Theorem infxpenc 9041

Description: A canonical version of infxpen 9037, by a completely different approach (although it uses infxpen 9037 via xpomen 9038). Using Cantor's normal form, we can show that 𝐴 ↑_𝑜 𝐵 respects equinumerosity (oef1o 8759), so that all the steps of (ω↑𝑊) · (ω↑𝑊) ≈ ω↑(2𝑊) ≈ (ω↑2)↑𝑊 ≈ ω↑𝑊 can be verified using bijections to do the ordinal commutations. (The assumption on 𝑁 can be satisfied using cnfcom3c 8767.) (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 7-Jul-2019.)

Hypotheses

Ref	Expression
infxpenc.1	⊢ (𝜑 → 𝐴 ∈ On)
infxpenc.2	⊢ (𝜑 → ω ⊆ 𝐴)
infxpenc.3	⊢ (𝜑 → 𝑊 ∈ (On ∖ 1𝑜))
infxpenc.4	⊢ (𝜑 → 𝐹:(ω ↑𝑜 2𝑜)–1-1-onto→ω)
infxpenc.5	⊢ (𝜑 → (𝐹‘∅) = ∅)
infxpenc.6	⊢ (𝜑 → 𝑁:𝐴–1-1-onto→(ω ↑𝑜 𝑊))
infxpenc.k	⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ ((ω ↑𝑜 2𝑜) ↑𝑚 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡( I ↾ 𝑊))))
infxpenc.h	⊢ 𝐻 = (((ω CNF 𝑊) ∘ 𝐾) ∘ ◡((ω ↑𝑜 2𝑜) CNF 𝑊))
infxpenc.l	⊢ 𝐿 = (𝑦 ∈ {𝑥 ∈ (ω ↑𝑚 (𝑊 ·𝑜 2𝑜)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ◡(𝑌 ∘ ◡𝑋))))
infxpenc.x	⊢ 𝑋 = (𝑧 ∈ 2𝑜, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·𝑜 𝑧) +𝑜 𝑤))
infxpenc.y	⊢ 𝑌 = (𝑧 ∈ 2𝑜, 𝑤 ∈ 𝑊 ↦ ((2𝑜 ·𝑜 𝑤) +𝑜 𝑧))
infxpenc.j	⊢ 𝐽 = (((ω CNF (2𝑜 ·𝑜 𝑊)) ∘ 𝐿) ∘ ◡(ω CNF (𝑊 ·𝑜 2𝑜)))
infxpenc.z	⊢ 𝑍 = (𝑥 ∈ (ω ↑𝑜 𝑊), 𝑦 ∈ (ω ↑𝑜 𝑊) ↦ (((ω ↑𝑜 𝑊) ·𝑜 𝑥) +𝑜 𝑦))
infxpenc.t	⊢ 𝑇 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝑁‘𝑥), (𝑁‘𝑦)⟩)
infxpenc.g	⊢ 𝐺 = (◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇))

Assertion

Ref	Expression
infxpenc	⊢ (𝜑 → 𝐺:(𝐴 × 𝐴)–1-1-onto→𝐴)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦   𝑥,𝑁,𝑦   𝜑,𝑥,𝑦   𝑥,𝑤,𝑦,𝑧,𝑊   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦

Allowed substitution hints:   𝜑(𝑧,𝑤)   𝐴(𝑧,𝑤)   𝑇(𝑥,𝑦,𝑧,𝑤)   𝐹(𝑧,𝑤)   𝐺(𝑥,𝑦,𝑧,𝑤)   𝐻(𝑥,𝑦,𝑧,𝑤)   𝐽(𝑥,𝑦,𝑧,𝑤)   𝐾(𝑥,𝑦,𝑧,𝑤)   𝐿(𝑥,𝑦,𝑧,𝑤)   𝑁(𝑧,𝑤)   𝑋(𝑧,𝑤)   𝑌(𝑧,𝑤)   𝑍(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem infxpenc

Step	Hyp	Ref	Expression
1		infxpenc.6	. . . 4 ⊢ (𝜑 → 𝑁:𝐴–1-1-onto→(ω ↑𝑜 𝑊))
2		f1ocnv 6290	. . . 4 ⊢ (𝑁:𝐴–1-1-onto→(ω ↑𝑜 𝑊) → ◡𝑁:(ω ↑𝑜 𝑊)–1-1-onto→𝐴)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → ◡𝑁:(ω ↑𝑜 𝑊)–1-1-onto→𝐴)
