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Theorem cantnfval 8842
 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 2825 . . . 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 8837 . . 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 6435 . 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 8838 . . . . 5 (𝜑 → dom (𝐴 CNF 𝐵) = {𝑔 ∈ (𝐴𝑚 𝐵) ∣ 𝑔 finSupp ∅})
97, 8syl5eq 2873 . . . 4 (𝜑𝑆 = {𝑔 ∈ (𝐴𝑚 𝐵) ∣ 𝑔 finSupp ∅})
106, 9eleqtrd 2908 . . 3 (𝜑𝐹 ∈ {𝑔 ∈ (𝐴𝑚 𝐵) ∣ 𝑔 finSupp ∅})
11 ovex 6937 . . . . . 6 (𝑓 supp ∅) ∈ V
12 eqid 2825 . . . . . . 7 OrdIso( E , (𝑓 supp ∅)) = OrdIso( E , (𝑓 supp ∅))
1312oiexg 8709 . . . . . 6 ((𝑓 supp ∅) ∈ V → OrdIso( E , (𝑓 supp ∅)) ∈ V)
1411, 13mp1i 13 . . . . 5 (𝑓 = 𝐹 → OrdIso( E , (𝑓 supp ∅)) ∈ V)
15 simpr 479 . . . . . . . . . . . . . . 15 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → = OrdIso( E , (𝑓 supp ∅)))
16 oveq1 6912 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝐹 → (𝑓 supp ∅) = (𝐹 supp ∅))
1716adantr 474 . . . . . . . . . . . . . . . 16 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (𝑓 supp ∅) = (𝐹 supp ∅))
18 oieq2 8687 . . . . . . . . . . . . . . . 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 2861 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → = OrdIso( E , (𝐹 supp ∅)))
21 cantnfcl.g . . . . . . . . . . . . . 14 𝐺 = OrdIso( E , (𝐹 supp ∅))
2220, 21syl6eqr 2879 . . . . . . . . . . . . 13 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → = 𝐺)
2322fveq1d 6435 . . . . . . . . . . . 12 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (𝑘) = (𝐺𝑘))
2423oveq2d 6921 . . . . . . . . . . 11 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (𝐴o (𝑘)) = (𝐴o (𝐺𝑘)))
25 simpl 476 . . . . . . . . . . . 12 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → 𝑓 = 𝐹)
2625, 23fveq12d 6440 . . . . . . . . . . 11 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (𝑓‘(𝑘)) = (𝐹‘(𝐺𝑘)))
2724, 26oveq12d 6923 . . . . . . . . . 10 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → ((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) = ((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))))
2827oveq1d 6920 . . . . . . . . 9 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧) = (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))
2928mpt2eq3dv 6981 . . . . . . . 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 2825 . . . . . . . 8 ∅ = ∅
31 seqomeq12 7815 . . . . . . . 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 582 . . . . . . 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 2879 . . . . . 6 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅) = 𝐻)
3522dmeqd 5558 . . . . . 6 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → dom = dom 𝐺)
3634, 35fveq12d 6440 . . . . 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 3784 . . . 4 (𝑓 = 𝐹OrdIso( E , (𝑓 supp ∅)) / (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅)‘dom ) = (𝐻‘dom 𝐺))
38 eqid 2825 . . . 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 6446 . . . 4 (𝐻‘dom 𝐺) ∈ V
4037, 38, 39fvmpt 6529 . . 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 2861 1 (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166  {crab 3121  Vcvv 3414  ⦋csb 3757  ∅c0 4144   class class class wbr 4873   ↦ cmpt 4952   E cep 5254  dom cdm 5342  Oncon0 5963  ‘cfv 6123  (class class class)co 6905   ↦ cmpt2 6907   supp csupp 7559  seq𝜔cseqom 7808   +o coa 7823   ·o comu 7824   ↑o coe 7825   ↑𝑚 cmap 8122   finSupp cfsupp 8544  OrdIsocoi 8683   CNF ccnf 8835 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-se 5302  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-isom 6132  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-seqom 7809  df-oi 8684  df-cnf 8836 This theorem is referenced by:  cantnfval2  8843  cantnfle  8845  cantnflt2  8847  cantnff  8848  cantnf0  8849  cantnfp1lem3  8854  cantnflem1  8863  cantnf  8867  cnfcom2  8876
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