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Theorem cantnfval 9693
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
cantnfs.s 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a (𝜑𝐴 ∈ On)
cantnfs.b (𝜑𝐵 ∈ On)
cantnfcl.g 𝐺 = OrdIso( E , (𝐹 supp ∅))
cantnfcl.f (𝜑𝐹𝑆)
cantnfval.h 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)
Assertion
Ref Expression
cantnfval (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺))
Distinct variable groups:   𝑧,𝑘,𝐵   𝐴,𝑘,𝑧   𝑘,𝐹,𝑧   𝑆,𝑘,𝑧   𝑘,𝐺,𝑧   𝜑,𝑘,𝑧
Allowed substitution hints:   𝐻(𝑧,𝑘)

Proof of Theorem cantnfval
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2725 . . . 4 {𝑔 ∈ (𝐴m 𝐵) ∣ 𝑔 finSupp ∅} = {𝑔 ∈ (𝐴m 𝐵) ∣ 𝑔 finSupp ∅}
2 cantnfs.a . . . 4 (𝜑𝐴 ∈ On)
3 cantnfs.b . . . 4 (𝜑𝐵 ∈ On)
41, 2, 3cantnffval 9688 . . 3 (𝜑 → (𝐴 CNF 𝐵) = (𝑓 ∈ {𝑔 ∈ (𝐴m 𝐵) ∣ 𝑔 finSupp ∅} ↦ OrdIso( E , (𝑓 supp ∅)) / (seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅)‘dom )))
54fveq1d 6898 . 2 (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = ((𝑓 ∈ {𝑔 ∈ (𝐴m 𝐵) ∣ 𝑔 finSupp ∅} ↦ OrdIso( E , (𝑓 supp ∅)) / (seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅)‘dom ))‘𝐹))
6 cantnfcl.f . . . 4 (𝜑𝐹𝑆)
7 cantnfs.s . . . . 5 𝑆 = dom (𝐴 CNF 𝐵)
81, 2, 3cantnfdm 9689 . . . . 5 (𝜑 → dom (𝐴 CNF 𝐵) = {𝑔 ∈ (𝐴m 𝐵) ∣ 𝑔 finSupp ∅})
97, 8eqtrid 2777 . . . 4 (𝜑𝑆 = {𝑔 ∈ (𝐴m 𝐵) ∣ 𝑔 finSupp ∅})
106, 9eleqtrd 2827 . . 3 (𝜑𝐹 ∈ {𝑔 ∈ (𝐴m 𝐵) ∣ 𝑔 finSupp ∅})
11 ovex 7452 . . . . . 6 (𝑓 supp ∅) ∈ V
12 eqid 2725 . . . . . . 7 OrdIso( E , (𝑓 supp ∅)) = OrdIso( E , (𝑓 supp ∅))
1312oiexg 9560 . . . . . 6 ((𝑓 supp ∅) ∈ V → OrdIso( E , (𝑓 supp ∅)) ∈ V)
1411, 13mp1i 13 . . . . 5 (𝑓 = 𝐹 → OrdIso( E , (𝑓 supp ∅)) ∈ V)
15 simpr 483 . . . . . . . . . . . . . . 15 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → = OrdIso( E , (𝑓 supp ∅)))
16 oveq1 7426 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝐹 → (𝑓 supp ∅) = (𝐹 supp ∅))
1716adantr 479 . . . . . . . . . . . . . . . 16 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (𝑓 supp ∅) = (𝐹 supp ∅))
18 oieq2 9538 . . . . . . . . . . . . . . . 16 ((𝑓 supp ∅) = (𝐹 supp ∅) → OrdIso( E , (𝑓 supp ∅)) = OrdIso( E , (𝐹 supp ∅)))
1917, 18syl 17 . . . . . . . . . . . . . . 15 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → OrdIso( E , (𝑓 supp ∅)) = OrdIso( E , (𝐹 supp ∅)))
2015, 19eqtrd 2765 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → = OrdIso( E , (𝐹 supp ∅)))
21 cantnfcl.g . . . . . . . . . . . . . 14 𝐺 = OrdIso( E , (𝐹 supp ∅))
2220, 21eqtr4di 2783 . . . . . . . . . . . . 13 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → = 𝐺)
2322fveq1d 6898 . . . . . . . . . . . 12 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (𝑘) = (𝐺𝑘))
2423oveq2d 7435 . . . . . . . . . . 11 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (𝐴o (𝑘)) = (𝐴o (𝐺𝑘)))
