MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cantnffval Structured version   Visualization version   GIF version

Theorem cantnffval 9114
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 7154 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥m 𝑦) = (𝐴m 𝐵))
43rabeqdv 3482 . . . . 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 2871 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → {𝑔 ∈ (𝑥m 𝑦) ∣ 𝑔 finSupp ∅} = 𝑆)
7 simp1l 1189 . . . . . . . . . . 11 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑘 ∈ V ∧ 𝑧 ∈ V) → 𝑥 = 𝐴)
87oveq1d 7160 . . . . . . . . . 10 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑘 ∈ V ∧ 𝑧 ∈ V) → (𝑥o (𝑘)) = (𝐴o (𝑘)))
98oveq1d 7160 . . . . . . . . 9 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑘 ∈ V ∧ 𝑧 ∈ V) → ((𝑥o (𝑘)) ·o (𝑓‘(𝑘))) = ((𝐴o (𝑘)) ·o (𝑓‘(𝑘))))
109oveq1d 7160 . . . . . . . 8 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑘 ∈ V ∧ 𝑧 ∈ V) → (((𝑥o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧) = (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧))
1110mpoeq3dva 7220 . . . . . . 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 2818 . . . . . . 7 ∅ = ∅
13 seqomeq12 8079 . . . . . . 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 6665 . . . . 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 3887 . . . 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 5142 . . 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 9113 . . 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 7178 . . . . 5 (𝐴m 𝐵) ∈ V
205, 19rabex2 5228 . . . 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 7294 . 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 396  w3a 1079   = wceq 1528  wcel 2105  {crab 3139  Vcvv 3492  csb 3880  c0 4288   class class class wbr 5057  cmpt 5137   E cep 5457  dom cdm 5548  Oncon0 6184  cfv 6348  (class class class)co 7145  cmpo 7147   supp csupp 7819  seqωcseqom 8072   +o coa 8088   ·o comu 8089  o coe 8090  m cmap 8395   finSupp cfsupp 8821  OrdIsocoi 8961   CNF ccnf 9112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-seqom 8073  df-cnf 9113
This theorem is referenced by:  cantnfdm  9115  cantnfval  9119  cantnff  9125
  Copyright terms: Public domain W3C validator