Theorem cantnfval 9651

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 2762	. . . 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 9646	. . 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 6884	. 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 9647	. . . . 5 ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅})
9	7, 8	eqtrid 2809	. . . 4 ⊢ (𝜑 → 𝑆 = {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅})
10	6, 9	eleqtrd 2864	. . 3 ⊢ (𝜑 → 𝐹 ∈ {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅})
11		ovex 7450	. . . . . 6 ⊢ (𝑓 supp ∅) ∈ V
12		eqid 2762	. . . . . . 7 ⊢ OrdIso( E , (𝑓 supp ∅)) = OrdIso( E , (𝑓 supp ∅))
13	12	oiexg 9511	. . . . . 6 ⊢ ((𝑓 supp ∅) ∈ V → OrdIso( E , (𝑓 supp ∅)) ∈ V)
14	11, 13	mp1i 14	. . . . 5 ⊢ (𝑓 = 𝐹 → OrdIso( E , (𝑓 supp ∅)) ∈ V)
15		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → ℎ = OrdIso( E , (𝑓 supp ∅)))
16		oveq1 7424	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 𝐹 → (𝑓 supp ∅) = (𝐹 supp ∅))
17	16	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → (𝑓 supp ∅) = (𝐹 supp ∅))
18		oieq2 9489	. . . . . . . . . . . . . . . 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 2797	. . . . . . . . . . . . . 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 2815	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → ℎ = 𝐺)
23	22	fveq1d 6884	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → (ℎ‘𝑘) = (𝐺‘𝑘))
24	23	oveq2d 7433	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → (𝐴 ↑o (ℎ‘𝑘)) = (𝐴 ↑o (𝐺‘𝑘)))
25		simpl 488	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → 𝑓 = 𝐹)
26	25, 23	fveq12d 6889	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → (𝑓‘(ℎ‘𝑘)) = (𝐹‘(𝐺‘𝑘)))
27	24, 26	oveq12d 7435	. . . . . . . . . 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 7432	. . . . . . . . 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 7496	. . . . . . . 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 2762	. . . . . . . 8 ⊢ ∅ = ∅
31		seqomeq12 8447	. . . . . . . 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 598	. . . . . . 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 2815	. . . . . 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 5893	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ ℎ = OrdIso( E , (𝑓 supp ∅))) → dom ℎ = dom 𝐺)
36	34, 35	fveq12d 6889	. . . . 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 3886	. . . 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 2762	. . . 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 6895	. . . 4 ⊢ (𝐻‘dom 𝐺) ∈ V
40	37, 38, 39	fvmpt 6990	. . 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 2797	1 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3414  Vcvv 3453  ⦋csb 3850  ∅c0 4282   class class class wbr 5107   ↦ cmpt 5190   E cep 5558  dom cdm 5659  Oncon0 6361  ‘cfv 6537  (class class class)co 7417   ∈ cmpo 7419   supp csupp 8162  seq_ωcseqom 8440   +_o coa 8456   ·_o comu 8457   ↑_o coe 8458   ↑_m cmap 8830   finSupp cfsupp 9335  OrdIsocoi 9485   CNF ccnf 9644

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-seqom 8441  df-oi 9486  df-cnf 9645

This theorem is used by:  cantnfval2  9652  cantnfle  9654  cantnflt2  9656  cantnff  9657  cantnf0  9658  cantnfp1lem3  9663  cantnflem1  9672  cantnf  9676  cnfcom2  9685