poimirlem3 - Mathbox for Brendan Leahy

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Theorem poimirlem3 38197

Description: Lemma for poimir 38227 to add an interior point to an admissible face on the back face of the cube. (Contributed by Brendan Leahy, 21-Aug-2020.)

**Hypotheses**
Ref	Expression
poimir.0	⊢ (𝜑 → 𝑁 ∈ ℕ)
poimirlem4.1	⊢ (𝜑 → 𝐾 ∈ ℕ)
poimirlem4.2	⊢ (𝜑 → 𝑀 ∈ ℕ₀)
poimirlem4.3	⊢ (𝜑 → 𝑀 < 𝑁)
poimirlem3.4	⊢ (𝜑 → 𝑇:(1...𝑀)⟶(0..^𝐾))
poimirlem3.5	⊢ (𝜑 → 𝑈:(1...𝑀)–1-1-onto→(1...𝑀))

**Assertion**
Ref	Expression
poimirlem3	⊢ (𝜑 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 → (⟨(𝑇 ∪ {⟨(𝑀 + 1), 0⟩}), (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ (((0..^𝐾) ↑_m (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_f + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1)))))

Distinct variable groups: 𝑓,𝑖,𝑗,𝑝 𝜑,𝑗 𝑗,𝑀 𝑗,𝑁 𝑇,𝑗 𝑈,𝑗 𝜑,𝑖,𝑝 𝐵,𝑓,𝑖,𝑗 𝑓,𝐾,𝑖,𝑗,𝑝 𝑓,𝑀,𝑖,𝑝 𝑓,𝑁,𝑖,𝑝 𝑇,𝑓,𝑖,𝑝 𝑈,𝑓,𝑖,𝑝 Allowed substitution hints: 𝜑(𝑓) 𝐵(𝑝)

Proof of Theorem poimirlem3 Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovexd 7446	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (1...𝑀) ∈ V)
2		poimirlem3.4	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑇:(1...𝑀)⟶(0..^𝐾))
3	2	ffnd 6707	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇 Fn (1...𝑀))
4	3	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → 𝑇 Fn (1...𝑀))
5		1ex 11203	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ V
6		fnconstg 6767	. . . . . . . . . . . . . . . . 17 ⊢ (1 ∈ V → ((𝑈 “ (1...𝑗)) × {1}) Fn (𝑈 “ (1...𝑗)))
7	5, 6	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈 “ (1...𝑗)) × {1}) Fn (𝑈 “ (1...𝑗))
8		c0ex 11200	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ V
9		fnconstg 6767	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ V → ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) Fn (𝑈 “ ((𝑗 + 1)...𝑀)))
10	8, 9	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) Fn (𝑈 “ ((𝑗 + 1)...𝑀))
11	7, 10	pm3.2i 475	. . . . . . . . . . . . . . 15 ⊢ (((𝑈 “ (1...𝑗)) × {1}) Fn (𝑈 “ (1...𝑗)) ∧ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) Fn (𝑈 “ ((𝑗 + 1)...𝑀)))
12		poimirlem3.5	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑈:(1...𝑀)–1-1-onto→(1...𝑀))
13		dff1o3 6828	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈:(1...𝑀)–1-1-onto→(1...𝑀) ↔ (𝑈:(1...𝑀)–onto→(1...𝑀) ∧ Fun ^◡𝑈))
14	13	simprbi 502	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈:(1...𝑀)–1-1-onto→(1...𝑀) → Fun ^◡𝑈)
15		imain 6622	. . . . . . . . . . . . . . . . 17 ⊢ (Fun ^◡𝑈 → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑀))))
16	12, 14, 15	3syl 19	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑀))))
17		elfznn0 13648	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℕ₀)
18	17	nn0red 12566	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℝ)
19	18	ltp1d 12145	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 < (𝑗 + 1))
20		fzdisj 13579	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑀)) = ∅)
21	19, 20	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑀)) = ∅)
22	21	imaeq2d 6063	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0...𝑀) → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = (𝑈 “ ∅))
23		ima0 6080	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 “ ∅) = ∅
24	22, 23	eqtrdi 2820	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (0...𝑀) → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = ∅)
25	16, 24	sylan9req 2825	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑀))) = ∅)
26		fnun 6650	. . . . . . . . . . . . . . 15 ⊢ (((((𝑈 “ (1...𝑗)) × {1}) Fn (𝑈 “ (1...𝑗)) ∧ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) Fn (𝑈 “ ((𝑗 + 1)...𝑀))) ∧ ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑀))) = ∅) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})) Fn ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑀))))
27	11, 25, 26	sylancr 598	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})) Fn ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑀))))
28		imaundi 6148	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))) = ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑀)))
29		nn0p1nn 12543	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ ℕ₀ → (𝑗 + 1) ∈ ℕ)
30		nnuz 12901	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℕ = (ℤ_≥‘1)
31	29, 30	eleqtrdi 2879	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ ℕ₀ → (𝑗 + 1) ∈ (ℤ_≥‘1))
32	17, 31	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈ (ℤ_≥‘1))
33		elfzuz3 13549	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0...𝑀) → 𝑀 ∈ (ℤ_≥‘𝑗))
34		fzsplit2 13577	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑗 + 1) ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ (ℤ_≥‘𝑗)) → (1...𝑀) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑀)))
35	32, 33, 34	syl2anc 595	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ (0...𝑀) → (1...𝑀) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑀)))
36	35	eqcomd 2775	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → ((1...𝑗) ∪ ((𝑗 + 1)...𝑀)) = (1...𝑀))
37	36	imaeq2d 6063	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0...𝑀) → (𝑈 “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))) = (𝑈 “ (1...𝑀)))
38		f1ofo 6829	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈:(1...𝑀)–1-1-onto→(1...𝑀) → 𝑈:(1...𝑀)–onto→(1...𝑀))
39		foima 6798	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈:(1...𝑀)–onto→(1...𝑀) → (𝑈 “ (1...𝑀)) = (1...𝑀))
40	12, 38, 39	3syl 19	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑈 “ (1...𝑀)) = (1...𝑀))
41	37, 40	sylan9eqr 2826	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑈 “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))) = (1...𝑀))
42	28, 41	eqtr3id 2818	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑀))) = (1...𝑀))
43	42	fneq2d 6630	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})) Fn ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑀))) ↔ (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀)))
44	27, 43	mpbid 235	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀))
45	1, 4, 44	offvalfv 7697	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) = (𝑛 ∈ (1...𝑀) ↦ ((𝑇‘𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))))
46		poimirlem4.2	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
47		nn0p1nn 12543	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ ℕ₀ → (𝑀 + 1) ∈ ℕ)
48	46, 47	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ)
