Theorem cantnfval 9577

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 2736	. . . 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 9572	. . 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 6836	. 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 9573	. . . . 5 ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅})
9	7, 8	eqtrid 2783	. . . 4 ⊢ (𝜑 → 𝑆 = {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅})
10	6, 9	eleqtrd 2838	. . 3 ⊢ (𝜑 → 𝐹 ∈ {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅})
11		ovex 7391	. . . . . 6 ⊢ (𝑓 supp ∅) ∈ V
12		eqid 2736	. . . . . . 7 ⊢ OrdIso( E , (𝑓 supp ∅)) = OrdIso( E , (𝑓 supp ∅))
13	12	oiexg 9440	. . . . . 6 ⊢ ((𝑓 supp ∅) ∈ V → OrdIso( E , (𝑓 supp ∅)) ∈ V)
14	11, 13	mp1i 13	. . . . 5 ⊢ (𝑓 = 𝐹 → OrdIso( E , (𝑓 supp ∅)) ∈ V)
15		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → ℎ = OrdIso( E , (𝑓 supp ∅)))
16		oveq1 7365	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 𝐹 → (𝑓 supp ∅) = (𝐹 supp ∅))
17	16	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → (𝑓 supp ∅) = (𝐹 supp ∅))
18		oieq2 9418	. . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . 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 2789	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → ℎ = 𝐺)
23	22	fveq1d 6836	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → (ℎ‘𝑘) = (𝐺‘𝑘))
24	23	oveq2d 7374	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → (𝐴 ↑o (ℎ‘𝑘)) = (𝐴 ↑o (𝐺‘𝑘)))
25		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → 𝑓 = 𝐹)
26	25, 23	fveq12d 6841	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → (𝑓‘(ℎ‘𝑘)) = (𝐹‘(𝐺‘𝑘)))
27	24, 26	oveq12d 7376	. . . . . . . . . 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 7373	. . . . . . . . 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 7437	. . . . . . . 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 2736	. . . . . . . 8 ⊢ ∅ = ∅
31		seqomeq12 8385	. . . . . . . 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 2789	. . . . . 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 5854	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → dom ℎ = dom 𝐺)
36	34, 35	fveq12d 6841	. . . . 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 3885	. . . 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 2736	. . . 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 6847	. . . 4 ⊢ (𝐻‘dom 𝐺) ∈ V
40	37, 38, 39	fvmpt 6941	. . 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 2771	1 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {crab 3399  Vcvv 3440  ⦋csb 3849  ∅c0 4285   class class class wbr 5098   ↦ cmpt 5179   E cep 5523  dom cdm 5624  Oncon0 6317  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360   supp csupp 8102  seq_ωcseqom 8378   +_o coa 8394   ·_o comu 8395   ↑_o coe 8396   ↑_m cmap 8763   finSupp cfsupp 9264  OrdIsocoi 9414   CNF ccnf 9570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-seqom 8379  df-oi 9415  df-cnf 9571

This theorem is referenced by:  cantnfval2  9578  cantnfle  9580  cantnflt2  9582  cantnff  9583  cantnf0  9584  cantnfp1lem3  9589  cantnflem1  9598  cantnf  9602  cnfcom2  9611