Theorem cantnfval 9625

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 2765	. . . 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 9620	. . 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 6873	. 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 9621	. . . . 5 ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅})
9	7, 8	eqtrid 2812	. . . 4 ⊢ (𝜑 → 𝑆 = {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅})
10	6, 9	eleqtrd 2867	. . 3 ⊢ (𝜑 → 𝐹 ∈ {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅})
11		ovex 7433	. . . . . 6 ⊢ (𝑓 supp ∅) ∈ V
12		eqid 2765	. . . . . . 7 ⊢ OrdIso( E , (𝑓 supp ∅)) = OrdIso( E , (𝑓 supp ∅))
13	12	oiexg 9485	. . . . . 6 ⊢ ((𝑓 supp ∅) ∈ V → OrdIso( E , (𝑓 supp ∅)) ∈ V)
14	11, 13	mp1i 14	. . . . 5 ⊢ (𝑓 = 𝐹 → OrdIso( E , (𝑓 supp ∅)) ∈ V)
15		simpr 489	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → ℎ = OrdIso( E , (𝑓 supp ∅)))
16		oveq1 7407	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 𝐹 → (𝑓 supp ∅) = (𝐹 supp ∅))
17	16	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → (𝑓 supp ∅) = (𝐹 supp ∅))
18		oieq2 9463	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 supp ∅) = (𝐹 supp ∅) → OrdIso( E , (𝑓 supp ∅)) = OrdIso( E , (𝐹 supp ∅)))
19	17, 18	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → OrdIso( E , (𝑓 supp ∅)) = OrdIso( E , (𝐹 supp ∅)))
20	15, 19	eqtrd 2800	. . . . . . . . . . . . . 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 2818	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → ℎ = 𝐺)
23	22	fveq1d 6873	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → (ℎ‘𝑘) = (𝐺‘𝑘))
24	23	oveq2d 7416	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → (𝐴 ↑o (ℎ‘𝑘)) = (𝐴 ↑o (𝐺‘𝑘)))
25		simpl 487	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → 𝑓 = 𝐹)
26	25, 23	fveq12d 6878	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → (𝑓‘(ℎ‘𝑘)) = (𝐹‘(𝐺‘𝑘)))
27	24, 26	oveq12d 7418	. . . . . . . . . 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 7415	. . . . . . . . 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 7479	. . . . . . . 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 2765	. . . . . . . 8 ⊢ ∅ = ∅
31		seqomeq12 8429	. . . . . . . 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 597	. . . . . . 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 2818	. . . . . 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 5886	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → dom ℎ = dom 𝐺)
36	34, 35	fveq12d 6878	. . . . 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 3891	. . . 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 2765	. . . 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 6884	. . . 4 ⊢ (𝐻‘dom 𝐺) ∈ V
40	37, 38, 39	fvmpt 6979	. . 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 18	. 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 2800	1 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  {crab 3417  Vcvv 3457  ⦋csb 3855  ∅c0 4288   class class class wbr 5105   ↦ cmpt 5186   E cep 5551  dom cdm 5652  Oncon0 6350  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402   supp csupp 8144  seq_ωcseqom 8422   +_o coa 8438   ·_o comu 8439   ↑_o coe 8440   ↑_m cmap 8812   finSupp cfsupp 9309  OrdIsocoi 9459   CNF ccnf 9618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-seqom 8423  df-oi 9460  df-cnf 9619

This theorem is referenced by:  cantnfval2  9626  cantnfle  9628  cantnflt2  9630  cantnff  9631  cantnf0  9632  cantnfp1lem3  9637  cantnflem1  9646  cantnf  9650  cnfcom2  9659