49	48	nnzd 12617	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
50		uzid 12877	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 + 1) ∈ ℤ → (𝑀 + 1) ∈ (ℤ_≥‘(𝑀 + 1)))
51		peano2uz 12925	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 + 1) ∈ (ℤ_≥‘(𝑀 + 1)) → ((𝑀 + 1) + 1) ∈ (ℤ_≥‘(𝑀 + 1)))
52	49, 50, 51	3syl 19	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑀 + 1) + 1) ∈ (ℤ_≥‘(𝑀 + 1)))
53		poimirlem4.3	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑀 < 𝑁)
54	46	nn0zd 12616	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ ℤ)
55		poimir.0	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑁 ∈ ℕ)
56	55	nnzd 12617	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℤ)
57		zltp1le 12644	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁))
58		peano2z 12635	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ ℤ → (𝑀 + 1) ∈ ℤ)
59		eluz 12876	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑀 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ_≥‘(𝑀 + 1)) ↔ (𝑀 + 1) ≤ 𝑁))
60	58, 59	sylan 591	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ_≥‘(𝑀 + 1)) ↔ (𝑀 + 1) ≤ 𝑁))
61	57, 60	bitr4d 285	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ 𝑁 ∈ (ℤ_≥‘(𝑀 + 1))))
62	54, 56, 61	syl2anc 595	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 < 𝑁 ↔ 𝑁 ∈ (ℤ_≥‘(𝑀 + 1))))
63	53, 62	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝑀 + 1)))
64		fzsplit2 13577	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑀 + 1) + 1) ∈ (ℤ_≥‘(𝑀 + 1)) ∧ 𝑁 ∈ (ℤ_≥‘(𝑀 + 1))) → ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)))
65	52, 63, 64	syl2anc 595	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)))
66		fzsn 13594	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 + 1) ∈ ℤ → ((𝑀 + 1)...(𝑀 + 1)) = {(𝑀 + 1)})
67	49, 66	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑀 + 1)...(𝑀 + 1)) = {(𝑀 + 1)})
68	67	uneq1d 4127	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)) = ({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)))
69	65, 68	eqtrd 2804	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) = ({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)))
70	69	xpeq1d 5691	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝑀 + 1)...𝑁) × {0}) = (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0}))
71		xpundir 5732	. . . . . . . . . . . . . . 15 ⊢ (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0}) = (({(𝑀 + 1)} × {0}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))
72		ovex 7444	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 + 1) ∈ V
73	72, 8	xpsn 7138	. . . . . . . . . . . . . . . 16 ⊢ ({(𝑀 + 1)} × {0}) = {⟨(𝑀 + 1), 0⟩}
74	73	uneq1i 4124	. . . . . . . . . . . . . . 15 ⊢ (({(𝑀 + 1)} × {0}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))
75	71, 74	eqtri 2792	. . . . . . . . . . . . . 14 ⊢ (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0}) = ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))
76	70, 75	eqtrdi 2820	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝑀 + 1)...𝑁) × {0}) = ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
77	76	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((𝑀 + 1)...𝑁) × {0}) = ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
78	45, 77	uneq12d 4129	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...𝑀) ↦ ((𝑇‘𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))) ∪ ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))))
79		unass 4131	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ (1...𝑀) ↦ ((𝑇‘𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...𝑀) ↦ ((𝑇‘𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))) ∪ ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
80	78, 79	eqtr4di 2822	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) = (((𝑛 ∈ (1...𝑀) ↦ ((𝑇‘𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
81		ovexd 7446	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (1...(𝑀 + 1)) ∈ V)
82	46	nn0red 12566	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑀 ∈ ℝ)
83	82	ltp1d 12145	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 < (𝑀 + 1))
84	48	nnred 12248	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑀 + 1) ∈ ℝ)
85	82, 84	ltnled 11357	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀))
86	83, 85	mpbid 235	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ¬ (𝑀 + 1) ≤ 𝑀)
87		elfzle2 13556	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 + 1) ∈ (1...𝑀) → (𝑀 + 1) ≤ 𝑀)
88	86, 87	nsyl 141	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ¬ (𝑀 + 1) ∈ (1...𝑀))
89		disjsn 4680	. . . . . . . . . . . . . . . . . 18 ⊢ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ↔ ¬ (𝑀 + 1) ∈ (1...𝑀))
90	88, 89	sylibr 237	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅)
91		eqid 2769	. . . . . . . . . . . . . . . . . . 19 ⊢ {⟨(𝑀 + 1), 0⟩} = {⟨(𝑀 + 1), 0⟩}
92	72, 8	fsn 7132	. . . . . . . . . . . . . . . . . . 19 ⊢ ({⟨(𝑀 + 1), 0⟩}:{(𝑀 + 1)}⟶{0} ↔ {⟨(𝑀 + 1), 0⟩} = {⟨(𝑀 + 1), 0⟩})
93	91, 92	mpbir 234	. . . . . . . . . . . . . . . . . 18 ⊢ {⟨(𝑀 + 1), 0⟩}:{(𝑀 + 1)}⟶{0}
94		fun 6741	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑇:(1...𝑀)⟶(0..^𝐾) ∧ {⟨(𝑀 + 1), 0⟩}:{(𝑀 + 1)}⟶{0}) ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅) → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):((1...𝑀) ∪ {(𝑀 + 1)})⟶((0..^𝐾) ∪ {0}))
95	93, 94	mpanl2 713	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇:(1...𝑀)⟶(0..^𝐾) ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅) → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):((1...𝑀) ∪ {(𝑀 + 1)})⟶((0..^𝐾) ∪ {0}))
96	2, 90, 95	syl2anc 595	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):((1...𝑀) ∪ {(𝑀 + 1)})⟶((0..^𝐾) ∪ {0}))
97		1z 12624	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℤ
98		nn0uz 12900	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℕ₀ = (ℤ_≥‘0)
99		1m1e0 12313	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 − 1) = 0
100	99	fveq2i 6885	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℤ_≥‘(1 − 1)) = (ℤ_≥‘0)
101	98, 100	eqtr4i 2795	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℕ₀ = (ℤ_≥‘(1 − 1))
102	46, 101	eleqtrdi 2879	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘(1 − 1)))
103		fzsuc2 13610	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ (ℤ_≥‘(1 − 1))) → (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)}))
104	97, 102, 103	sylancr 598	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)}))
