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Theorem cantnfval 8509
 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 2621 . . . 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 8504 . . 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 6150 . 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 8505 . . . . 5 (𝜑 → dom (𝐴 CNF 𝐵) = {𝑔 ∈ (𝐴𝑚 𝐵) ∣ 𝑔 finSupp ∅})
97, 8syl5eq 2667 . . . 4 (𝜑𝑆 = {𝑔 ∈ (𝐴𝑚 𝐵) ∣ 𝑔 finSupp ∅})
106, 9eleqtrd 2700 . . 3 (𝜑𝐹 ∈ {𝑔 ∈ (𝐴𝑚 𝐵) ∣ 𝑔 finSupp ∅})
11 ovex 6632 . . . . . 6 (𝑓 supp ∅) ∈ V
12 eqid 2621 . . . . . . 7 OrdIso( E , (𝑓 supp ∅)) = OrdIso( E , (𝑓 supp ∅))
1312oiexg 8384 . . . . . 6 ((𝑓 supp ∅) ∈ V → OrdIso( E , (𝑓 supp ∅)) ∈ V)
1411, 13mp1i 13 . . . . 5 (𝑓 = 𝐹 → OrdIso( E , (𝑓 supp ∅)) ∈ V)
15 simpr 477 . . . . . . . . . . . . . . 15 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → = OrdIso( E , (𝑓 supp ∅)))
16 oveq1 6611 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝐹 → (𝑓 supp ∅) = (𝐹 supp ∅))
1716adantr 481 . . . . . . . . . . . . . . . 16 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (𝑓 supp ∅) = (𝐹 supp ∅))
18 oieq2 8362 . . . . . . . . . . . . . . . 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 2655 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → = OrdIso( E , (𝐹 supp ∅)))
21 cantnfcl.g . . . . . . . . . . . . . 14 𝐺 = OrdIso( E , (𝐹 supp ∅))
2220, 21syl6eqr 2673 . . . . . . . . . . . . 13 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → = 𝐺)
2322fveq1d 6150 . . . . . . . . . . . 12 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (𝑘) = (𝐺𝑘))
2423oveq2d 6620 . . . . . . . . . . 11 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (𝐴𝑜 (𝑘)) = (𝐴𝑜 (𝐺𝑘)))
25 simpl 473 . . . . . . . . . . . 12 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → 𝑓 = 𝐹)
2625, 23fveq12d 6154 . . . . . . . . . . 11 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (𝑓‘(𝑘)) = (𝐹‘(𝐺𝑘)))
2724, 26oveq12d 6622 . . . . . . . . . 10 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → ((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) = ((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))))
2827oveq1d 6619 . . . . . . . . 9 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧) = (((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧))
2928mpt2eq3dv 6674 . . . . . . . 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 2621 . . . . . . . 8 ∅ = ∅
31 seqomeq12 7494 . . . . . . . 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 693 . . . . . . 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 2673 . . . . . 6 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅) = 𝐻)
3522dmeqd 5286 . . . . . 6 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → dom = dom 𝐺)
3634, 35fveq12d 6154 . . . . 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 3541 . . . 4 (𝑓 = 𝐹OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝑘)) ·𝑜 (𝑓‘(𝑘))) +𝑜 𝑧)), ∅)‘dom ) = (𝐻‘dom 𝐺))
38 eqid 2621 . . . 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 6158 . . . 4 (𝐻‘dom 𝐺) ∈ V
4037, 38, 39fvmpt 6239 . . 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 2655 1 (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  {crab 2911  Vcvv 3186  ⦋csb 3514  ∅c0 3891   class class class wbr 4613   ↦ cmpt 4673   E cep 4983  dom cdm 5074  Oncon0 5682  ‘cfv 5847  (class class class)co 6604   ↦ cmpt2 6606   supp csupp 7240  seq𝜔cseqom 7487   +𝑜 coa 7502   ·𝑜 comu 7503   ↑𝑜 coe 7504   ↑𝑚 cmap 7802   finSupp cfsupp 8219  OrdIsocoi 8358   CNF ccnf 8502 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-seqom 7488  df-oi 8359  df-cnf 8503 This theorem is referenced by:  cantnfval2  8510  cantnfle  8512  cantnflt2  8514  cantnff  8515  cantnf0  8516  cantnfp1lem3  8521  cantnflem1  8530  cantnf  8534  cnfcom2  8543
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