Theorem cantnffval 8443

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
cantnffval.s	⊢ 𝑆 = {𝑔 ∈ (𝐴 ↑𝑚 𝐵) ∣ 𝑔 finSupp ∅}
cantnffval.a	⊢ (𝜑 → 𝐴 ∈ On)
cantnffval.b	⊢ (𝜑 → 𝐵 ∈ On)

Assertion

Ref	Expression
cantnffval	⊢ (𝜑 → (𝐴 CNF 𝐵) = (𝑓 ∈ 𝑆 ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (ℎ‘𝑘)) ·𝑜 (𝑓‘(ℎ‘𝑘))) +𝑜 𝑧)), ∅)‘dom ℎ)))

Distinct variable groups:   𝑓,𝑔,ℎ,𝑘,𝑧,𝐴   𝐵,𝑓,𝑔,ℎ,𝑘,𝑧   𝑆,𝑓

Allowed substitution hints:   𝜑(𝑧,𝑓,𝑔,ℎ,𝑘)   𝑆(𝑧,𝑔,ℎ,𝑘)

Proof of Theorem cantnffval

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cantnffval.a	. 2 ⊢ (𝜑 → 𝐴 ∈ On)
2		cantnffval.b	. 2 ⊢ (𝜑 → 𝐵 ∈ On)
3		oveq12 6558	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ↑𝑚 𝑦) = (𝐴 ↑𝑚 𝐵))
4	3	rabeqdv 3167	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → {𝑔 ∈ (𝑥 ↑𝑚 𝑦) ∣ 𝑔 finSupp ∅} = {𝑔 ∈ (𝐴 ↑𝑚 𝐵) ∣ 𝑔 finSupp ∅})
5		cantnffval.s	. . . . 5 ⊢ 𝑆 = {𝑔 ∈ (𝐴 ↑𝑚 𝐵) ∣ 𝑔 finSupp ∅}
6	4, 5	syl6eqr 2662	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → {𝑔 ∈ (𝑥 ↑𝑚 𝑦) ∣ 𝑔 finSupp ∅} = 𝑆)
7		simp1l 1078	. . . . . . . . . . 11 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑘 ∈ V ∧ 𝑧 ∈ V) → 𝑥 = 𝐴)
8	7	oveq1d 6564	. . . . . . . . . 10 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑘 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ↑𝑜 (ℎ‘𝑘)) = (𝐴 ↑𝑜 (ℎ‘𝑘)))
9	8	oveq1d 6564	. . . . . . . . 9 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑘 ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ↑𝑜 (ℎ‘𝑘)) ·𝑜 (𝑓‘(ℎ‘𝑘))) = ((𝐴 ↑𝑜 (ℎ‘𝑘)) ·𝑜 (𝑓‘(ℎ‘𝑘))))
10	9	oveq1d 6564	. . . . . . . 8 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑘 ∈ V ∧ 𝑧 ∈ V) → (((𝑥 ↑𝑜 (ℎ‘𝑘)) ·𝑜 (𝑓‘(ℎ‘𝑘))) +𝑜 𝑧) = (((𝐴 ↑𝑜 (ℎ‘𝑘)) ·𝑜 (𝑓‘(ℎ‘𝑘))) +𝑜 𝑧))
11	10	mpt2eq3dva 6617	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥 ↑𝑜 (ℎ‘𝑘)) ·𝑜 (𝑓‘(ℎ‘𝑘))) +𝑜 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (ℎ‘𝑘)) ·𝑜 (𝑓‘(ℎ‘𝑘))) +𝑜 𝑧)))
12		eqid 2610	. . . . . . 7 ⊢ ∅ = ∅
13		seqomeq12 7436	. . . . . . 7 ⊢ (((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥 ↑𝑜 (ℎ‘𝑘)) ·𝑜 (𝑓‘(ℎ‘𝑘))) +𝑜 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (ℎ‘𝑘)) ·𝑜 (𝑓‘(ℎ‘𝑘))) +𝑜 𝑧)) ∧ ∅ = ∅) → seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥 ↑𝑜 (ℎ‘𝑘)) ·𝑜 (𝑓‘(ℎ‘𝑘))) +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (ℎ‘𝑘)) ·𝑜 (𝑓‘(ℎ‘𝑘))) +𝑜 𝑧)), ∅))