105	104	eqcomd 2775	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((1...𝑀) ∪ {(𝑀 + 1)}) = (1...(𝑀 + 1)))
106		poimirlem4.1	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐾 ∈ ℕ)
107		lbfzo0 13728	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∈ (0..^𝐾) ↔ 𝐾 ∈ ℕ)
108	106, 107	sylibr 237	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 ∈ (0..^𝐾))
109	108	snssd 4755	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → {0} ⊆ (0..^𝐾))
110		ssequn2 4148	. . . . . . . . . . . . . . . . . 18 ⊢ ({0} ⊆ (0..^𝐾) ↔ ((0..^𝐾) ∪ {0}) = (0..^𝐾))
111	109, 110	sylib 221	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((0..^𝐾) ∪ {0}) = (0..^𝐾))
112	105, 111	feq23d 6701	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):((1...𝑀) ∪ {(𝑀 + 1)})⟶((0..^𝐾) ∪ {0}) ↔ (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):(1...(𝑀 + 1))⟶(0..^𝐾)))
113	96, 112	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):(1...(𝑀 + 1))⟶(0..^𝐾))
114	113	ffnd 6707	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) Fn (1...(𝑀 + 1)))
115	114	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) Fn (1...(𝑀 + 1)))
116		fnconstg 6767	. . . . . . . . . . . . . . . . 17 ⊢ (1 ∈ V → (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)))
117	5, 116	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗))
118		fnconstg 6767	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ V → (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))))
119	8, 118	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))
120	117, 119	pm3.2i 475	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∧ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))))
121	72, 72	f1osn 6863	. . . . . . . . . . . . . . . . . . 19 ⊢ {⟨(𝑀 + 1), (𝑀 + 1)⟩}:{(𝑀 + 1)}–1-1-onto→{(𝑀 + 1)}
122		f1oun 6841	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑈:(1...𝑀)–1-1-onto→(1...𝑀) ∧ {⟨(𝑀 + 1), (𝑀 + 1)⟩}:{(𝑀 + 1)}–1-1-onto→{(𝑀 + 1)}) ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅)) → (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)}))
123	121, 122	mpanl2 713	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑈:(1...𝑀)–1-1-onto→(1...𝑀) ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅)) → (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)}))
124	12, 90, 90, 123	syl12anc 849	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)}))
125		dff1o3 6828	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)}) ↔ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–onto→((1...𝑀) ∪ {(𝑀 + 1)}) ∧ Fun ^◡(𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})))
126	125	simprbi 502	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)}) → Fun ^◡(𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
127		imain 6622	. . . . . . . . . . . . . . . . 17 ⊢ (Fun ^◡(𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∩ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))))
128	124, 126, 127	3syl 19	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∩ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))))
129		fzdisj 13579	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1))) = ∅)
130	19, 129	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1))) = ∅)
131	130	imaeq2d 6063	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0...𝑀) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ∅))
132		ima0 6080	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ∅) = ∅
133	131, 132	eqtrdi 2820	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (0...𝑀) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = ∅)
134	128, 133	sylan9req 2825	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∩ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅)
135		fnun 6650	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∧ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))) ∧ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∩ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅) → ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∪ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))))
136	120, 134, 135	sylancr 598	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∪ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))))
137		f1ofo 6829	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)}) → (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–onto→((1...𝑀) ∪ {(𝑀 + 1)}))
138		foima 6798	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–onto→((1...𝑀) ∪ {(𝑀 + 1)}) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑀) ∪ {(𝑀 + 1)})) = ((1...𝑀) ∪ {(𝑀 + 1)}))
139	124, 137, 138	3syl 19	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑀) ∪ {(𝑀 + 1)})) = ((1...𝑀) ∪ {(𝑀 + 1)}))
140	104	imaeq2d 6063	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...(𝑀 + 1))) = ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑀) ∪ {(𝑀 + 1)})))
141	139, 140, 104	3eqtr4d 2814	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...(𝑀 + 1))) = (1...(𝑀 + 1)))
142		peano2uz 12925	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ (ℤ_≥‘𝑗) → (𝑀 + 1) ∈ (ℤ_≥‘𝑗))
143	33, 142	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈ (ℤ_≥‘𝑗))
144		fzsplit2 13577	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑗 + 1) ∈ (ℤ_≥‘1) ∧ (𝑀 + 1) ∈ (ℤ_≥‘𝑗)) → (1...(𝑀 + 1)) = ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1))))
145	32, 143, 144	syl2anc 595	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → (1...(𝑀 + 1)) = ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1))))
146	145	imaeq2d 6063	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0...𝑀) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...(𝑀 + 1))) = ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1)))))
147	141, 146	sylan9req 2825	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (1...(𝑀 + 1)) = ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1)))))
148		imaundi 6148	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1)))) = (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∪ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))))
149	147, 148	eqtrdi 2820	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (1...(𝑀 + 1)) = (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∪ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))))
150	149	fneq2d 6630	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (1...(𝑀 + 1)) ↔ ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∪ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))))))
