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Theorem cantnffval 9655
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 7415 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥m 𝑦) = (𝐴m 𝐵))
43rabeqdv 3448 . . . . 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 2791 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → {𝑔 ∈ (𝑥m 𝑦) ∣ 𝑔 finSupp ∅} = 𝑆)
7 simp1l 1198 . . . . . . . . . . 11 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑘 ∈ V ∧ 𝑧 ∈ V) → 𝑥 = 𝐴)
87oveq1d 7421 . . . . . . . . . 10 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑘 ∈ V ∧ 𝑧 ∈ V) → (𝑥o (𝑘)) = (𝐴o (𝑘)))
98oveq1d 7421 . . . . . . . . 9 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑘 ∈ V ∧ 𝑧 ∈ V) → ((𝑥o (𝑘)) ·o (𝑓‘(𝑘))) = ((𝐴o (𝑘)) ·o (𝑓‘(𝑘))))
109oveq1d 7421 . . . . . . . 8 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑘 ∈ V ∧ 𝑧 ∈ V) → (((𝑥o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧) = (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧))
1110mpoeq3dva 7483 . . . . . . 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 2733 . . . . . . 7 ∅ = ∅
13 seqomeq12 8451 . . . . . . 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 587 . . . . . 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 6891 . . . . 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 3900 . . . 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 5239 . . 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 9654 . . 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 7439 . . . . 5 (𝐴m 𝐵) ∈ V
205, 19rabex2 5334 . . . 4 𝑆 ∈ V
2120mptex 7222 . . 3 (𝑓𝑆OrdIso( E , (𝑓 supp ∅)) / (seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅)‘dom )) ∈ V
2217, 18, 21ovmpoa 7560 . 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 585 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 397  w3a 1088   = wceq 1542  wcel 2107  {crab 3433  Vcvv 3475  csb 3893  c0 4322   class class class wbr 5148  cmpt 5231   E cep 5579  dom cdm 5676  Oncon0 6362  cfv 6541  (class class class)co 7406  cmpo 7408   supp csupp 8143  seqωcseqom 8444   +o coa 8460   ·o comu 8461  o coe 8462  m cmap 8817   finSupp cfsupp 9358  OrdIsocoi 9501   CNF ccnf 9653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-seqom 8445  df-cnf 9654
This theorem is referenced by:  cantnfdm  9656  cantnfval  9660  cantnff  9666
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