4		infxpenc.4	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(ω ↑𝑜 2𝑜)–1-1-onto→ω)
5		f1oi 6315	. . . . . . . . 9 ⊢ ( I ↾ 𝑊):𝑊–1-1-onto→𝑊
6	5	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ( I ↾ 𝑊):𝑊–1-1-onto→𝑊)
7		omelon 8707	. . . . . . . . . . 11 ⊢ ω ∈ On
8	7	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ω ∈ On)
9		2on 7722	. . . . . . . . . 10 ⊢ 2𝑜 ∈ On
10		oecl 7771	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ 2𝑜 ∈ On) → (ω ↑𝑜 2𝑜) ∈ On)
11	8, 9, 10	sylancl 566	. . . . . . . . 9 ⊢ (𝜑 → (ω ↑𝑜 2𝑜) ∈ On)
12	9	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2𝑜 ∈ On)
13		peano1 7232	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
14	13	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ∅ ∈ ω)
15		oen0 7820	. . . . . . . . . 10 ⊢ (((ω ∈ On ∧ 2𝑜 ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑𝑜 2𝑜))
16	8, 12, 14, 15	syl21anc 1475	. . . . . . . . 9 ⊢ (𝜑 → ∅ ∈ (ω ↑𝑜 2𝑜))
17		ondif1 7735	. . . . . . . . 9 ⊢ ((ω ↑𝑜 2𝑜) ∈ (On ∖ 1𝑜) ↔ ((ω ↑𝑜 2𝑜) ∈ On ∧ ∅ ∈ (ω ↑𝑜 2𝑜)))
18	11, 16, 17	sylanbrc 564	. . . . . . . 8 ⊢ (𝜑 → (ω ↑𝑜 2𝑜) ∈ (On ∖ 1𝑜))
19		infxpenc.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ (On ∖ 1𝑜))
20	19	eldifad 3735	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ On)
21		infxpenc.5	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘∅) = ∅)
22		infxpenc.k	. . . . . . . 8 ⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ ((ω ↑𝑜 2𝑜) ↑𝑚 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡( I ↾ 𝑊))))
23		infxpenc.h	. . . . . . . 8 ⊢ 𝐻 = (((ω CNF 𝑊) ∘ 𝐾) ∘ ◡((ω ↑𝑜 2𝑜) CNF 𝑊))
24	4, 6, 18, 20, 8, 20, 21, 22, 23	oef1o 8759	. . . . . . 7 ⊢ (𝜑 → 𝐻:((ω ↑𝑜 2𝑜) ↑𝑜 𝑊)–1-1-onto→(ω ↑𝑜 𝑊))
25		f1oi 6315	. . . . . . . . . 10 ⊢ ( I ↾ ω):ω–1-1-onto→ω
26	25	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ( I ↾ ω):ω–1-1-onto→ω)
27		infxpenc.x	. . . . . . . . . . 11 ⊢ 𝑋 = (𝑧 ∈ 2𝑜, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·𝑜 𝑧) +𝑜 𝑤))
28		infxpenc.y	. . . . . . . . . . 11 ⊢ 𝑌 = (𝑧 ∈ 2𝑜, 𝑤 ∈ 𝑊 ↦ ((2𝑜 ·𝑜 𝑤) +𝑜 𝑧))
29	27, 28	omf1o 8219	. . . . . . . . . 10 ⊢ ((𝑊 ∈ On ∧ 2𝑜 ∈ On) → (𝑌 ∘ ◡𝑋):(𝑊 ·𝑜 2𝑜)–1-1-onto→(2𝑜 ·𝑜 𝑊))
