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Theorem infxpenc 8793
Description: A canonical version of infxpen 8789, by a completely different approach (although it uses infxpen 8789 via xpomen 8790). Using Cantor's normal form, we can show that 𝐴𝑜 𝐵 respects equinumerosity (oef1o 8547), 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 8555.) (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 6111 . . . 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 6136 . . . . . . . . 9 ( I ↾ 𝑊):𝑊1-1-onto𝑊
65a1i 11 . . . . . . . 8 (𝜑 → ( I ↾ 𝑊):𝑊1-1-onto𝑊)
7 omelon 8495 . . . . . . . . . . 11 ω ∈ On
87a1i 11 . . . . . . . . . 10 (𝜑 → ω ∈ On)
9 2on 7520 . . . . . . . . . 10 2𝑜 ∈ On
10 oecl 7569 . . . . . . . . . 10 ((ω ∈ On ∧ 2𝑜 ∈ On) → (ω ↑𝑜 2𝑜) ∈ On)
118, 9, 10sylancl 693 . . . . . . . . 9 (𝜑 → (ω ↑𝑜 2𝑜) ∈ On)
129a1i 11 . . . . . . . . . 10 (𝜑 → 2𝑜 ∈ On)
13 peano1 7039 . . . . . . . . . . 11 ∅ ∈ ω
1413a1i 11 . . . . . . . . . 10 (𝜑 → ∅ ∈ ω)
15 oen0 7618 . . . . . . . . . 10 (((ω ∈ On ∧ 2𝑜 ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑𝑜 2𝑜))
168, 12, 14, 15syl21anc 1322 . . . . . . . . 9 (𝜑 → ∅ ∈ (ω ↑𝑜 2𝑜))
17 ondif1 7533 . . . . . . . . 9 ((ω ↑𝑜 2𝑜) ∈ (On ∖ 1𝑜) ↔ ((ω ↑𝑜 2𝑜) ∈ On ∧ ∅ ∈ (ω ↑𝑜 2𝑜)))
1811, 16, 17sylanbrc 697 . . . . . . . 8 (𝜑 → (ω ↑𝑜 2𝑜) ∈ (On ∖ 1𝑜))
19 infxpenc.3 . . . . . . . . 9 (𝜑𝑊 ∈ (On ∖ 1𝑜))
2019eldifad 3571 . . . . . . . 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 8547 . . . . . . 7 (𝜑𝐻:((ω ↑𝑜 2𝑜) ↑𝑜 𝑊)–1-1-onto→(ω ↑𝑜 𝑊))
25 f1oi 6136 . . . . . . . . . 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 8015 . . . . . . . . . 10 ((𝑊 ∈ On ∧ 2𝑜 ∈ On) → (𝑌𝑋):(𝑊 ·𝑜 2𝑜)–1-1-onto→(2𝑜 ·𝑜 𝑊))
3020, 9, 29sylancl 693 . . . . . . . . 9 (𝜑 → (𝑌𝑋):(𝑊 ·𝑜 2𝑜)–1-1-onto→(2𝑜 ·𝑜 𝑊))
31 ondif1 7533 . . . . . . . . . . 11 (ω ∈ (On ∖ 1𝑜) ↔ (ω ∈ On ∧ ∅ ∈ ω))
327, 13, 31mpbir2an 954 . . . . . . . . . 10 ω ∈ (On ∖ 1𝑜)
3332a1i 11 . . . . . . . . 9 (𝜑 → ω ∈ (On ∖ 1𝑜))
34 omcl 7568 . . . . . . . . . 10 ((𝑊 ∈ On ∧ 2𝑜 ∈ On) → (𝑊 ·𝑜 2𝑜) ∈ On)
3520, 9, 34sylancl 693 . . . . . . . . 9 (𝜑 → (𝑊 ·𝑜 2𝑜) ∈ On)
36 omcl 7568 . . . . . . . . . 10 ((2𝑜 ∈ On ∧ 𝑊 ∈ On) → (2𝑜 ·𝑜 𝑊) ∈ On)
3712, 20, 36syl2anc 692 . . . . . . . . 9 (𝜑 → (2𝑜 ·𝑜 𝑊) ∈ On)
38 fvresi 6399 . . . . . . . . . 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 8547 . . . . . . . 8 (𝜑𝐽:(ω ↑𝑜 (𝑊 ·𝑜 2𝑜))–1-1-onto→(ω ↑𝑜 (2𝑜 ·𝑜 𝑊)))
43 oeoe 7631 . . . . . . . . . 10 ((ω ∈ On ∧ 2𝑜 ∈ On ∧ 𝑊 ∈ On) → ((ω ↑𝑜 2𝑜) ↑𝑜 𝑊) = (ω ↑𝑜 (2𝑜 ·𝑜 𝑊)))
448, 12, 20, 43syl3anc 1323 . . . . . . . . 9 (𝜑 → ((ω ↑𝑜 2𝑜) ↑𝑜 𝑊) = (ω ↑𝑜 (2𝑜 ·𝑜 𝑊)))
45 f1oeq3 6091 . . . . . . . . 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 6121 . . . . . . 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 692 . . . . . 6 (𝜑 → (𝐻𝐽):(ω ↑𝑜 (𝑊 ·𝑜 2𝑜))–1-1-onto→(ω ↑𝑜 𝑊))
50 df-2o 7513 . . . . . . . . . . . 12 2𝑜 = suc 1𝑜
5150oveq2i 6621 . . . . . . . . . . 11 (𝑊 ·𝑜 2𝑜) = (𝑊 ·𝑜 suc 1𝑜)
52 1on 7519 . . . . . . . . . . . 12 1𝑜 ∈ On
53 omsuc 7558 . . . . . . . . . . . 12 ((𝑊 ∈ On ∧ 1𝑜 ∈ On) → (𝑊 ·𝑜 suc 1𝑜) = ((𝑊 ·𝑜 1𝑜) +𝑜 𝑊))
5420, 52, 53sylancl 693 . . . . . . . . . . 11 (𝜑 → (𝑊 ·𝑜 suc 1𝑜) = ((𝑊 ·𝑜 1𝑜) +𝑜 𝑊))
