

This theorem is referenced by: poimirlem13 37965

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