151	136, 150	mpbird 260	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (1...(𝑀 + 1)))
152	81, 115, 151	offvalfv 7697	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_f + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) = (𝑛 ∈ (1...(𝑀 + 1)) ↦ (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) + (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))))
153		imadmrn 6073	. . . . . . . . . . . . . . . . . . 19 ⊢ (({(𝑀 + 1)} × {(𝑀 + 1)}) “ dom ({(𝑀 + 1)} × {(𝑀 + 1)})) = ran ({(𝑀 + 1)} × {(𝑀 + 1)})
154	72, 72	xpsn 7138	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({(𝑀 + 1)} × {(𝑀 + 1)}) = {⟨(𝑀 + 1), (𝑀 + 1)⟩}
155	154	imaeq1i 6060	. . . . . . . . . . . . . . . . . . . 20 ⊢ (({(𝑀 + 1)} × {(𝑀 + 1)}) “ dom ({(𝑀 + 1)} × {(𝑀 + 1)})) = ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ dom ({(𝑀 + 1)} × {(𝑀 + 1)}))
156		dmxpid 5921	. . . . . . . . . . . . . . . . . . . . 21 ⊢ dom ({(𝑀 + 1)} × {(𝑀 + 1)}) = {(𝑀 + 1)}
157	156	imaeq2i 6061	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ dom ({(𝑀 + 1)} × {(𝑀 + 1)})) = ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ {(𝑀 + 1)})
158	155, 157	eqtri 2792	. . . . . . . . . . . . . . . . . . 19 ⊢ (({(𝑀 + 1)} × {(𝑀 + 1)}) “ dom ({(𝑀 + 1)} × {(𝑀 + 1)})) = ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ {(𝑀 + 1)})
159		rnxpid 6172	. . . . . . . . . . . . . . . . . . 19 ⊢ ran ({(𝑀 + 1)} × {(𝑀 + 1)}) = {(𝑀 + 1)}
160	153, 158, 159	3eqtr3ri 2801	. . . . . . . . . . . . . . . . . 18 ⊢ {(𝑀 + 1)} = ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ {(𝑀 + 1)})
161		eluzp1p1 12890	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ (ℤ_≥‘𝑗) → (𝑀 + 1) ∈ (ℤ_≥‘(𝑗 + 1)))
162		eluzfz2 13560	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑀 + 1) ∈ (ℤ_≥‘(𝑗 + 1)) → (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1)))
163	33, 161, 162	3syl 19	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1)))
164	163	snssd 4755	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ (0...𝑀) → {(𝑀 + 1)} ⊆ ((𝑗 + 1)...(𝑀 + 1)))
165		imass2 6105	. . . . . . . . . . . . . . . . . . 19 ⊢ ({(𝑀 + 1)} ⊆ ((𝑗 + 1)...(𝑀 + 1)) → ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ {(𝑀 + 1)}) ⊆ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))))
166	164, 165	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ {(𝑀 + 1)}) ⊆ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))))
167	160, 166	eqsstrid 3981	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0...𝑀) → {(𝑀 + 1)} ⊆ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))))
168	72	snid 4631	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 + 1) ∈ {(𝑀 + 1)}
169		ssel 3937	. . . . . . . . . . . . . . . . 17 ⊢ ({(𝑀 + 1)} ⊆ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) → ((𝑀 + 1) ∈ {(𝑀 + 1)} → (𝑀 + 1) ∈ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1)))))
170	167, 168, 169	mpisyl 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))))
171		elun2 4142	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 + 1) ∈ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) → (𝑀 + 1) ∈ ((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) ∪ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1)))))
172	170, 171	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈ ((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) ∪ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1)))))
173		imaundir 6149	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) = ((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) ∪ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))))
174	172, 173	eleqtrrdi 2880	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))))
175	174	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑀 + 1) ∈ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))))
176	8	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → 0 ∈ V)
177	105	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((1...𝑀) ∪ {(𝑀 + 1)}) = (1...(𝑀 + 1)))
178		fveq2 6882	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (𝑀 + 1) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) = ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)))
179	72, 8	fnsn 6595	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {⟨(𝑀 + 1), 0⟩} Fn {(𝑀 + 1)}
180		fvun2 6974	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑇 Fn (1...𝑀) ∧ {⟨(𝑀 + 1), 0⟩} Fn {(𝑀 + 1)} ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ (𝑀 + 1) ∈ {(𝑀 + 1)})) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = ({⟨(𝑀 + 1), 0⟩}‘(𝑀 + 1)))
181	179, 180	mp3an2 1475	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑇 Fn (1...𝑀) ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ (𝑀 + 1) ∈ {(𝑀 + 1)})) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = ({⟨(𝑀 + 1), 0⟩}‘(𝑀 + 1)))
182	168, 181	mpanr2 716	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑇 Fn (1...𝑀) ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = ({⟨(𝑀 + 1), 0⟩}‘(𝑀 + 1)))
183	3, 90, 182	syl2anc 595	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = ({⟨(𝑀 + 1), 0⟩}‘(𝑀 + 1)))
184	72, 8	fvsn 7180	. . . . . . . . . . . . . . . . . 18 ⊢ ({⟨(𝑀 + 1), 0⟩}‘(𝑀 + 1)) = 0
185	183, 184	eqtrdi 2820	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0)
186	178, 185	sylan9eqr 2826	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 = (𝑀 + 1)) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) = 0)
187	186	adantlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 = (𝑀 + 1)) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) = 0)
188		fveq2 6882	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑀 + 1) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛) = (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)))
189		fvun2 6974	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∧ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) ∧ ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∩ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅ ∧ (𝑀 + 1) ∈ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))))) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)))
190	117, 119, 189	mp3an12 1477	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∩ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅ ∧ (𝑀 + 1) ∈ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)))
191	134, 175, 190	syl2anc 595	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)))
192	8	fvconst2 7203	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 + 1) ∈ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) → ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)) = 0)
