Theorem cantnffval 9553

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 7355	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ↑m 𝑦) = (𝐴 ↑m 𝐵))
4	3	rabeqdv 3410	. . . . 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	eqtr4di 2784	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → {𝑔 ∈ (𝑥 ↑m 𝑦) ∣ 𝑔 finSupp ∅} = 𝑆)
7		simp1l 1198	. . . . . . . . . . 11 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑘 ∈ V ∧ 𝑧 ∈ V) → 𝑥 = 𝐴)
8	7	oveq1d 7361	. . . . . . . . . 10 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑘 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ↑o (ℎ‘𝑘)) = (𝐴 ↑o (ℎ‘𝑘)))
9	8	oveq1d 7361	. . . . . . . . 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 7361	. . . . . . . 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	mpoeq3dva 7423	. . . . . . 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 2731	. . . . . . 7 ⊢ ∅ = ∅
13		seqomeq12 8373	. . . . . . 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 586	. . . . . 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 6824	. . . . 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 3857	. . . 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 5178	. . 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 9552	. . 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 7379	. . . . 5 ⊢ (𝐴 ↑m 𝐵) ∈ V
20	5, 19	rabex2 5279	. . . 4 ⊢ 𝑆 ∈ V
21	20	mptex 7157	. . 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	ovmpoa 7501	. 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 584	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 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  {crab 3395  Vcvv 3436  ⦋csb 3850  ∅c0 4283   class class class wbr 5091   ↦ cmpt 5172   E cep 5515  dom cdm 5616  Oncon0 6306  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348   supp csupp 8090  seq_ωcseqom 8366   +_o coa 8382   ·_o comu 8383   ↑_o coe 8384   ↑_m cmap 8750   finSupp cfsupp 9245  OrdIsocoi 9395   CNF ccnf 9551

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-seqom 8367  df-cnf 9552

This theorem is referenced by:  cantnfdm  9554  cantnfval  9558  cantnff  9564