30	20, 9, 29	sylancl 566	. . . . . . . . 9 ⊢ (𝜑 → (𝑌 ∘ ◡𝑋):(𝑊 ·𝑜 2𝑜)–1-1-onto→(2𝑜 ·𝑜 𝑊))
31		ondif1 7735	. . . . . . . . . . 11 ⊢ (ω ∈ (On ∖ 1𝑜) ↔ (ω ∈ On ∧ ∅ ∈ ω))
32	7, 13, 31	mpbir2an 682	. . . . . . . . . 10 ⊢ ω ∈ (On ∖ 1𝑜)
33	32	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ω ∈ (On ∖ 1𝑜))
34		omcl 7770	. . . . . . . . . 10 ⊢ ((𝑊 ∈ On ∧ 2𝑜 ∈ On) → (𝑊 ·𝑜 2𝑜) ∈ On)
35	20, 9, 34	sylancl 566	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 ·𝑜 2𝑜) ∈ On)
36		omcl 7770	. . . . . . . . . 10 ⊢ ((2𝑜 ∈ On ∧ 𝑊 ∈ On) → (2𝑜 ·𝑜 𝑊) ∈ On)
37	12, 20, 36	syl2anc 565	. . . . . . . . 9 ⊢ (𝜑 → (2𝑜 ·𝑜 𝑊) ∈ On)
38		fvresi 6583	. . . . . . . . . 10 ⊢ (∅ ∈ ω → (( I ↾ ω)‘∅) = ∅)
39	13, 38	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → (( I ↾ ω)‘∅) = ∅)
40		infxpenc.l	. . . . . . . . 9 ⊢ 𝐿 = (𝑦 ∈ {𝑥 ∈ (ω ↑𝑚 (𝑊 ·𝑜 2𝑜)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ◡(𝑌 ∘ ◡𝑋))))
41		infxpenc.j	. . . . . . . . 9 ⊢ 𝐽 = (((ω CNF (2𝑜 ·𝑜 𝑊)) ∘ 𝐿) ∘ ◡(ω CNF (𝑊 ·𝑜 2𝑜)))
42	26, 30, 33, 35, 8, 37, 39, 40, 41	oef1o 8759	. . . . . . . 8 ⊢ (𝜑 → 𝐽:(ω ↑𝑜 (𝑊 ·𝑜 2𝑜))–1-1-onto→(ω ↑𝑜 (2𝑜 ·𝑜 𝑊)))
43		oeoe 7833	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ 2𝑜 ∈ On ∧ 𝑊 ∈ On) → ((ω ↑𝑜 2𝑜) ↑𝑜 𝑊) = (ω ↑𝑜 (2𝑜 ·𝑜 𝑊)))
44	8, 12, 20, 43	syl3anc 1476	. . . . . . . . 9 ⊢ (𝜑 → ((ω ↑𝑜 2𝑜) ↑𝑜 𝑊) = (ω ↑𝑜 (2𝑜 ·𝑜 𝑊)))
45		f1oeq3 6270	. . . . . . . . 9 ⊢ (((ω ↑𝑜 2𝑜) ↑𝑜 𝑊) = (ω ↑𝑜 (2𝑜 ·𝑜 𝑊)) → (𝐽:(ω ↑𝑜 (𝑊 ·𝑜 2𝑜))–1-1-onto→((ω ↑𝑜 2𝑜) ↑𝑜 𝑊) ↔ 𝐽:(ω ↑𝑜 (𝑊 ·𝑜 2𝑜))–1-1-onto→(ω ↑𝑜 (2𝑜 ·𝑜 𝑊))))
46	44, 45	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐽:(ω ↑𝑜 (𝑊 ·𝑜 2𝑜))–1-1-onto→((ω ↑𝑜 2𝑜) ↑𝑜 𝑊) ↔ 𝐽:(ω ↑𝑜 (𝑊 ·𝑜 2𝑜))–1-1-onto→(ω ↑𝑜 (2𝑜 ·𝑜 𝑊))))
47	42, 46	mpbird 247	. . . . . . 7 ⊢ (𝜑 → 𝐽:(ω ↑𝑜 (𝑊 ·𝑜 2𝑜))–1-1-onto→((ω ↑𝑜 2𝑜) ↑𝑜 𝑊))
48		f1oco 6300	. . . . . . 7 ⊢ ((𝐻:((ω ↑𝑜 2𝑜) ↑𝑜 𝑊)–1-1-onto→(ω ↑𝑜 𝑊) ∧ 𝐽:(ω ↑𝑜 (𝑊 ·𝑜 2𝑜))–1-1-onto→((ω ↑𝑜 2𝑜) ↑𝑜 𝑊)) → (𝐻 ∘ 𝐽):(ω ↑𝑜 (𝑊 ·𝑜 2𝑜))–1-1-onto→(ω ↑𝑜 𝑊))
49	24, 47, 48	syl2anc 565	. . . . . 6 ⊢ (𝜑 → (𝐻 ∘ 𝐽):(ω ↑𝑜 (𝑊 ·𝑜 2𝑜))–1-1-onto→(ω ↑𝑜 𝑊))
50		df-2o 7714	. . . . . . . . . . . 12 ⊢ 2𝑜 = suc 1𝑜
51	50	oveq2i 6804	. . . . . . . . . . 11 ⊢ (𝑊 ·𝑜 2𝑜) = (𝑊 ·𝑜 suc 1𝑜)
52		1on 7720	. . . . . . . . . . . 12 ⊢ 1𝑜 ∈ On