193	174, 192	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)) = 0)
194	193	adantl 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)) = 0)
195	191, 194	eqtrd 2804	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = 0)
196	188, 195	sylan9eqr 2826	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 = (𝑀 + 1)) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛) = 0)
197	187, 196	oveq12d 7429	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 = (𝑀 + 1)) → (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) + (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛)) = (0 + 0))
198		00id 11385	. . . . . . . . . . . . . 14 ⊢ (0 + 0) = 0
199	197, 198	eqtrdi 2820	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 = (𝑀 + 1)) → (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) + (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛)) = 0)
200	175, 176, 177, 199	fmptapd 7170	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑛 ∈ (1...𝑀) ↦ (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) + (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) = (𝑛 ∈ (1...(𝑀 + 1)) ↦ (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) + (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))))
201	3, 90	jca 520	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑇 Fn (1...𝑀) ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅))
202		fvun1 6973	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑇 Fn (1...𝑀) ∧ {⟨(𝑀 + 1), 0⟩} Fn {(𝑀 + 1)} ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ 𝑛 ∈ (1...𝑀))) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) = (𝑇‘𝑛))
203	179, 202	mp3an2 1475	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑇 Fn (1...𝑀) ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ 𝑛 ∈ (1...𝑀))) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) = (𝑇‘𝑛))
204	203	anassrs 472	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇 Fn (1...𝑀) ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅) ∧ 𝑛 ∈ (1...𝑀)) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) = (𝑇‘𝑛))
205	201, 204	sylan 591	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) = (𝑇‘𝑛))
206	205	adantlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) = (𝑇‘𝑛))
207		fvres 6901	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (1...𝑀) → ((((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) ↾ (1...𝑀))‘𝑛) = (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))
208	207	eqcomd 2775	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (1...𝑀) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛) = ((((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) ↾ (1...𝑀))‘𝑛))
209		resundir 5994	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) ↾ (1...𝑀)) = (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ↾ (1...𝑀)) ∪ ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)))
210		relxp 5680	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Rel ((𝑈 “ (1...𝑗)) × {1})
211		dmxpss 6170	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ dom ((𝑈 “ (1...𝑗)) × {1}) ⊆ (𝑈 “ (1...𝑗))
212		imassrn 6074	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑈 “ (1...𝑗)) ⊆ ran 𝑈
213	211, 212	sstri 3952	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ dom ((𝑈 “ (1...𝑗)) × {1}) ⊆ ran 𝑈
214		f1of 6821	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑈:(1...𝑀)–1-1-onto→(1...𝑀) → 𝑈:(1...𝑀)⟶(1...𝑀))
215		frn 6714	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑈:(1...𝑀)⟶(1...𝑀) → ran 𝑈 ⊆ (1...𝑀))
216	12, 214, 215	3syl 19	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ran 𝑈 ⊆ (1...𝑀))
217	213, 216	sstrid 3954	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → dom ((𝑈 “ (1...𝑗)) × {1}) ⊆ (1...𝑀))
218		relssres 6022	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((Rel ((𝑈 “ (1...𝑗)) × {1}) ∧ dom ((𝑈 “ (1...𝑗)) × {1}) ⊆ (1...𝑀)) → (((𝑈 “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ((𝑈 “ (1...𝑗)) × {1}))
219	210, 217, 218	sylancr 598	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (((𝑈 “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ((𝑈 “ (1...𝑗)) × {1}))
220	219	adantr 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((𝑈 “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ((𝑈 “ (1...𝑗)) × {1}))
221		imassrn 6074	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ⊆ ran {⟨(𝑀 + 1), (𝑀 + 1)⟩}
222	72	rnsnop 6226	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ran {⟨(𝑀 + 1), (𝑀 + 1)⟩} = {(𝑀 + 1)}
223	221, 222	sseqtri 3991	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ⊆ {(𝑀 + 1)}
224		ssrin 4200	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ⊆ {(𝑀 + 1)} → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ∩ (1...𝑀)) ⊆ ({(𝑀 + 1)} ∩ (1...𝑀)))
225	223, 224	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ∩ (1...𝑀)) ⊆ ({(𝑀 + 1)} ∩ (1...𝑀))
226		incom 4168	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ({(𝑀 + 1)} ∩ (1...𝑀)) = ((1...𝑀) ∩ {(𝑀 + 1)})
227	226, 90	eqtrid 2816	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ({(𝑀 + 1)} ∩ (1...𝑀)) = ∅)
228	225, 227	sseqtrid 3985	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ∩ (1...𝑀)) ⊆ ∅)
229		ss0 4364	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ∩ (1...𝑀)) ⊆ ∅ → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ∩ (1...𝑀)) = ∅)
230	228, 229	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ∩ (1...𝑀)) = ∅)
231		fnconstg 6767	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1 ∈ V → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) Fn ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)))
232		fnresdisj 6656	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) Fn ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) → ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ∩ (1...𝑀)) = ∅ ↔ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ∅))
233	5, 231, 232	mp2b 10	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ∩ (1...𝑀)) = ∅ ↔ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ∅)
234	230, 233	sylib 221	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ∅)
235	234	adantr 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ∅)
236	220, 235	uneq12d 4129	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((((𝑈 “ (1...𝑗)) × {1}) ↾ (1...𝑀)) ∪ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) ↾ (1...𝑀))) = (((𝑈 “ (1...𝑗)) × {1}) ∪ ∅))