14	11, 12, 13	sylancl 693	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥 ↑𝑜 (ℎ‘𝑘)) ·𝑜 (𝑓‘(ℎ‘𝑘))) +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (ℎ‘𝑘)) ·𝑜 (𝑓‘(ℎ‘𝑘))) +𝑜 𝑧)), ∅))
15	14	fveq1d 6105	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥 ↑𝑜 (ℎ‘𝑘)) ·𝑜 (𝑓‘(ℎ‘𝑘))) +𝑜 𝑧)), ∅)‘dom ℎ) = (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (ℎ‘𝑘)) ·𝑜 (𝑓‘(ℎ‘𝑘))) +𝑜 𝑧)), ∅)‘dom ℎ))
16	15	csbeq2dv 3944	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥 ↑𝑜 (ℎ‘𝑘)) ·𝑜 (𝑓‘(ℎ‘𝑘))) +𝑜 𝑧)), ∅)‘dom ℎ) = ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (ℎ‘𝑘)) ·𝑜 (𝑓‘(ℎ‘𝑘))) +𝑜 𝑧)), ∅)‘dom ℎ))
17	6, 16	mpteq12dv 4663	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑓 ∈ {𝑔 ∈ (𝑥 ↑𝑚 𝑦) ∣ 𝑔 finSupp ∅} ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥 ↑𝑜 (ℎ‘𝑘)) ·𝑜 (𝑓‘(ℎ‘𝑘))) +𝑜 𝑧)), ∅)‘dom ℎ)) = (𝑓 ∈ 𝑆 ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (ℎ‘𝑘)) ·𝑜 (𝑓‘(ℎ‘𝑘))) +𝑜 𝑧)), ∅)‘dom ℎ)))
18		df-cnf 8442	. . 3 ⊢ CNF = (𝑥 ∈ On, 𝑦 ∈ On ↦ (𝑓 ∈ {𝑔 ∈ (𝑥 ↑𝑚 𝑦) ∣ 𝑔 finSupp ∅} ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥 ↑𝑜 (ℎ‘𝑘)) ·𝑜 (𝑓‘(ℎ‘𝑘))) +𝑜 𝑧)), ∅)‘dom ℎ)))
19		ovex 6577	. . . . 5 ⊢ (𝐴 ↑𝑚 𝐵) ∈ V
20	5, 19	rabex2 4742	. . . 4 ⊢ 𝑆 ∈ V
21	20	mptex 6390	. . 3 ⊢ (𝑓 ∈ 𝑆 ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (ℎ‘𝑘)) ·𝑜 (𝑓‘(ℎ‘𝑘))) +𝑜 𝑧)), ∅)‘dom ℎ)) ∈ V
22	17, 18, 21	ovmpt2a 6689	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 CNF 𝐵) = (𝑓 ∈ 𝑆 ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (ℎ‘𝑘)) ·𝑜 (𝑓‘(ℎ‘𝑘))) +𝑜 𝑧)), ∅)‘dom ℎ)))
23	1, 2, 22	syl2anc 691	1 ⊢ (𝜑 → (𝐴 CNF 𝐵) = (𝑓 ∈ 𝑆 ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (ℎ‘𝑘)) ·𝑜 (𝑓‘(ℎ‘𝑘))) +𝑜 𝑧)), ∅)‘dom ℎ)))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  ⦋csb 3499  ∅c0 3874   class class class wbr 4583   ↦ cmpt 4643   E cep 4947  dom cdm 5038  Oncon0 5640  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   supp csupp 7182  seq_𝜔cseqom 7429   +_𝑜 coa 7444   ·_𝑜 comu 7445   ↑_𝑜 coe 7446   ↑_𝑚 cmap 7744   finSupp cfsupp 8158  OrdIsocoi 8297   CNF ccnf 8441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seqom 7430  df-cnf 8442

This theorem is referenced by:  cantnfdm  8444  cantnfval  8448  cantnff  8454