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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 𝐴𝑜 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 (𝜑𝐴 ∈ On)
infxpenc.2 (𝜑 → ω ⊆ 𝐴)
infxpenc.3 (𝜑𝑊 ∈ (On ∖ 1𝑜))
infxpenc.4 (𝜑𝐹:(ω ↑𝑜 2𝑜)–1-1-onto→ω)
infxpenc.5 (𝜑 → (𝐹‘∅) = ∅)
infxpenc.6 (𝜑𝑁:𝐴1-1-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(𝑌𝑋))))
infxpenc.x 𝑋 = (𝑧 ∈ 2𝑜, 𝑤𝑊 ↦ ((𝑊 ·𝑜 𝑧) +𝑜 𝑤))
infxpenc.y 𝑌 = (𝑧 ∈ 2𝑜, 𝑤𝑊 ↦ ((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 = (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝑁𝑥), (𝑁𝑦)⟩)
infxpenc.g 𝐺 = (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇))
Assertion
Ref Expression
infxpenc (𝜑𝐺:(𝐴 × 𝐴)–1-1-onto𝐴)
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
StepHypRef Expression
1 infxpenc.6 . . . 4 (𝜑𝑁:𝐴1-1-onto→(ω ↑𝑜 𝑊))
2 f1ocnv 6290 . . . 4 (𝑁:𝐴1-1-onto→(ω ↑𝑜 𝑊) → 𝑁:(ω ↑𝑜 𝑊)–1-1-onto𝐴)
31, 2syl 17 . . 3 (𝜑𝑁:(ω ↑𝑜 𝑊)–1-1-onto𝐴)
4 infxpenc.4 . . . . . . . 8 (𝜑𝐹:(ω ↑𝑜 2𝑜)–1-1-onto→ω)
5 f1oi 6315 . . . . . . . . 9 ( I ↾ 𝑊):𝑊1-1-onto𝑊
65a1i 11 . . . . . . . 8 (𝜑 → ( I ↾ 𝑊):𝑊1-1-onto𝑊)
7 omelon 8707 . . . . . . . . . . 11 ω ∈ On
87a1i 11 . . . . . . . . . 10 (𝜑 → ω ∈ On)
9 2on 7722 . . . . . . . . . 10 2𝑜 ∈ On
10 oecl 7771 . . . . . . . . . 10 ((ω ∈ On ∧ 2𝑜 ∈ On) → (ω ↑𝑜 2𝑜) ∈ On)
118, 9, 10sylancl 566 . . . . . . . . 9 (𝜑 → (ω ↑𝑜 2𝑜) ∈ On)
129a1i 11 . . . . . . . . . 10 (𝜑 → 2𝑜 ∈ On)
13 peano1 7232 . . . . . . . . . . 11 ∅ ∈ ω
1413a1i 11 . . . . . . . . . 10 (𝜑 → ∅ ∈ ω)
15 oen0 7820 . . . . . . . . . 10 (((ω ∈ On ∧ 2𝑜 ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑𝑜 2𝑜))
168, 12, 14, 15syl21anc 1475 . . . . . . . . 9 (𝜑 → ∅ ∈ (ω ↑𝑜 2𝑜))
17 ondif1 7735 . . . . . . . . 9 ((ω ↑𝑜 2𝑜) ∈ (On ∖ 1𝑜) ↔ ((ω ↑𝑜 2𝑜) ∈ On ∧ ∅ ∈ (ω ↑𝑜 2𝑜)))
1811, 16, 17sylanbrc 564 . . . . . . . 8 (𝜑 → (ω ↑𝑜 2𝑜) ∈ (On ∖ 1𝑜))
19 infxpenc.3 . . . . . . . . 9 (𝜑𝑊 ∈ (On ∖ 1𝑜))
2019eldifad 3735 . . . . . . . 8 (𝜑𝑊 ∈ 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))
244, 6, 18, 20, 8, 20, 21, 22, 23oef1o 8759 . . . . . . 7 (𝜑𝐻:((ω ↑𝑜 2𝑜) ↑𝑜 𝑊)–1-1-onto→(ω ↑𝑜 𝑊))
25 f1oi 6315 . . . . . . . . . 10 ( I ↾ ω):ω–1-1-onto→ω