237		imaundir 6149	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) = ((𝑈 “ (1...𝑗)) ∪ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)))
238	237	xpeq1i 5688	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) = (((𝑈 “ (1...𝑗)) ∪ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗))) × {1})
239		xpundir 5732	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑈 “ (1...𝑗)) ∪ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗))) × {1}) = (((𝑈 “ (1...𝑗)) × {1}) ∪ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}))
240	238, 239	eqtri 2792	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) = (((𝑈 “ (1...𝑗)) × {1}) ∪ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}))
241	240	reseq1i 5975	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ((((𝑈 “ (1...𝑗)) × {1}) ∪ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1})) ↾ (1...𝑀))
242		resundir 5994	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑈 “ (1...𝑗)) × {1}) ∪ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1})) ↾ (1...𝑀)) = ((((𝑈 “ (1...𝑗)) × {1}) ↾ (1...𝑀)) ∪ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) ↾ (1...𝑀)))
243	241, 242	eqtr2i 2793	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑈 “ (1...𝑗)) × {1}) ↾ (1...𝑀)) ∪ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) ↾ (1...𝑀))) = ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ↾ (1...𝑀))
244		un0 4356	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑈 “ (1...𝑗)) × {1}) ∪ ∅) = ((𝑈 “ (1...𝑗)) × {1})
245	236, 243, 244	3eqtr3g 2827	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ((𝑈 “ (1...𝑗)) × {1}))
246		f1odm 6825	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑈:(1...𝑀)–1-1-onto→(1...𝑀) → dom 𝑈 = (1...𝑀))
247	12, 246	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → dom 𝑈 = (1...𝑀))
248	247	ineq2d 4179	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → (((𝑗 + 1)...(𝑀 + 1)) ∩ dom 𝑈) = (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑀)))
249	248	reseq2d 5979	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (𝑈 ↾ (((𝑗 + 1)...(𝑀 + 1)) ∩ dom 𝑈)) = (𝑈 ↾ (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑀))))
250		resindm 6030	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑈 ↾ (((𝑗 + 1)...(𝑀 + 1)) ∩ dom 𝑈)) = (𝑈 ↾ ((𝑗 + 1)...(𝑀 + 1)))
251	249, 250	eqtr3di 2819	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑈 ↾ (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑀))) = (𝑈 ↾ ((𝑗 + 1)...(𝑀 + 1))))
252	35	ineq2d 4179	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ (0...𝑀) → (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑀)) = (((𝑗 + 1)...(𝑀 + 1)) ∩ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))))
253		fzssp1 13595	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑗 + 1)...𝑀) ⊆ ((𝑗 + 1)...(𝑀 + 1))
254		sseqin2 4182	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑗 + 1)...𝑀) ⊆ ((𝑗 + 1)...(𝑀 + 1)) ↔ (((𝑗 + 1)...(𝑀 + 1)) ∩ ((𝑗 + 1)...𝑀)) = ((𝑗 + 1)...𝑀))
255	253, 254	mpbi 233	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑗 + 1)...(𝑀 + 1)) ∩ ((𝑗 + 1)...𝑀)) = ((𝑗 + 1)...𝑀)
256	255	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 ∈ (0...𝑀) → (((𝑗 + 1)...(𝑀 + 1)) ∩ ((𝑗 + 1)...𝑀)) = ((𝑗 + 1)...𝑀))
257		incom 4168	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑗)) = ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))
258	257, 130	eqtrid 2816	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 ∈ (0...𝑀) → (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑗)) = ∅)
259	256, 258	uneq12d 4129	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ (0...𝑀) → ((((𝑗 + 1)...(𝑀 + 1)) ∩ ((𝑗 + 1)...𝑀)) ∪ (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑗))) = (((𝑗 + 1)...𝑀) ∪ ∅))
260		uncom 4118	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑗 + 1)...(𝑀 + 1)) ∩ ((𝑗 + 1)...𝑀)) ∪ (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑗))) = ((((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑗)) ∪ (((𝑗 + 1)...(𝑀 + 1)) ∩ ((𝑗 + 1)...𝑀)))
261		indi 4243	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑗 + 1)...(𝑀 + 1)) ∩ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))) = ((((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑗)) ∪ (((𝑗 + 1)...(𝑀 + 1)) ∩ ((𝑗 + 1)...𝑀)))
262	260, 261	eqtr4i 2795	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑗 + 1)...(𝑀 + 1)) ∩ ((𝑗 + 1)...𝑀)) ∪ (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑗))) = (((𝑗 + 1)...(𝑀 + 1)) ∩ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀)))
263		un0 4356	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑗 + 1)...𝑀) ∪ ∅) = ((𝑗 + 1)...𝑀)
264	259, 262, 263	3eqtr3g 2827	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ (0...𝑀) → (((𝑗 + 1)...(𝑀 + 1)) ∩ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))) = ((𝑗 + 1)...𝑀))
265	252, 264	eqtrd 2804	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ (0...𝑀) → (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑀)) = ((𝑗 + 1)...𝑀))
266	265	reseq2d 5979	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ (0...𝑀) → (𝑈 ↾ (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑀))) = (𝑈 ↾ ((𝑗 + 1)...𝑀)))
267	251, 266	sylan9req 2825	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑈 ↾ ((𝑗 + 1)...(𝑀 + 1))) = (𝑈 ↾ ((𝑗 + 1)...𝑀)))
268	267	rneqd 5929	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ran (𝑈 ↾ ((𝑗 + 1)...(𝑀 + 1))) = ran (𝑈 ↾ ((𝑗 + 1)...𝑀)))
269		df-ima 5675	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) = ran (𝑈 ↾ ((𝑗 + 1)...(𝑀 + 1)))
270		df-ima 5675	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑈 “ ((𝑗 + 1)...𝑀)) = ran (𝑈 ↾ ((𝑗 + 1)...𝑀))
271	268, 269, 270	3eqtr4g 2829	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) = (𝑈 “ ((𝑗 + 1)...𝑀)))
272	271	xpeq1d 5691	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))
273	272	reseq1d 5978	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) = (((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ↾ (1...𝑀)))
274		relxp 5680	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Rel ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})
275		dmxpss 6170	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ dom ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ⊆ (𝑈 “ ((𝑗 + 1)...𝑀))
276		imassrn 6074	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑈 “ ((𝑗 + 1)...𝑀)) ⊆ ran 𝑈
277	275, 276	sstri 3952	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ dom ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ⊆ ran 𝑈