53		omsuc 7760	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ On ∧ 1𝑜 ∈ On) → (𝑊 ·𝑜 suc 1𝑜) = ((𝑊 ·𝑜 1𝑜) +𝑜 𝑊))
54	20, 52, 53	sylancl 566	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑊 ·𝑜 suc 1𝑜) = ((𝑊 ·𝑜 1𝑜) +𝑜 𝑊))
55	51, 54	syl5eq 2817	. . . . . . . . . 10 ⊢ (𝜑 → (𝑊 ·𝑜 2𝑜) = ((𝑊 ·𝑜 1𝑜) +𝑜 𝑊))
56		om1 7776	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ On → (𝑊 ·𝑜 1𝑜) = 𝑊)
57	20, 56	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑊 ·𝑜 1𝑜) = 𝑊)
58	57	oveq1d 6808	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑊 ·𝑜 1𝑜) +𝑜 𝑊) = (𝑊 +𝑜 𝑊))
59	55, 58	eqtrd 2805	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 ·𝑜 2𝑜) = (𝑊 +𝑜 𝑊))
60	59	oveq2d 6809	. . . . . . . 8 ⊢ (𝜑 → (ω ↑𝑜 (𝑊 ·𝑜 2𝑜)) = (ω ↑𝑜 (𝑊 +𝑜 𝑊)))
61		oeoa 7831	. . . . . . . . 9 ⊢ ((ω ∈ On ∧ 𝑊 ∈ On ∧ 𝑊 ∈ On) → (ω ↑𝑜 (𝑊 +𝑜 𝑊)) = ((ω ↑𝑜 𝑊) ·𝑜 (ω ↑𝑜 𝑊)))
62	8, 20, 20, 61	syl3anc 1476	. . . . . . . 8 ⊢ (𝜑 → (ω ↑𝑜 (𝑊 +𝑜 𝑊)) = ((ω ↑𝑜 𝑊) ·𝑜 (ω ↑𝑜 𝑊)))
63	60, 62	eqtrd 2805	. . . . . . 7 ⊢ (𝜑 → (ω ↑𝑜 (𝑊 ·𝑜 2𝑜)) = ((ω ↑𝑜 𝑊) ·𝑜 (ω ↑𝑜 𝑊)))
64		f1oeq2 6269	. . . . . . 7 ⊢ ((ω ↑𝑜 (𝑊 ·𝑜 2𝑜)) = ((ω ↑𝑜 𝑊) ·𝑜 (ω ↑𝑜 𝑊)) → ((𝐻 ∘ 𝐽):(ω ↑𝑜 (𝑊 ·𝑜 2𝑜))–1-1-onto→(ω ↑𝑜 𝑊) ↔ (𝐻 ∘ 𝐽):((ω ↑𝑜 𝑊) ·𝑜 (ω ↑𝑜 𝑊))–1-1-onto→(ω ↑𝑜 𝑊)))
65	63, 64	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝐻 ∘ 𝐽):(ω ↑𝑜 (𝑊 ·𝑜 2𝑜))–1-1-onto→(ω ↑𝑜 𝑊) ↔ (𝐻 ∘ 𝐽):((ω ↑𝑜 𝑊) ·𝑜 (ω ↑𝑜 𝑊))–1-1-onto→(ω ↑𝑜 𝑊)))
66	49, 65	mpbid 222	. . . . 5 ⊢ (𝜑 → (𝐻 ∘ 𝐽):((ω ↑𝑜 𝑊) ·𝑜 (ω ↑𝑜 𝑊))–1-1-onto→(ω ↑𝑜 𝑊))
67		oecl 7771	. . . . . . 7 ⊢ ((ω ∈ On ∧ 𝑊 ∈ On) → (ω ↑𝑜 𝑊) ∈ On)
68	8, 20, 67	syl2anc 565	. . . . . 6 ⊢ (𝜑 → (ω ↑𝑜 𝑊) ∈ On)
69		infxpenc.z	. . . . . . 7 ⊢ 𝑍 = (𝑥 ∈ (ω ↑𝑜 𝑊), 𝑦 ∈ (ω ↑𝑜 𝑊) ↦ (((ω ↑𝑜 𝑊) ·𝑜 𝑥) +𝑜 𝑦))
70	69	omxpenlem 8217	. . . . . 6 ⊢ (((ω ↑𝑜 𝑊) ∈ On ∧ (ω ↑𝑜 𝑊) ∈ On) → 𝑍:((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊))–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (ω ↑𝑜 𝑊)))
71	68, 68, 70	syl2anc 565	. . . . 5 ⊢ (𝜑 → 𝑍:((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊))–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (ω ↑𝑜 𝑊)))
