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Theorem cantnffval 9704
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 7441 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥m 𝑦) = (𝐴m 𝐵))
43rabeqdv 3451 . . . . 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, 5eqtr4di 2794 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → {𝑔 ∈ (𝑥m 𝑦) ∣ 𝑔 finSupp ∅} = 𝑆)
7 simp1l 1197 . . . . . . . . . . 11 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑘 ∈ V ∧ 𝑧 ∈ V) → 𝑥 = 𝐴)
87oveq1d 7447 . . . . . . . . . 10 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑘 ∈ V ∧ 𝑧 ∈ V) → (𝑥o (𝑘)) = (𝐴o (𝑘)))
98oveq1d 7447 . . . . . . . . 9 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑘 ∈ V ∧ 𝑧 ∈ V) → ((𝑥o (𝑘)) ·o (𝑓‘(𝑘))) = ((𝐴o (𝑘)) ·o (𝑓‘(𝑘))))
109oveq1d 7447 . . . . . . . 8 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑘 ∈ V ∧ 𝑧 ∈ V) → (((𝑥o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧) = (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧))
1110mpoeq3dva 7511 . . . . . . 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 2736 . . . . . . 7 ∅ = ∅
13 seqomeq12 8495 . . . . . . 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 586 . . . . . 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 6907 . . . . 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 3905 . . . 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 5232 . . 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 9703 . . 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 7465 . . . . 5 (𝐴m 𝐵) ∈ V
205, 19rabex2 5340 . . . 4 𝑆 ∈ V
2120mptex 7244 . . 3 (𝑓𝑆OrdIso( E , (𝑓 supp ∅)) / (seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅)‘dom )) ∈ V
2217, 18, 21ovmpoa 7589 . 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 584 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 395  w3a 1086   = wceq 1539  wcel 2107  {crab 3435  Vcvv 3479  csb 3898  c0 4332   class class class wbr 5142  cmpt 5224   E cep 5582  dom cdm 5684  Oncon0 6383  cfv 6560  (class class class)co 7432  cmpo 7434   supp csupp 8186  seqωcseqom 8488   +o coa 8504   ·o comu 8505  o coe 8506  m cmap 8867   finSupp cfsupp 9402  OrdIsocoi 9550   CNF ccnf 9702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-seqom 8489  df-cnf 9703
This theorem is referenced by:  cantnfdm  9705  cantnfval  9709  cantnff  9715
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