278	277, 216	sstrid 3954	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → dom ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ⊆ (1...𝑀))
279		relssres 6022	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((Rel ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ∧ dom ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ⊆ (1...𝑀)) → (((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ↾ (1...𝑀)) = ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))
280	274, 278, 279	sylancr 598	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ↾ (1...𝑀)) = ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))
281	280	adantr 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ↾ (1...𝑀)) = ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))
282	273, 281	eqtrd 2804	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) = ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))
283		imassrn 6074	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ⊆ ran {⟨(𝑀 + 1), (𝑀 + 1)⟩}
284	283, 222	sseqtri 3991	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ⊆ {(𝑀 + 1)}
285		ssrin 4200	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ⊆ {(𝑀 + 1)} → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ∩ (1...𝑀)) ⊆ ({(𝑀 + 1)} ∩ (1...𝑀)))
286	284, 285	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ∩ (1...𝑀)) ⊆ ({(𝑀 + 1)} ∩ (1...𝑀))
287	286, 227	sseqtrid 3985	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ∩ (1...𝑀)) ⊆ ∅)
288		ss0 4364	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ∩ (1...𝑀)) ⊆ ∅ → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ∩ (1...𝑀)) = ∅)
289	287, 288	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ∩ (1...𝑀)) = ∅)
290		fnconstg 6767	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0 ∈ V → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))))
291		fnresdisj 6656	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) → ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ∩ (1...𝑀)) = ∅ ↔ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) = ∅))
292	8, 290, 291	mp2b 10	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ∩ (1...𝑀)) = ∅ ↔ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) = ∅)
293	289, 292	sylib 221	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) = ∅)
294	293	adantr 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) = ∅)
295	282, 294	uneq12d 4129	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) ∪ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀))) = (((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ∪ ∅))
296	173	xpeq1i 5688	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = (((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) ∪ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1)))) × {0})
297		xpundir 5732	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) ∪ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1)))) × {0}) = (((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ∪ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))
298	296, 297	eqtri 2792	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = (((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ∪ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))
299	298	reseq1i 5975	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) = ((((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ∪ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) ↾ (1...𝑀))
300		resundir 5994	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ∪ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) ↾ (1...𝑀)) = ((((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) ∪ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)))
301	299, 300	eqtr2i 2793	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) ∪ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀))) = ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀))
302		un0 4356	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ∪ ∅) = ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})
303	295, 301, 302	3eqtr3g 2827	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) = ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))
304	245, 303	uneq12d 4129	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ↾ (1...𝑀)) ∪ ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀))) = (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})))
305	209, 304	eqtrid 2816	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) ↾ (1...𝑀)) = (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})))
306	305	fveq1d 6884	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) ↾ (1...𝑀))‘𝑛) = ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))
307	208, 306	sylan9eqr 2826	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛) = ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))
308	206, 307	oveq12d 7429	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) + (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛)) = ((𝑇‘𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛)))
309	308	mpteq2dva 5206	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑛 ∈ (1...𝑀) ↦ (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) + (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) = (𝑛 ∈ (1...𝑀) ↦ ((𝑇‘𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))))
310	309	uneq1d 4127	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑛 ∈ (1...𝑀) ↦ (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) + (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) = ((𝑛 ∈ (1...𝑀) ↦ ((𝑇‘𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}))
311	152, 200, 310	3eqtr2d 2810	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_f + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) = ((𝑛 ∈ (1...𝑀) ↦ ((𝑇‘𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}))
312	311	uneq1d 4127	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_f + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = (((𝑛 ∈ (1...𝑀) ↦ ((𝑇‘𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
313	80, 312	eqtr4d 2807	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) = (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_f + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
314	313	csbeq1d 3863	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ⦋((𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_f + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)