72		f1oco 6300	. . . . 5 ⊢ (((𝐻 ∘ 𝐽):((ω ↑𝑜 𝑊) ·𝑜 (ω ↑𝑜 𝑊))–1-1-onto→(ω ↑𝑜 𝑊) ∧ 𝑍:((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊))–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (ω ↑𝑜 𝑊))) → ((𝐻 ∘ 𝐽) ∘ 𝑍):((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊))–1-1-onto→(ω ↑𝑜 𝑊))
73	66, 71, 72	syl2anc 565	. . . 4 ⊢ (𝜑 → ((𝐻 ∘ 𝐽) ∘ 𝑍):((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊))–1-1-onto→(ω ↑𝑜 𝑊))
74		f1of 6278	. . . . . . . . . 10 ⊢ (𝑁:𝐴–1-1-onto→(ω ↑𝑜 𝑊) → 𝑁:𝐴⟶(ω ↑𝑜 𝑊))
75	1, 74	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑁:𝐴⟶(ω ↑𝑜 𝑊))
76	75	feqmptd 6391	. . . . . . . 8 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐴 ↦ (𝑁‘𝑥)))
77		f1oeq1 6268	. . . . . . . 8 ⊢ (𝑁 = (𝑥 ∈ 𝐴 ↦ (𝑁‘𝑥)) → (𝑁:𝐴–1-1-onto→(ω ↑𝑜 𝑊) ↔ (𝑥 ∈ 𝐴 ↦ (𝑁‘𝑥)):𝐴–1-1-onto→(ω ↑𝑜 𝑊)))
78	76, 77	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑁:𝐴–1-1-onto→(ω ↑𝑜 𝑊) ↔ (𝑥 ∈ 𝐴 ↦ (𝑁‘𝑥)):𝐴–1-1-onto→(ω ↑𝑜 𝑊)))
79	1, 78	mpbid 222	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝑁‘𝑥)):𝐴–1-1-onto→(ω ↑𝑜 𝑊))
80	75	feqmptd 6391	. . . . . . . 8 ⊢ (𝜑 → 𝑁 = (𝑦 ∈ 𝐴 ↦ (𝑁‘𝑦)))
81		f1oeq1 6268	. . . . . . . 8 ⊢ (𝑁 = (𝑦 ∈ 𝐴 ↦ (𝑁‘𝑦)) → (𝑁:𝐴–1-1-onto→(ω ↑𝑜 𝑊) ↔ (𝑦 ∈ 𝐴 ↦ (𝑁‘𝑦)):𝐴–1-1-onto→(ω ↑𝑜 𝑊)))
82	80, 81	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑁:𝐴–1-1-onto→(ω ↑𝑜 𝑊) ↔ (𝑦 ∈ 𝐴 ↦ (𝑁‘𝑦)):𝐴–1-1-onto→(ω ↑𝑜 𝑊)))
83	1, 82	mpbid 222	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ (𝑁‘𝑦)):𝐴–1-1-onto→(ω ↑𝑜 𝑊))
84	79, 83	xpf1o 8278	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝑁‘𝑥), (𝑁‘𝑦)⟩):(𝐴 × 𝐴)–1-1-onto→((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊)))
85		infxpenc.t	. . . . . 6 ⊢ 𝑇 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝑁‘𝑥), (𝑁‘𝑦)⟩)
86		f1oeq1 6268	. . . . . 6 ⊢ (𝑇 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝑁‘𝑥), (𝑁‘𝑦)⟩) → (𝑇:(𝐴 × 𝐴)–1-1-onto→((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊)) ↔ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝑁‘𝑥), (𝑁‘𝑦)⟩):(𝐴 × 𝐴)–1-1-onto→((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊))))
87	85, 86	ax-mp 5	. . . . 5 ⊢ (𝑇:(𝐴 × 𝐴)–1-1-onto→((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊)) ↔ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝑁‘𝑥), (𝑁‘𝑦)⟩):(𝐴 × 𝐴)–1-1-onto→((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊)))
