Theorem cantnfval 9426

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	⊢ (𝜑 → 𝐴 ∈ On)
cantnfs.b	⊢ (𝜑 → 𝐵 ∈ On)
cantnfcl.g	⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))
cantnfcl.f	⊢ (𝜑 → 𝐹 ∈ 𝑆)
cantnfval.h	⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +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 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2738	. . . 4 ⊢ {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅} = {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅}
2		cantnfs.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ On)
3		cantnfs.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ On)
4	1, 2, 3	cantnffval 9421	. . 3 ⊢ (𝜑 → (𝐴 CNF 𝐵) = (𝑓 ∈ {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅} ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ)))
5	4	fveq1d 6776	. 2 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = ((𝑓 ∈ {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅} ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ))‘𝐹))
6		cantnfcl.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
7		cantnfs.s	. . . . 5 ⊢ 𝑆 = dom (𝐴 CNF 𝐵)
8	1, 2, 3	cantnfdm 9422	. . . . 5 ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅})
9	7, 8	eqtrid 2790	. . . 4 ⊢ (𝜑 → 𝑆 = {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅})
10	6, 9	eleqtrd 2841	. . 3 ⊢ (𝜑 → 𝐹 ∈ {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅})
11		ovex 7308	. . . . . 6 ⊢ (𝑓 supp ∅) ∈ V
12		eqid 2738	. . . . . . 7 ⊢ OrdIso( E , (𝑓 supp ∅)) = OrdIso( E , (𝑓 supp ∅))
13	12	oiexg 9294	. . . . . 6 ⊢ ((𝑓 supp ∅) ∈ V → OrdIso( E , (𝑓 supp ∅)) ∈ V)
14	11, 13	mp1i 13	. . . . 5 ⊢ (𝑓 = 𝐹 → OrdIso( E , (𝑓 supp ∅)) ∈ V)
15		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → ℎ = OrdIso( E , (𝑓 supp ∅)))
16		oveq1 7282	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 𝐹 → (𝑓 supp ∅) = (𝐹 supp ∅))
17	16	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → (𝑓 supp ∅) = (𝐹 supp ∅))
18		oieq2 9272	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 supp ∅) = (𝐹 supp ∅) → OrdIso( E , (𝑓 supp ∅)) = OrdIso( E , (𝐹 supp ∅)))
19	17, 18	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → OrdIso( E , (𝑓 supp ∅)) = OrdIso( E , (𝐹 supp ∅)))
20	15, 19	eqtrd 2778	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → ℎ = OrdIso( E , (𝐹 supp ∅)))
21		cantnfcl.g	. . . . . . . . . . . . . 14 ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))
22	20, 21	eqtr4di 2796	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → ℎ = 𝐺)
23	22	fveq1d 6776	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → (ℎ‘𝑘) = (𝐺‘𝑘))
24	23	oveq2d 7291	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → (𝐴 ↑o (ℎ‘𝑘)) = (𝐴 ↑o (𝐺‘𝑘)))
25		simpl 483	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → 𝑓 = 𝐹)
26	25, 23	fveq12d 6781	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → (𝑓‘(ℎ‘𝑘)) = (𝐹‘(𝐺‘𝑘)))
27	24, 26	oveq12d 7293	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → ((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) = ((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))))
28	27	oveq1d 7290	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧) = (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧))
29	28	mpoeq3dv 7354	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → (𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)))
30		eqid 2738	. . . . . . . 8 ⊢ ∅ = ∅
31		seqomeq12 8285	. . . . . . . 8 ⊢ (((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)) ∧ ∅ = ∅) → seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅))
32	29, 30, 31	sylancl 586	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅))
33		cantnfval.h	. . . . . . 7 ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)
34	32, 33	eqtr4di 2796	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅) = 𝐻)
35	22	dmeqd 5814	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → dom ℎ = dom 𝐺)
36	34, 35	fveq12d 6781	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → (seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ) = (𝐻‘dom 𝐺))
37	14, 36	csbied 3870	. . . 4 ⊢ (𝑓 = 𝐹 → ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ) = (𝐻‘dom 𝐺))
38		eqid 2738	. . . 4 ⊢ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅} ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ)) = (𝑓 ∈ {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅} ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ))
39		fvex 6787	. . . 4 ⊢ (𝐻‘dom 𝐺) ∈ V
40	37, 38, 39	fvmpt 6875	. . 3 ⊢ (𝐹 ∈ {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅} → ((𝑓 ∈ {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅} ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ))‘𝐹) = (𝐻‘dom 𝐺))
41	10, 40	syl 17	. 2 ⊢ (𝜑 → ((𝑓 ∈ {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅} ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ))‘𝐹) = (𝐻‘dom 𝐺))
42	5, 41	eqtrd 2778	1 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {crab 3068  Vcvv 3432  ⦋csb 3832  ∅c0 4256   class class class wbr 5074   ↦ cmpt 5157   E cep 5494  dom cdm 5589  Oncon0 6266  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277   supp csupp 7977  seq_ωcseqom 8278   +_o coa 8294   ·_o comu 8295   ↑_o coe 8296   ↑_m cmap 8615   finSupp cfsupp 9128  OrdIsocoi 9268   CNF ccnf 9419

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-seqom 8279  df-oi 9269  df-cnf 9420

This theorem is referenced by:  cantnfval2  9427  cantnfle  9429  cantnflt2  9431  cantnff  9432  cantnf0  9433  cantnfp1lem3  9438  cantnflem1  9447  cantnf  9451  cnfcom2  9460