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Theorem infxpenc2lem3 10030
Description: Lemma for infxpenc2 10031. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 7-Jul-2019.)
Hypotheses
Ref Expression
infxpenc2.1 (πœ‘ β†’ 𝐴 ∈ On)
infxpenc2.2 (πœ‘ β†’ βˆ€π‘ ∈ 𝐴 (Ο‰ βŠ† 𝑏 β†’ βˆƒπ‘€ ∈ (On βˆ– 1o)(π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀)))
infxpenc2.3 π‘Š = (β—‘(π‘₯ ∈ (On βˆ– 1o) ↦ (Ο‰ ↑o π‘₯))β€˜ran (π‘›β€˜π‘))
infxpenc2.4 (πœ‘ β†’ 𝐹:(Ο‰ ↑o 2o)–1-1-ontoβ†’Ο‰)
infxpenc2.5 (πœ‘ β†’ (πΉβ€˜βˆ…) = βˆ…)
Assertion
Ref Expression
infxpenc2lem3 (πœ‘ β†’ βˆƒπ‘”βˆ€π‘ ∈ 𝐴 (Ο‰ βŠ† 𝑏 β†’ (π‘”β€˜π‘):(𝑏 Γ— 𝑏)–1-1-onto→𝑏))
Distinct variable groups:   𝑔,𝑏,𝑛,𝑀,π‘₯,𝐴   πœ‘,𝑏,𝑀,π‘₯   𝑔,π‘Š,𝑀,π‘₯   𝑔,𝐹,π‘₯
Allowed substitution hints:   πœ‘(𝑔,𝑛)   𝐹(𝑀,𝑛,𝑏)   π‘Š(𝑛,𝑏)

Proof of Theorem infxpenc2lem3
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 infxpenc2.1 . 2 (πœ‘ β†’ 𝐴 ∈ On)
2 infxpenc2.2 . 2 (πœ‘ β†’ βˆ€π‘ ∈ 𝐴 (Ο‰ βŠ† 𝑏 β†’ βˆƒπ‘€ ∈ (On βˆ– 1o)(π‘›β€˜π‘):𝑏–1-1-ontoβ†’(Ο‰ ↑o 𝑀)))
3 infxpenc2.3 . 2 π‘Š = (β—‘(π‘₯ ∈ (On βˆ– 1o) ↦ (Ο‰ ↑o π‘₯))β€˜ran (π‘›β€˜π‘))
4 infxpenc2.4 . 2 (πœ‘ β†’ 𝐹:(Ο‰ ↑o 2o)–1-1-ontoβ†’Ο‰)
5 infxpenc2.5 . 2 (πœ‘ β†’ (πΉβ€˜βˆ…) = βˆ…)
6 eqid 2727 . 2 (𝑦 ∈ {π‘₯ ∈ ((Ο‰ ↑o 2o) ↑m π‘Š) ∣ π‘₯ finSupp βˆ…} ↦ (𝐹 ∘ (𝑦 ∘ β—‘( I β†Ύ π‘Š)))) = (𝑦 ∈ {π‘₯ ∈ ((Ο‰ ↑o 2o) ↑m π‘Š) ∣ π‘₯ finSupp βˆ…} ↦ (𝐹 ∘ (𝑦 ∘ β—‘( I β†Ύ π‘Š))))
7 eqid 2727 . 2 (((Ο‰ CNF π‘Š) ∘ (𝑦 ∈ {π‘₯ ∈ ((Ο‰ ↑o 2o) ↑m π‘Š) ∣ π‘₯ finSupp βˆ…} ↦ (𝐹 ∘ (𝑦 ∘ β—‘( I β†Ύ π‘Š))))) ∘ β—‘((Ο‰ ↑o 2o) CNF π‘Š)) = (((Ο‰ CNF π‘Š) ∘ (𝑦 ∈ {π‘₯ ∈ ((Ο‰ ↑o 2o) ↑m π‘Š) ∣ π‘₯ finSupp βˆ…} ↦ (𝐹 ∘ (𝑦 ∘ β—‘( I β†Ύ π‘Š))))) ∘ β—‘((Ο‰ ↑o 2o) CNF π‘Š))