2625a1i 11 . . . . . . . . 9 (𝜑 → ( I ↾ ω):ω–1-1-onto→ω)
27 infxpenc.x . . . . . . . . . . 11 𝑋 = (𝑧 ∈ 2𝑜, 𝑤𝑊 ↦ ((𝑊 ·𝑜 𝑧) +𝑜 𝑤))
28 infxpenc.y . . . . . . . . . . 11 𝑌 = (𝑧 ∈ 2𝑜, 𝑤𝑊 ↦ ((2𝑜 ·𝑜 𝑤) +𝑜 𝑧))
2927, 28omf1o 8219 . . . . . . . . . 10 ((𝑊 ∈ On ∧ 2𝑜 ∈ On) → (𝑌𝑋):(𝑊 ·𝑜 2𝑜)–1-1-onto→(2𝑜 ·𝑜 𝑊))
3020, 9, 29sylancl 566 . . . . . . . . 9 (𝜑 → (𝑌𝑋):(𝑊 ·𝑜 2𝑜)–1-1-onto→(2𝑜 ·𝑜 𝑊))
31 ondif1 7735 . . . . . . . . . . 11 (ω ∈ (On ∖ 1𝑜) ↔ (ω ∈ On ∧ ∅ ∈ ω))
327, 13, 31mpbir2an 682 . . . . . . . . . 10 ω ∈ (On ∖ 1𝑜)
3332a1i 11 . . . . . . . . 9 (𝜑 → ω ∈ (On ∖ 1𝑜))
34 omcl 7770 . . . . . . . . . 10 ((𝑊 ∈ On ∧ 2𝑜 ∈ On) → (𝑊 ·𝑜 2𝑜) ∈ On)
3520, 9, 34sylancl 566 . . . . . . . . 9 (𝜑 → (𝑊 ·𝑜 2𝑜) ∈ On)
36 omcl 7770 . . . . . . . . . 10 ((2𝑜 ∈ On ∧ 𝑊 ∈ On) → (2𝑜 ·𝑜 𝑊) ∈ On)
3712, 20, 36syl2anc 565 . . . . . . . . 9 (𝜑 → (2𝑜 ·𝑜 𝑊) ∈ On)
38 fvresi 6583 . . . . . . . . . 10 (∅ ∈ ω → (( I ↾ ω)‘∅) = ∅)
3913, 38mp1i 13 . . . . . . . . 9 (𝜑 → (( I ↾ ω)‘∅) = ∅)
40 infxpenc.l . . . . . . . . 9 𝐿 = (𝑦 ∈ {𝑥 ∈ (ω ↑𝑚 (𝑊 ·𝑜 2𝑜)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦(𝑌𝑋))))
41 infxpenc.j . . . . . . . . 9 𝐽 = (((ω CNF (2𝑜 ·𝑜 𝑊)) ∘ 𝐿) ∘ (ω CNF (𝑊 ·𝑜 2𝑜)))
4226, 30, 33, 35, 8, 37, 39, 40, 41oef1o 8759 . . . . . . . 8 (𝜑𝐽:(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)))
448, 12, 20, 43syl3anc 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))))
4644, 45syl 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))))
4742, 46mpbird 247 . . . . . . 7 (𝜑𝐽:(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)) → (𝐻𝐽):(ω ↑𝑜 (𝑊 ·𝑜 2𝑜))–1-1-onto→(ω ↑𝑜 𝑊))
4924, 47, 48syl2anc 565 . . . . . 6 (𝜑 → (𝐻𝐽):(ω ↑𝑜 (𝑊 ·𝑜 2𝑜))–1-1-onto→(ω ↑𝑜 𝑊))
50 df-2o 7714 . . . . . . . . . . . 12 2𝑜 = suc 1𝑜
5150oveq2i 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))
5420, 52, 53sylancl 566 . . . . . . . . . . 11 (𝜑 → (𝑊 ·𝑜 suc 1𝑜) = ((𝑊 ·𝑜 1𝑜) +𝑜 𝑊))
5551, 54syl5eq 2817 . . . . . . . . . 10 (𝜑 → (𝑊 ·𝑜 2𝑜) = ((𝑊 ·𝑜 1𝑜) +𝑜 𝑊))
56 om1 7776 . . . . . . . . . . . 12 (𝑊 ∈ On → (𝑊 ·𝑜 1𝑜) = 𝑊)
5720, 56syl 17 . . . . . . . . . . 11 (𝜑 → (𝑊 ·𝑜 1𝑜) = 𝑊)
5857oveq1d 6808 . . . . . . . . . 10 (𝜑 → ((𝑊 ·𝑜 1𝑜) +𝑜 𝑊) = (𝑊 +𝑜 𝑊))
