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Theorem cantnffval 9159
 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 7159 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥m 𝑦) = (𝐴m 𝐵))
43rabeqdv 3397 . . . . 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 2811 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → {𝑔 ∈ (𝑥m 𝑦) ∣ 𝑔 finSupp ∅} = 𝑆)
7 simp1l 1194 . . . . . . . . . . 11 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑘 ∈ V ∧ 𝑧 ∈ V) → 𝑥 = 𝐴)
87oveq1d 7165 . . . . . . . . . 10 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑘 ∈ V ∧ 𝑧 ∈ V) → (𝑥o (𝑘)) = (𝐴o (𝑘)))
98oveq1d 7165 . . . . . . . . 9 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑘 ∈ V ∧ 𝑧 ∈ V) → ((𝑥o (𝑘)) ·o (𝑓‘(𝑘))) = ((𝐴o (𝑘)) ·o (𝑓‘(𝑘))))
109oveq1d 7165 . . . . . . . 8 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑘 ∈ V ∧ 𝑧 ∈ V) → (((𝑥o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧) = (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧))
1110mpoeq3dva 7225 . . . . . . 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 2758 . . . . . . 7 ∅ = ∅
13 seqomeq12 8100 . . . . . . 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 589 . . . . . 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 6660 . . . . 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 3812 . . . 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 5117 . . 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 9158 . . 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 7183 . . . . 5 (𝐴m 𝐵) ∈ V
205, 19rabex2 5204 . . . 4 𝑆 ∈ V
2120mptex 6977 . . 3 (𝑓𝑆OrdIso( E , (𝑓 supp ∅)) / (seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅)‘dom )) ∈ V
2217, 18, 21ovmpoa 7300 . 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 587 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 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {crab 3074  Vcvv 3409  ⦋csb 3805  ∅c0 4225   class class class wbr 5032   ↦ cmpt 5112   E cep 5434  dom cdm 5524  Oncon0 6169  ‘cfv 6335  (class class class)co 7150   ∈ cmpo 7152   supp csupp 7835  seqωcseqom 8093   +o coa 8109   ·o comu 8110   ↑o coe 8111   ↑m cmap 8416   finSupp cfsupp 8866  OrdIsocoi 9006   CNF ccnf 9157 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-seqom 8094  df-cnf 9158 This theorem is referenced by:  cantnfdm  9160  cantnfval  9164  cantnff  9170
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