8 eqid 2727 . 2 (𝑦 ∈ {π‘₯ ∈ (Ο‰ ↑m (π‘Š Β·o 2o)) ∣ π‘₯ finSupp βˆ…} ↦ (( I β†Ύ Ο‰) ∘ (𝑦 ∘ β—‘((𝑧 ∈ 2o, 𝑀 ∈ π‘Š ↦ ((2o Β·o 𝑀) +o 𝑧)) ∘ β—‘(𝑧 ∈ 2o, 𝑀 ∈ π‘Š ↦ ((π‘Š Β·o 𝑧) +o 𝑀)))))) = (𝑦 ∈ {π‘₯ ∈ (Ο‰ ↑m (π‘Š Β·o 2o)) ∣ π‘₯ finSupp βˆ…} ↦ (( I β†Ύ Ο‰) ∘ (𝑦 ∘ β—‘((𝑧 ∈ 2o, 𝑀 ∈ π‘Š ↦ ((2o Β·o 𝑀) +o 𝑧)) ∘ β—‘(𝑧 ∈ 2o, 𝑀 ∈ π‘Š ↦ ((π‘Š Β·o 𝑧) +o 𝑀))))))
9 eqid 2727 . 2 (𝑧 ∈ 2o, 𝑀 ∈ π‘Š ↦ ((π‘Š Β·o 𝑧) +o 𝑀)) = (𝑧 ∈ 2o, 𝑀 ∈ π‘Š ↦ ((π‘Š Β·o 𝑧) +o 𝑀))
10 eqid 2727 . 2 (𝑧 ∈ 2o, 𝑀 ∈ π‘Š ↦ ((2o Β·o 𝑀) +o 𝑧)) = (𝑧 ∈ 2o, 𝑀 ∈ π‘Š ↦ ((2o Β·o 𝑀) +o 𝑧))
11 eqid 2727 . 2 (((Ο‰ CNF (2o Β·o π‘Š)) ∘ (𝑦 ∈ {π‘₯ ∈ (Ο‰ ↑m (π‘Š Β·o 2o)) ∣ π‘₯ finSupp βˆ…} ↦ (( I β†Ύ Ο‰) ∘ (𝑦 ∘ β—‘((𝑧 ∈ 2o, 𝑀 ∈ π‘Š ↦ ((2o Β·o 𝑀) +o 𝑧)) ∘ β—‘(𝑧 ∈ 2o, 𝑀 ∈ π‘Š ↦ ((π‘Š Β·o 𝑧) +o 𝑀))))))) ∘ β—‘(Ο‰ CNF (π‘Š Β·o 2o))) = (((Ο‰ CNF (2o Β·o π‘Š)) ∘ (𝑦 ∈ {π‘₯ ∈ (Ο‰ ↑m (π‘Š Β·o 2o)) ∣ π‘₯ finSupp βˆ…} ↦ (( I β†Ύ Ο‰) ∘ (𝑦 ∘ β—‘((𝑧 ∈ 2o, 𝑀 ∈ π‘Š ↦ ((2o Β·o 𝑀) +o 𝑧)) ∘ β—‘(𝑧 ∈ 2o, 𝑀 ∈ π‘Š ↦ ((π‘Š Β·o 𝑧) +o 𝑀))))))) ∘ β—‘(Ο‰ CNF (π‘Š Β·o 2o)))
12 eqid 2727 . 2 (π‘₯ ∈ (Ο‰ ↑o π‘Š), 𝑦 ∈ (Ο‰ ↑o π‘Š) ↦ (((Ο‰ ↑o π‘Š) Β·o π‘₯) +o 𝑦)) = (π‘₯ ∈ (Ο‰ ↑o π‘Š), 𝑦 ∈ (Ο‰ ↑o π‘Š) ↦ (((Ο‰ ↑o π‘Š) Β·o π‘₯) +o 𝑦))
13 eqid 2727 . 2 (π‘₯ ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ ⟨((π‘›β€˜π‘)β€˜π‘₯), ((π‘›β€˜π‘)β€˜π‘¦)⟩) = (π‘₯ ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ ⟨((π‘›β€˜π‘)β€˜π‘₯), ((π‘›β€˜π‘)β€˜π‘¦)⟩)
14 eqid 2727 . 2 (β—‘(π‘›β€˜π‘) ∘ ((((((Ο‰ CNF π‘Š) ∘ (𝑦 ∈ {π‘₯ ∈ ((Ο‰ ↑o 2o) ↑m π‘Š) ∣ π‘₯ finSupp βˆ…} ↦ (𝐹 ∘ (𝑦 ∘ β—‘( I β†Ύ π‘Š))))) ∘ β—‘((Ο‰ ↑o 2o) CNF π‘Š)) ∘ (((Ο‰ CNF (2o Β·o π‘Š)) ∘ (𝑦 ∈ {π‘₯ ∈ (Ο‰ ↑m (π‘Š Β·o 2o)) ∣ π‘₯ finSupp βˆ…} ↦ (( I β†Ύ Ο‰) ∘ (𝑦 ∘ β—‘((𝑧 ∈ 2o, 𝑀 ∈ π‘Š ↦ ((2o Β·o 𝑀) +o 𝑧)) ∘ β—‘(𝑧 ∈ 2o, 𝑀 ∈ π‘Š ↦ ((π‘Š Β·o 𝑧) +o 𝑀))))))) ∘ β—‘(Ο‰ CNF (π‘Š Β·o 2o)))) ∘ (π‘₯ ∈ (Ο‰ ↑o π‘Š), 𝑦 ∈ (Ο‰ ↑o π‘Š) ↦ (((Ο‰ ↑o π‘Š) Β·o π‘₯) +o 𝑦))) ∘ (π‘₯ ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ ⟨((π‘›β€˜π‘)β€˜π‘₯), ((π‘›β€˜π‘)β€˜π‘¦)⟩))) = (β—‘(π‘›β€˜π‘) ∘ ((((((Ο‰ CNF π‘Š) ∘ (𝑦 ∈ {π‘₯ ∈ ((Ο‰ ↑o 2o) ↑m π‘Š) ∣ π‘₯ finSupp βˆ…} ↦ (𝐹 ∘ (𝑦 ∘ β—‘( I β†Ύ π‘Š))))) ∘ β—‘((Ο‰ ↑o 2o) CNF π‘Š)) ∘ (((Ο‰ CNF (2o Β·o π‘Š)) ∘ (𝑦 ∈ {π‘₯ ∈ (Ο‰ ↑m (π‘Š Β·o 2o)) ∣ π‘₯ finSupp βˆ…} ↦ (( I β†Ύ Ο‰) ∘ (𝑦 ∘ β—‘((𝑧 ∈ 2o, 𝑀 ∈ π‘Š ↦ ((2o Β·o 𝑀) +o 𝑧)) ∘ β—‘(𝑧 ∈ 2o, 𝑀 ∈ π‘Š ↦ ((π‘Š Β·o 𝑧) +o 𝑀))))))) ∘ β—‘(Ο‰ CNF (π‘Š Β·o 2o)))) ∘ (π‘₯ ∈ (Ο‰ ↑o π‘Š), 𝑦 ∈ (Ο‰ ↑o π‘Š) ↦ (((Ο‰ ↑o π‘Š) Β·o π‘₯) +o 𝑦))) ∘ (π‘₯ ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ ⟨((π‘›β€˜π‘)β€˜π‘₯), ((π‘›β€˜π‘)β€˜π‘¦)⟩)))
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14infxpenc2lem2 10029 1 (πœ‘ β†’ βˆƒπ‘”βˆ€π‘ ∈ 𝐴 (Ο‰ βŠ† 𝑏 β†’ (π‘”β€˜π‘):(𝑏 Γ— 𝑏)–1-1-onto→𝑏))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1534  βˆƒwex 1774   ∈ wcel 2099  βˆ€wral 3056  βˆƒwrex 3065  {crab 3427   βˆ– cdif 3941   βŠ† wss 3944  βˆ…c0 4318  βŸ¨cop 4630   class class class wbr 5142   ↦ cmpt 5225   I cid 5569   Γ— cxp 5670  β—‘ccnv 5671  ran crn 5673   β†Ύ cres 5674   ∘ ccom 5676  Oncon0 6363  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7414   ∈ cmpo 7416  Ο‰com 7862  1oc1o 8471  2oc2o 8472   +o coa 8475   Β·o comu 8476   ↑o coe 8477   ↑m cmap 8834   finSupp cfsupp 9375   CNF ccnf 9670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-inf2 9650
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-1st 7985  df-2nd 7986  df-supp 8158  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-seqom 8460  df-1o 8478  df-2o 8479  df-oadd 8482  df-omul 8483  df-oexp 8484  df-er 8716  df-map 8836  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-fsupp 9376  df-oi 9519  df-cnf 9671
This theorem is referenced by:  infxpenc2  10031
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