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Theorem cantnfval 9120
 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 2826 . . . 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 9115 . . 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 6669 . 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 9116 . . . . 5 (𝜑 → dom (𝐴 CNF 𝐵) = {𝑔 ∈ (𝐴m 𝐵) ∣ 𝑔 finSupp ∅})
97, 8syl5eq 2873 . . . 4 (𝜑𝑆 = {𝑔 ∈ (𝐴m 𝐵) ∣ 𝑔 finSupp ∅})
106, 9eleqtrd 2920 . . 3 (𝜑𝐹 ∈ {𝑔 ∈ (𝐴m 𝐵) ∣ 𝑔 finSupp ∅})
11 ovex 7181 . . . . . 6 (𝑓 supp ∅) ∈ V
12 eqid 2826 . . . . . . 7 OrdIso( E , (𝑓 supp ∅)) = OrdIso( E , (𝑓 supp ∅))
1312oiexg 8988 . . . . . 6 ((𝑓 supp ∅) ∈ V → OrdIso( E , (𝑓 supp ∅)) ∈ V)
1411, 13mp1i 13 . . . . 5 (𝑓 = 𝐹 → OrdIso( E , (𝑓 supp ∅)) ∈ V)
15 simpr 485 . . . . . . . . . . . . . . 15 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → = OrdIso( E , (𝑓 supp ∅)))
16 oveq1 7155 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝐹 → (𝑓 supp ∅) = (𝐹 supp ∅))
1716adantr 481 . . . . . . . . . . . . . . . 16 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (𝑓 supp ∅) = (𝐹 supp ∅))
18 oieq2 8966 . . . . . . . . . . . . . . . 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 6669 . . . . . . . . . . . 12 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (𝑘) = (𝐺𝑘))
2423oveq2d 7164 . . . . . . . . . . 11 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (𝐴o (𝑘)) = (𝐴o (𝐺𝑘)))
25 simpl 483 . . . . . . . . . . . 12 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → 𝑓 = 𝐹)
2625, 23fveq12d 6674 . . . . . . . . . . 11 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (𝑓‘(𝑘)) = (𝐹‘(𝐺𝑘)))
2724, 26oveq12d 7166 . . . . . . . . . 10 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → ((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) = ((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))))
2827oveq1d 7163 . . . . . . . . 9 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧) = (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))
2928mpoeq3dv 7225 . . . . . . . 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 2826 . . . . . . . 8 ∅ = ∅
31 seqomeq12 8081 . . . . . . . 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 586 . . . . . . 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 5773 . . . . . 6 ((𝑓 = 𝐹 = OrdIso( E , (𝑓 supp ∅))) → dom = dom 𝐺)
3634, 35fveq12d 6674 . . . . 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 3923 . . . 4 (𝑓 = 𝐹OrdIso( E , (𝑓 supp ∅)) / (seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝑘)) ·o (𝑓‘(𝑘))) +o 𝑧)), ∅)‘dom ) = (𝐻‘dom 𝐺))
38 eqid 2826 . . . 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 6680 . . . 4 (𝐻‘dom 𝐺) ∈ V
4037, 38, 39fvmpt 6765 . . 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 396   = wceq 1530   ∈ wcel 2107  {crab 3147  Vcvv 3500  ⦋csb 3887  ∅c0 4295   class class class wbr 5063   ↦ cmpt 5143   E cep 5463  dom cdm 5554  Oncon0 6189  ‘cfv 6352  (class class class)co 7148   ∈ cmpo 7150   supp csupp 7821  seqωcseqom 8074   +o coa 8090   ·o comu 8091   ↑o coe 8092   ↑m cmap 8396   finSupp cfsupp 8822  OrdIsocoi 8962   CNF ccnf 9113 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-isom 6361  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-seqom 8075  df-oi 8963  df-cnf 9114 This theorem is referenced by:  cantnfval2  9121  cantnfle  9123  cantnflt2  9125  cantnff  9126  cantnf0  9127  cantnfp1lem3  9132  cantnflem1  9141  cantnf  9145  cnfcom2  9154
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