88	84, 87	sylibr 224	. . . 4 ⊢ (𝜑 → 𝑇:(𝐴 × 𝐴)–1-1-onto→((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊)))
89		f1oco 6300	. . . 4 ⊢ ((((𝐻 ∘ 𝐽) ∘ 𝑍):((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊))–1-1-onto→(ω ↑𝑜 𝑊) ∧ 𝑇:(𝐴 × 𝐴)–1-1-onto→((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊))) → (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇):(𝐴 × 𝐴)–1-1-onto→(ω ↑𝑜 𝑊))
90	73, 88, 89	syl2anc 565	. . 3 ⊢ (𝜑 → (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇):(𝐴 × 𝐴)–1-1-onto→(ω ↑𝑜 𝑊))
91		f1oco 6300	. . 3 ⊢ ((◡𝑁:(ω ↑𝑜 𝑊)–1-1-onto→𝐴 ∧ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇):(𝐴 × 𝐴)–1-1-onto→(ω ↑𝑜 𝑊)) → (◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto→𝐴)
92	3, 90, 91	syl2anc 565	. 2 ⊢ (𝜑 → (◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto→𝐴)
93		infxpenc.g	. . 3 ⊢ 𝐺 = (◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇))
94		f1oeq1 6268	. . 3 ⊢ (𝐺 = (◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇)) → (𝐺:(𝐴 × 𝐴)–1-1-onto→𝐴 ↔ (◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto→𝐴))
95	93, 94	ax-mp 5	. 2 ⊢ (𝐺:(𝐴 × 𝐴)–1-1-onto→𝐴 ↔ (◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto→𝐴)
96	92, 95	sylibr 224	1 ⊢ (𝜑 → 𝐺:(𝐴 × 𝐴)–1-1-onto→𝐴)

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1631   ∈ wcel 2145  {crab 3065   ∖ cdif 3720   ⊆ wss 3723  ∅c0 4063  ⟨cop 4322   class class class wbr 4786   ↦ cmpt 4863   I cid 5156   × cxp 5247  ^◡ccnv 5248   ↾ cres 5251   ∘ ccom 5253  Oncon0 5866  suc csuc 5868  ⟶wf 6027  –1-1-onto→wf1o 6030  ‘cfv 6031  (class class class)co 6793   ↦ cmpt2 6795  ωcom 7212  1_𝑜c1o 7706  2_𝑜c2o 7707   +_𝑜 coa 7710   ·_𝑜 comu 7711   ↑_𝑜 coe 7712   ↑_𝑚 cmap 8009   finSupp cfsupp 8431   CNF ccnf 8722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-supp 7447  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-seqom 7696  df-1o 7713  df-2o 7714  df-oadd 7717  df-omul 7718  df-oexp 7719  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fsupp 8432  df-oi 8571  df-cnf 8723

This theorem is referenced by:  infxpenc2lem2  9043