25 simpl 481 . . . . . . . . . . . 12 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → 𝑓 = 𝐹)
2625, 23fveq12d 6903 . . . . . . . . . . 11 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (𝑓‘(𝑘)) = (𝐹‘(𝐺𝑘)))
2724, 26oveq12d 7437 . . . . . . . . . 10 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → ((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) = ((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))))
2827oveq1d 7434 . . . . . . . . 9 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧) = (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))
2928mpoeq3dv 7499 . . . . . . . 8 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)))
30 eqid 2725 . . . . . . . 8 ∅ = ∅
31 seqomeq12 8475 . . . . . . . 8 (((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)) ∧ ∅ = ∅) → seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅))
3229, 30, 31sylancl 584 . . . . . . 7 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅))
33 cantnfval.h . . . . . . 7 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)
3432, 33eqtr4di 2783 . . . . . 6 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅) = 𝐻)
3522dmeqd 5908 . . . . . 6 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → dom = dom 𝐺)
3634, 35fveq12d 6903 . . . . 5 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅)‘dom ) = (𝐻‘dom 𝐺))
3714, 36csbied 3927 . . . 4 (𝑓 = 𝐹OrdIso( E , (𝑓 supp ∅)) / (seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅)‘dom ) = (𝐻‘dom 𝐺))
38 eqid 2725 . . . 4 (𝑓 ∈ {𝑔 ∈ (𝐴m 𝐵) ∣ 𝑔 finSupp ∅} ↦ OrdIso( E , (𝑓 supp ∅)) / (seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅)‘dom )) = (𝑓 ∈ {𝑔 ∈ (𝐴m 𝐵) ∣ 𝑔 finSupp ∅} ↦ OrdIso( E , (𝑓 supp ∅)) / (seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅)‘dom ))
39 fvex 6909 . . . 4 (𝐻‘dom 𝐺) ∈ V
4037, 38, 39fvmpt 7004 . . 3 (𝐹 ∈ {𝑔 ∈ (𝐴m 𝐵) ∣ 𝑔 finSupp ∅} → ((𝑓 ∈ {𝑔 ∈ (𝐴m 𝐵) ∣ 𝑔 finSupp ∅} ↦ OrdIso( E , (𝑓 supp ∅)) / (seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅)‘dom ))‘𝐹) = (𝐻‘dom 𝐺))
4110, 40syl 17 . 2 (𝜑 → ((𝑓 ∈ {𝑔 ∈ (𝐴m 𝐵) ∣ 𝑔 finSupp ∅} ↦ OrdIso( E , (𝑓 supp ∅)) / (seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅)‘dom ))‘𝐹) = (𝐻‘dom 𝐺))
425, 41eqtrd 2765 1 (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  {crab 3418  Vcvv 3461  csb 3889  c0 4322   class class class wbr 5149  cmpt 5232   E cep 5581  dom cdm 5678  Oncon0 6371  cfv 6549  (class class class)co 7419  cmpo 7421   supp csupp 8165  seqωcseqom 8468   +o coa 8484   ·o comu 8485  o coe 8486  m cmap 8845   finSupp cfsupp 9387  OrdIsocoi 9534   CNF ccnf 9686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-seqom 8469  df-oi 9535  df-cnf 9687
This theorem is referenced by:  cantnfval2  9694  cantnfle  9696  cantnflt2  9698  cantnff  9699  cantnf0  9700  cantnfp1lem3  9705  cantnflem1  9714  cantnf  9718  cnfcom2  9727
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