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Theorem cantnffval 8504
 Description: The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
Hypotheses
Ref Expression
cantnffval.s 𝑆 = {𝑔 ∈ (𝐴𝑚 𝐵) ∣ 𝑔 finSupp ∅}
cantnffval.a (𝜑𝐴 ∈ On)
cantnffval.b (𝜑𝐵 ∈ On)
Assertion
Ref Expression
cantnffval (𝜑 → (𝐴 CNF 𝐵) = (𝑓𝑆OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅)‘dom )))
Distinct variable groups:   𝑓,𝑔,,𝑘,𝑧,𝐴   𝐵,𝑓,𝑔,,𝑘,𝑧   𝑆,𝑓
Allowed substitution hints:   𝜑(𝑧,𝑓,𝑔,,𝑘)   𝑆(𝑧,𝑔,,𝑘)

Proof of Theorem cantnffval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnffval.a . 2 (𝜑𝐴 ∈ On)
2 cantnffval.b . 2 (𝜑𝐵 ∈ On)
3 oveq12 6613 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑚 𝑦) = (𝐴𝑚 𝐵))
43rabeqdv 3180 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → {𝑔 ∈ (𝑥𝑚 𝑦) ∣ 𝑔 finSupp ∅} = {𝑔 ∈ (𝐴𝑚 𝐵) ∣ 𝑔 finSupp ∅})
5 cantnffval.s . . . . 5 𝑆 = {𝑔 ∈ (𝐴𝑚 𝐵) ∣ 𝑔 finSupp ∅}
64, 5syl6eqr 2673 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → {𝑔 ∈ (𝑥𝑚 𝑦) ∣ 𝑔 finSupp ∅} = 𝑆)
7 simp1l 1083 . . . . . . . . . . 11 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑘 ∈ V ∧ 𝑧 ∈ V) → 𝑥 = 𝐴)
87oveq1d 6619 . . . . . . . . . 10 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑘 ∈ V ∧ 𝑧 ∈ V) → (𝑥𝑜 (𝑘)) = (𝐴𝑜 (𝑘)))
98oveq1d 6619 . . . . . . . . 9 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑘 ∈ V ∧ 𝑧 ∈ V) → ((𝑥𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) = ((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))))
109oveq1d 6619 . . . . . . . 8 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑘 ∈ V ∧ 𝑧 ∈ V) → (((𝑥𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧) = (((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧))
1110mpt2eq3dva 6672 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)))
12 eqid 2621 . . . . . . 7 ∅ = ∅
13 seqomeq12 7494 . . . . . . 7 (((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)) ∧ ∅ = ∅) → seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅))
1411, 12, 13sylancl 693 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅))
1514fveq1d 6150 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅)‘dom ) = (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅)‘dom ))
1615csbeq2dv 3964 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅)‘dom ) = OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅)‘dom ))
176, 16mpteq12dv 4693 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑓 ∈ {𝑔 ∈ (𝑥𝑚 𝑦) ∣ 𝑔 finSupp ∅} ↦ OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅)‘dom )) = (𝑓𝑆OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅)‘dom )))
18 df-cnf 8503 . . 3 CNF = (𝑥 ∈ On, 𝑦 ∈ On ↦ (𝑓 ∈ {𝑔 ∈ (𝑥𝑚 𝑦) ∣ 𝑔 finSupp ∅} ↦ OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅)‘dom )))
19 ovex 6632 . . . . 5 (𝐴𝑚 𝐵) ∈ V
205, 19rabex2 4775 . . . 4 𝑆 ∈ V
2120mptex 6440 . . 3 (𝑓𝑆OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅)‘dom )) ∈ V
2217, 18, 21ovmpt2a 6744 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 CNF 𝐵) = (𝑓𝑆OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅)‘dom )))
231, 2, 22syl2anc 692 1 (𝜑 → (𝐴 CNF 𝐵) = (𝑓𝑆OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅)‘dom )))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  {crab 2911  Vcvv 3186  ⦋csb 3514  ∅c0 3891   class class class wbr 4613   ↦ cmpt 4673   E cep 4983  dom cdm 5074  Oncon0 5682  ‘cfv 5847  (class class class)co 6604   ↦ cmpt2 6606   supp csupp 7240  seq𝜔cseqom 7487   +𝑜 coa 7502   ·𝑜 comu 7503   ↑𝑜 coe 7504   ↑𝑚 cmap 7802   finSupp cfsupp 8219  OrdIsocoi 8358   CNF ccnf 8502 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-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  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-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-seqom 7488  df-cnf 8503 This theorem is referenced by:  cantnfdm  8505  cantnfval  8509  cantnff  8515
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