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Theorem cantnfval 8729
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 2771 . . . 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 8724 . . 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 6334 . 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 8725 . . . . 5 (𝜑 → dom (𝐴 CNF 𝐵) = {𝑔 ∈ (𝐴𝑚 𝐵) ∣ 𝑔 finSupp ∅})
97, 8syl5eq 2817 . . . 4 (𝜑𝑆 = {𝑔 ∈ (𝐴𝑚 𝐵) ∣ 𝑔 finSupp ∅})
106, 9eleqtrd 2852 . . 3 (𝜑𝐹 ∈ {𝑔 ∈ (𝐴𝑚 𝐵) ∣ 𝑔 finSupp ∅})
11 ovex 6823 . . . . . 6 (𝑓 supp ∅) ∈ V
12 eqid 2771 . . . . . . 7 OrdIso( E , (𝑓 supp ∅)) = OrdIso( E , (𝑓 supp ∅))
1312oiexg 8596 . . . . . 6 ((𝑓 supp ∅) ∈ V → OrdIso( E , (𝑓 supp ∅)) ∈ V)
1411, 13mp1i 13 . . . . 5 (𝑓 = 𝐹 → OrdIso( E , (𝑓 supp ∅)) ∈ V)
15 simpr 471 . . . . . . . . . . . . . . 15 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → = OrdIso( E , (𝑓 supp ∅)))
16 oveq1 6800 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝐹 → (𝑓 supp ∅) = (𝐹 supp ∅))
1716adantr 466 . . . . . . . . . . . . . . . 16 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (𝑓 supp ∅) = (𝐹 supp ∅))
18 oieq2 8574 . . . . . . . . . . . . . . . 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 2805 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → = OrdIso( E , (𝐹 supp ∅)))
21 cantnfcl.g . . . . . . . . . . . . . 14 𝐺 = OrdIso( E , (𝐹 supp ∅))
2220, 21syl6eqr 2823 . . . . . . . . . . . . 13 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → = 𝐺)
2322fveq1d 6334 . . . . . . . . . . . 12 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (𝑘) = (𝐺𝑘))
2423oveq2d 6809 . . . . . . . . . . 11 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (𝐴𝑜 (𝑘)) = (𝐴𝑜 (𝐺𝑘)))
25 simpl 468 . . . . . . . . . . . 12 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → 𝑓 = 𝐹)
2625, 23fveq12d 6338 . . . . . . . . . . 11 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (𝑓‘(𝑘)) = (𝐹‘(𝐺𝑘)))
2724, 26oveq12d 6811 . . . . . . . . . 10 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → ((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) = ((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))))
2827oveq1d 6808 . . . . . . . . 9 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧) = (((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧))
2928mpt2eq3dv 6868 . . . . . . . 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 2771 . . . . . . . 8 ∅ = ∅
31 seqomeq12 7702 . . . . . . . 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 574 . . . . . . 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, 33syl6eqr 2823 . . . . . 6 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅) = 𝐻)
3522dmeqd 5464 . . . . . 6 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → dom = dom 𝐺)
3634, 35fveq12d 6338 . . . . 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 3709 . . . 4 (𝑓 = 𝐹OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅)‘dom ) = (𝐻‘dom 𝐺))
38 eqid 2771 . . . 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 6342 . . . 4 (𝐻‘dom 𝐺) ∈ V
4037, 38, 39fvmpt 6424 . . 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 2805 1 (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  {crab 3065  Vcvv 3351  csb 3682  c0 4063   class class class wbr 4786  cmpt 4863   E cep 5161  dom cdm 5249  Oncon0 5866  cfv 6031  (class class class)co 6793  cmpt2 6795   supp csupp 7446  seq𝜔cseqom 7695   +𝑜 coa 7710   ·𝑜 comu 7711  𝑜 coe 7712  𝑚 cmap 8009   finSupp cfsupp 8431  OrdIsocoi 8570   CNF ccnf 8722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-seqom 7696  df-oi 8571  df-cnf 8723
This theorem is referenced by:  cantnfval2  8730  cantnfle  8732  cantnflt2  8734  cantnff  8735  cantnf0  8736  cantnfp1lem3  8741  cantnflem1  8750  cantnf  8754  cnfcom2  8763
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