5955, 58eqtrd 2805 . . . . . . . . 9 (𝜑 → (𝑊 ·𝑜 2𝑜) = (𝑊 +𝑜 𝑊))
6059oveq2d 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)))
628, 20, 20, 61syl3anc 1476 . . . . . . . 8 (𝜑 → (ω ↑𝑜 (𝑊 +𝑜 𝑊)) = ((ω ↑𝑜 𝑊) ·𝑜 (ω ↑𝑜 𝑊)))
6360, 62eqtrd 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)) → ((𝐻𝐽):(ω ↑𝑜 (𝑊 ·𝑜 2𝑜))–1-1-onto→(ω ↑𝑜 𝑊) ↔ (𝐻𝐽):((ω ↑𝑜 𝑊) ·𝑜 (ω ↑𝑜 𝑊))–1-1-onto→(ω ↑𝑜 𝑊)))
6563, 64syl 17 . . . . . 6 (𝜑 → ((𝐻𝐽):(ω ↑𝑜 (𝑊 ·𝑜 2𝑜))–1-1-onto→(ω ↑𝑜 𝑊) ↔ (𝐻𝐽):((ω ↑𝑜 𝑊) ·𝑜 (ω ↑𝑜 𝑊))–1-1-onto→(ω ↑𝑜 𝑊)))
6649, 65mpbid 222 . . . . 5 (𝜑 → (𝐻𝐽):((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)
688, 20, 67syl2anc 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))
7069omxpenlem 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)))
7168, 68, 70syl2anc 565 . . . . 5 (𝜑𝑍:((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊))–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (ω ↑𝑜 𝑊)))
72 f1oco 6300 . . . . 5 (((𝐻𝐽):((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))) → ((𝐻𝐽) ∘ 𝑍):((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊))–1-1-onto→(ω ↑𝑜 𝑊))
7366, 71, 72syl2anc 565 . . . 4 (𝜑 → ((𝐻𝐽) ∘ 𝑍):((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊))–1-1-onto→(ω ↑𝑜 𝑊))
74 f1of 6278 . . . . . . . . . 10 (𝑁:𝐴1-1-onto→(ω ↑𝑜 𝑊) → 𝑁:𝐴⟶(ω ↑𝑜 𝑊))
751, 74syl 17 . . . . . . . . 9 (𝜑𝑁:𝐴⟶(ω ↑𝑜 𝑊))
7675feqmptd 6391 . . . . . . . 8 (𝜑𝑁 = (𝑥𝐴 ↦ (𝑁𝑥)))
77 f1oeq1 6268 . . . . . . . 8 (𝑁 = (𝑥𝐴 ↦ (𝑁𝑥)) → (𝑁:𝐴1-1-onto→(ω ↑𝑜 𝑊) ↔ (𝑥𝐴 ↦ (𝑁𝑥)):𝐴1-1-onto→(ω ↑𝑜 𝑊)))
7876, 77syl 17 . . . . . . 7 (𝜑 → (𝑁:𝐴1-1-onto→(ω ↑𝑜 𝑊) ↔ (𝑥𝐴 ↦ (𝑁𝑥)):𝐴1-1-onto→(ω ↑𝑜 𝑊)))
791, 78mpbid 222 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (𝑁𝑥)):𝐴1-1-onto→(ω ↑𝑜 𝑊))
8075feqmptd 6391 . . . . . . . 8 (𝜑𝑁 = (𝑦𝐴 ↦ (𝑁𝑦)))
81 f1oeq1 6268 . . . . . . . 8 (𝑁 = (𝑦𝐴 ↦ (𝑁𝑦)) → (𝑁:𝐴1-1-onto→(ω ↑𝑜 𝑊) ↔ (𝑦𝐴 ↦ (𝑁𝑦)):𝐴1-1-onto→(ω ↑𝑜 𝑊)))
8280, 81syl 17 . . . . . . 7 (𝜑 → (𝑁:𝐴1-1-onto→(ω ↑𝑜 𝑊) ↔ (𝑦𝐴 ↦ (𝑁𝑦)):𝐴1-1-onto→(ω ↑𝑜 𝑊)))
831, 82mpbid 222 . . . . . 6 (𝜑 → (𝑦𝐴 ↦ (𝑁𝑦)):𝐴1-1-onto→(ω ↑𝑜 𝑊))
8479, 83xpf1o 8278 . . . . 5 (𝜑 → (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝑁𝑥), (𝑁𝑦)⟩):(𝐴 × 𝐴)–1-1-onto→((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊)))
