Theorem cantnfval 9628

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 2730	. . . 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 9623	. . 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 6863	. 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 9624	. . . . 5 ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅})
9	7, 8	eqtrid 2777	. . . 4 ⊢ (𝜑 → 𝑆 = {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅})
10	6, 9	eleqtrd 2831	. . 3 ⊢ (𝜑 → 𝐹 ∈ {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅})
11		ovex 7423	. . . . . 6 ⊢ (𝑓 supp ∅) ∈ V
12		eqid 2730	. . . . . . 7 ⊢ OrdIso( E , (𝑓 supp ∅)) = OrdIso( E , (𝑓 supp ∅))
13	12	oiexg 9495	. . . . . 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 7397	. . . . . . . . . . . . . . . . 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 9473	. . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → ℎ = 𝐺)
23	22	fveq1d 6863	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → (ℎ‘𝑘) = (𝐺‘𝑘))
24	23	oveq2d 7406	. . . . . . . . . . 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 6868	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → (𝑓‘(ℎ‘𝑘)) = (𝐹‘(𝐺‘𝑘)))
27	24, 26	oveq12d 7408	. . . . . . . . . 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 7405	. . . . . . . . 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 7471	. . . . . . . 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 2730	. . . . . . . 8 ⊢ ∅ = ∅
31		seqomeq12 8425	. . . . . . . 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 2783	. . . . . 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 5872	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → dom ℎ = dom 𝐺)
36	34, 35	fveq12d 6868	. . . . 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 3901	. . . 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 2730	. . . 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 6874	. . . 4 ⊢ (𝐻‘dom 𝐺) ∈ V
40	37, 38, 39	fvmpt 6971	. . 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 2765	1 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3408  Vcvv 3450  ⦋csb 3865  ∅c0 4299   class class class wbr 5110   ↦ cmpt 5191   E cep 5540  dom cdm 5641  Oncon0 6335  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392   supp csupp 8142  seq_ωcseqom 8418   +_o coa 8434   ·_o comu 8435   ↑_o coe 8436   ↑_m cmap 8802   finSupp cfsupp 9319  OrdIsocoi 9469   CNF ccnf 9621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-seqom 8419  df-oi 9470  df-cnf 9622

This theorem is referenced by:  cantnfval2  9629  cantnfle  9631  cantnflt2  9633  cantnff  9634  cantnf0  9635  cantnfp1lem3  9640  cantnflem1  9649  cantnf  9653  cnfcom2  9662