315	314	eqeq2d 2780	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑖 = ⦋((𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_f + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
316	315	rexbidva 3193	. . . . . 6 ⊢ (𝜑 → (∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_f + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
317	316	ralbidv 3194	. . . . 5 ⊢ (𝜑 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_f + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
318	317	biimpd 232	. . . 4 ⊢ (𝜑 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 → ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_f + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
319		f1ofn 6822	. . . . . . . 8 ⊢ (𝑈:(1...𝑀)–1-1-onto→(1...𝑀) → 𝑈 Fn (1...𝑀))
320	12, 319	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑈 Fn (1...𝑀))
321	72, 72	fnsn 6595	. . . . . . . . 9 ⊢ {⟨(𝑀 + 1), (𝑀 + 1)⟩} Fn {(𝑀 + 1)}
322		fvun2 6974	. . . . . . . . 9 ⊢ ((𝑈 Fn (1...𝑀) ∧ {⟨(𝑀 + 1), (𝑀 + 1)⟩} Fn {(𝑀 + 1)} ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ (𝑀 + 1) ∈ {(𝑀 + 1)})) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = ({⟨(𝑀 + 1), (𝑀 + 1)⟩}‘(𝑀 + 1)))
323	321, 322	mp3an2 1475	. . . . . . . 8 ⊢ ((𝑈 Fn (1...𝑀) ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ (𝑀 + 1) ∈ {(𝑀 + 1)})) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = ({⟨(𝑀 + 1), (𝑀 + 1)⟩}‘(𝑀 + 1)))
324	168, 323	mpanr2 716	. . . . . . 7 ⊢ ((𝑈 Fn (1...𝑀) ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = ({⟨(𝑀 + 1), (𝑀 + 1)⟩}‘(𝑀 + 1)))
325	320, 90, 324	syl2anc 595	. . . . . 6 ⊢ (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = ({⟨(𝑀 + 1), (𝑀 + 1)⟩}‘(𝑀 + 1)))
326	72, 72	fvsn 7180	. . . . . 6 ⊢ ({⟨(𝑀 + 1), (𝑀 + 1)⟩}‘(𝑀 + 1)) = (𝑀 + 1)
327	325, 326	eqtrdi 2820	. . . . 5 ⊢ (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1))
328	185, 327	jca 520	. . . 4 ⊢ (𝜑 → (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1)))
329	318, 328	jctird 535	. . 3 ⊢ (𝜑 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_f + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1)))))
330		3anass 1109	. . 3 ⊢ ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_f + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1)) ↔ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_f + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1))))
331	329, 330	imbitrrdi 255	. 2 ⊢ (𝜑 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_f + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1))))
332	2, 93	jctir 529	. . . . . 6 ⊢ (𝜑 → (𝑇:(1...𝑀)⟶(0..^𝐾) ∧ {⟨(𝑀 + 1), 0⟩}:{(𝑀 + 1)}⟶{0}))
333	332, 90, 94	syl2anc 595	. . . . 5 ⊢ (𝜑 → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):((1...𝑀) ∪ {(𝑀 + 1)})⟶((0..^𝐾) ∪ {0}))
334	333, 112	mpbid 235	. . . 4 ⊢ (𝜑 → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):(1...(𝑀 + 1))⟶(0..^𝐾))
335		ovex 7444	. . . . 5 ⊢ (0..^𝐾) ∈ V
336		ovex 7444	. . . . 5 ⊢ (1...(𝑀 + 1)) ∈ V
337	335, 336	elmap 8869	. . . 4 ⊢ ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∈ ((0..^𝐾) ↑_m (1...(𝑀 + 1))) ↔ (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):(1...(𝑀 + 1))⟶(0..^𝐾))
338	334, 337	sylibr 237	. . 3 ⊢ (𝜑 → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∈ ((0..^𝐾) ↑_m (1...(𝑀 + 1))))
339		ovex 7444	. . . . . . . 8 ⊢ (1...𝑀) ∈ V
340		f1oexrnex 7924	. . . . . . . 8 ⊢ ((𝑈:(1...𝑀)–1-1-onto→(1...𝑀) ∧ (1...𝑀) ∈ V) → 𝑈 ∈ V)
341	12, 339, 340	sylancl 597	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ V)
342		snex 5411	. . . . . . 7 ⊢ {⟨(𝑀 + 1), (𝑀 + 1)⟩} ∈ V
343		unexg 7742	. . . . . . 7 ⊢ ((𝑈 ∈ V ∧ {⟨(𝑀 + 1), (𝑀 + 1)⟩} ∈ V) → (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ V)
344	341, 342, 343	sylancl 597	. . . . . 6 ⊢ (𝜑 → (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ V)
345		f1oeq1 6809	. . . . . . 7 ⊢ (𝑓 = (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) → (𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) ↔ (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))))
346	345	elabg 3642	. . . . . 6 ⊢ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ V → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))} ↔ (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))))
347	344, 346	syl 18	. . . . 5 ⊢ (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))} ↔ (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))))
348		f1oeq23 6812	. . . . . 6 ⊢ (((1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)}) ∧ (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)})) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) ↔ (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)})))
349	104, 104, 348	syl2anc 595	. . . . 5 ⊢ (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) ↔ (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)})))
350	347, 349	bitrd 282	. . . 4 ⊢ (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))} ↔ (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)})))
351	124, 350	mpbird 260	. . 3 ⊢ (𝜑 → (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})
352	338, 351	opelxpd 5701	. 2 ⊢ (𝜑 → ⟨(𝑇 ∪ {⟨(𝑀 + 1), 0⟩}), (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ (((0..^𝐾) ↑_m (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}))
353	331, 352	jctild 534	1 ⊢ (𝜑 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((𝑇 ∘_f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 → (⟨(𝑇 ∪ {⟨(𝑀 + 1), 0⟩}), (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ (((0..^𝐾) ↑_m (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_f + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 {cab 2747 ∀wral 3085 ∃wrex 3095 Vcvv 3461 ⦋csb 3859 ∪ cun 3909 ∩ cin 3910 ⊆ wss 3911 ∅c0 4292 {csn 4592 ⟨cop 4598 class class class wbr 5111 ↦ cmpt 5194 × cxp 5660 ^◡ccnv 5661 dom cdm 5662 ran crn 5663 ↾ cres 5664 “ cima 5665 Rel wrel 5667 Fun wfun 6531 Fn wfn 6532 ⟶wf 6533 –onto→wfo 6535 –1-1-onto→wf1o 6536 ‘cfv 6537 (class class class)co 7411 ∘_f cof 7673 ↑_m cmap 8824 0cc0 11100 1c1 11101 + caddc 11103 < clt 11243 ≤ cle 11244 − cmin 11441 ℕcn 12233 ℕ₀cn0 12504 ℤcz 12591 ℤ_≥cuz 12862 ...cfz 13535 ..^cfzo 13682

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-reu 3376 df-rab 3423 df-v 3463 df-sbc 3752 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4491 df-pw 4567 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-er 8694 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-nn 12234 df-n0 12505 df-z 12592 df-uz 12863 df-fz 13536 df-fzo 13683

This theorem is referenced by: poimirlem4 38198

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