85 infxpenc.t . . . . . 6 𝑇 = (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝑁𝑥), (𝑁𝑦)⟩)
86 f1oeq1 6268 . . . . . 6 (𝑇 = (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝑁𝑥), (𝑁𝑦)⟩) → (𝑇:(𝐴 × 𝐴)–1-1-onto→((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊)) ↔ (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝑁𝑥), (𝑁𝑦)⟩):(𝐴 × 𝐴)–1-1-onto→((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊))))
8785, 86ax-mp 5 . . . . 5 (𝑇:(𝐴 × 𝐴)–1-1-onto→((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊)) ↔ (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝑁𝑥), (𝑁𝑦)⟩):(𝐴 × 𝐴)–1-1-onto→((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊)))
8884, 87sylibr 224 . . . 4 (𝜑𝑇:(𝐴 × 𝐴)–1-1-onto→((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊)))
89 f1oco 6300 . . . 4 ((((𝐻𝐽) ∘ 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))) → (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇):(𝐴 × 𝐴)–1-1-onto→(ω ↑𝑜 𝑊))
9073, 88, 89syl2anc 565 . . 3 (𝜑 → (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇):(𝐴 × 𝐴)–1-1-onto→(ω ↑𝑜 𝑊))
91 f1oco 6300 . . 3 ((𝑁:(ω ↑𝑜 𝑊)–1-1-onto𝐴 ∧ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇):(𝐴 × 𝐴)–1-1-onto→(ω ↑𝑜 𝑊)) → (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto𝐴)
923, 90, 91syl2anc 565 . 2 (𝜑 → (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto𝐴)
93 infxpenc.g . . 3 𝐺 = (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇))
94 f1oeq1 6268 . . 3 (𝐺 = (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇)) → (𝐺:(𝐴 × 𝐴)–1-1-onto𝐴 ↔ (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto𝐴))
9593, 94ax-mp 5 . 2 (𝐺:(𝐴 × 𝐴)–1-1-onto𝐴 ↔ (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto𝐴)
9692, 95sylibr 224 1 (𝜑𝐺:(𝐴 × 𝐴)–1-1-onto𝐴)
Colors of variables: wff setvar class
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-ontowf1o 6030  cfv 6031  (class class class)co 6793  cmpt2 6795  ωcom 7212  1𝑜c1o 7706  2𝑜c2o 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
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