5551, 54syl5eq 2667 . . . . . . . . . 10 (𝜑 → (𝑊 ·𝑜 2𝑜) = ((𝑊 ·𝑜 1𝑜) +𝑜 𝑊))
56 om1 7574 . . . . . . . . . . . 12 (𝑊 ∈ On → (𝑊 ·𝑜 1𝑜) = 𝑊)
5720, 56syl 17 . . . . . . . . . . 11 (𝜑 → (𝑊 ·𝑜 1𝑜) = 𝑊)
5857oveq1d 6625 . . . . . . . . . 10 (𝜑 → ((𝑊 ·𝑜 1𝑜) +𝑜 𝑊) = (𝑊 +𝑜 𝑊))
5955, 58eqtrd 2655 . . . . . . . . 9 (𝜑 → (𝑊 ·𝑜 2𝑜) = (𝑊 +𝑜 𝑊))
6059oveq2d 6626 . . . . . . . 8 (𝜑 → (ω ↑𝑜 (𝑊 ·𝑜 2𝑜)) = (ω ↑𝑜 (𝑊 +𝑜 𝑊)))
61 oeoa 7629 . . . . . . . . 9 ((ω ∈ On ∧ 𝑊 ∈ On ∧ 𝑊 ∈ On) → (ω ↑𝑜 (𝑊 +𝑜 𝑊)) = ((ω ↑𝑜 𝑊) ·𝑜 (ω ↑𝑜 𝑊)))
628, 20, 20, 61syl3anc 1323 . . . . . . . 8 (𝜑 → (ω ↑𝑜 (𝑊 +𝑜 𝑊)) = ((ω ↑𝑜 𝑊) ·𝑜 (ω ↑𝑜 𝑊)))
6360, 62eqtrd 2655 . . . . . . 7 (𝜑 → (ω ↑𝑜 (𝑊 ·𝑜 2𝑜)) = ((ω ↑𝑜 𝑊) ·𝑜 (ω ↑𝑜 𝑊)))
64 f1oeq2 6090 . . . . . . 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 7569 . . . . . . 7 ((ω ∈ On ∧ 𝑊 ∈ On) → (ω ↑𝑜 𝑊) ∈ On)
688, 20, 67syl2anc 692 . . . . . 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 8013 . . . . . 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 692 . . . . 5 (𝜑𝑍:((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊))–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (ω ↑𝑜 𝑊)))
72 f1oco 6121 . . . . 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 692 . . . 4 (𝜑 → ((𝐻𝐽) ∘ 𝑍):((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊))–1-1-onto→(ω ↑𝑜 𝑊))
74 f1of 6099 . . . . . . . . . 10 (𝑁:𝐴1-1-onto→(ω ↑𝑜 𝑊) → 𝑁:𝐴⟶(ω ↑𝑜 𝑊))
751, 74syl 17 . . . . . . . . 9 (𝜑𝑁:𝐴⟶(ω ↑𝑜 𝑊))
7675feqmptd 6211 . . . . . . . 8 (𝜑𝑁 = (𝑥𝐴 ↦ (𝑁𝑥)))
77 f1oeq1 6089 . . . . . . . 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 6211 . . . . . . . 8 (𝜑𝑁 = (𝑦𝐴 ↦ (𝑁𝑦)))
81 f1oeq1 6089 . . . . . . . 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 8074 . . . . 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 6089 . . . . . 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 6121 . . . 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 692 . . 3 (𝜑 → (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇):(𝐴 × 𝐴)–1-1-onto→(ω ↑𝑜 𝑊))
91 f1oco 6121 . . 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 692 . 2 (𝜑 → (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto𝐴)
93 infxpenc.g . . 3 𝐺 = (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇))
94 f1oeq1 6089 . . 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 1480  wcel 1987  {crab 2911  cdif 3556  wss 3559  c0 3896  cop 4159   class class class wbr 4618  cmpt 4678   I cid 4989   × cxp 5077  ccnv 5078  cres 5081  ccom 5083  Oncon0 5687  suc csuc 5689  wf 5848  1-1-ontowf1o 5851  cfv 5852  (class class class)co 6610  cmpt2 6612  ωcom 7019  1𝑜c1o 7505  2𝑜c2o 7506   +𝑜 coa 7509   ·𝑜 comu 7510  𝑜 coe 7511  𝑚 cmap 7809   finSupp cfsupp 8227   CNF ccnf 8510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-supp 7248  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-seqom 7495  df-1o 7512  df-2o 7513  df-oadd 7516  df-omul 7517  df-oexp 7518  df-er 7694  df-map 7811  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-fsupp 8228  df-oi 8367  df-cnf 8511
This theorem is referenced by:  infxpenc2lem2  8795
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