Theorem naddcnfid1 43399

Description: Identity law for component-wise ordinal addition of Cantor normal forms. (Contributed by RP, 3-Jan-2025.)

Assertion

Ref	Expression
naddcnfid1	⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → (𝐹 ∘f +o (𝑋 × {∅})) = 𝐹)

Proof of Theorem naddcnfid1

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano1 7819	. . . 4 ⊢ ∅ ∈ ω
2		fconst6g 6712	. . . 4 ⊢ (∅ ∈ ω → (𝑋 × {∅}):𝑋⟶ω)
3	1, 2	mp1i 13	. . 3 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}):𝑋⟶ω)
4		simpl 482	. . . 4 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑋 ∈ On)
5	1	a1i 11	. . . 4 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ∅ ∈ ω)
6	4, 5	fczfsuppd 9270	. . 3 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}) finSupp ∅)
7		simpr 484	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑆 = dom (ω CNF 𝑋))
8	7	eleq2d 2817	. . . 4 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ 𝑆 ↔ (𝑋 × {∅}) ∈ dom (ω CNF 𝑋)))
9		eqid 2731	. . . . 5 ⊢ dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
10		omelon 9536	. . . . . 6 ⊢ ω ∈ On
11	10	a1i 11	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ω ∈ On)
12	9, 11, 4	cantnfs 9556	. . . 4 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ dom (ω CNF 𝑋) ↔ ((𝑋 × {∅}):𝑋⟶ω ∧ (𝑋 × {∅}) finSupp ∅)))
13	8, 12	bitrd 279	. . 3 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ((𝑋 × {∅}) ∈ 𝑆 ↔ ((𝑋 × {∅}):𝑋⟶ω ∧ (𝑋 × {∅}) finSupp ∅)))
14	3, 6, 13	mpbir2and 713	. 2 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝑋 × {∅}) ∈ 𝑆)
15	7	eleq2d 2817	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ 𝑆 ↔ 𝐹 ∈ dom (ω CNF 𝑋)))
16	9, 11, 4	cantnfs 9556	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ dom (ω CNF 𝑋) ↔ (𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅)))
17	15, 16	bitrd 279	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ 𝑆 ↔ (𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅)))
18	17	simprbda 498	. . . . . 6 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → 𝐹:𝑋⟶ω)
19	18	ffnd 6652	. . . . 5 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → 𝐹 Fn 𝑋)
20	19	adantr 480	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → 𝐹 Fn 𝑋)
21	2	ffnd 6652	. . . . 5 ⊢ (∅ ∈ ω → (𝑋 × {∅}) Fn 𝑋)
22	1, 21	mp1i 13	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → (𝑋 × {∅}) Fn 𝑋)
23		simplll 774	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → 𝑋 ∈ On)
24		inidm 4177	. . . 4 ⊢ (𝑋 ∩ 𝑋) = 𝑋
25	20, 22, 23, 23, 24	offn 7623	. . 3 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → (𝐹 ∘f +o (𝑋 × {∅})) Fn 𝑋)
26	20	adantr 480	. . . . 5 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → 𝐹 Fn 𝑋)
27	1, 21	mp1i 13	. . . . 5 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → (𝑋 × {∅}) Fn 𝑋)
28		simp-4l 782	. . . . 5 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → 𝑋 ∈ On)
29		simpr 484	. . . . 5 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
30		fnfvof 7627	. . . . 5 ⊢ (((𝐹 Fn 𝑋 ∧ (𝑋 × {∅}) Fn 𝑋) ∧ (𝑋 ∈ On ∧ 𝑥 ∈ 𝑋)) → ((𝐹 ∘f +o (𝑋 × {∅}))‘𝑥) = ((𝐹‘𝑥) +o ((𝑋 × {∅})‘𝑥)))
31	26, 27, 28, 29, 30	syl22anc 838	. . . 4 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → ((𝐹 ∘f +o (𝑋 × {∅}))‘𝑥) = ((𝐹‘𝑥) +o ((𝑋 × {∅})‘𝑥)))
32		fvconst2g 7136	. . . . . 6 ⊢ ((∅ ∈ ω ∧ 𝑥 ∈ 𝑋) → ((𝑋 × {∅})‘𝑥) = ∅)
33	1, 29, 32	sylancr 587	. . . . 5 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → ((𝑋 × {∅})‘𝑥) = ∅)
34	33	oveq2d 7362	. . . 4 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) +o ((𝑋 × {∅})‘𝑥)) = ((𝐹‘𝑥) +o ∅))
35	18	adantr 480	. . . . . 6 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → 𝐹:𝑋⟶ω)
36	35	ffvelcdmda 7017	. . . . 5 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ω)
37		nna0 8519	. . . . 5 ⊢ ((𝐹‘𝑥) ∈ ω → ((𝐹‘𝑥) +o ∅) = (𝐹‘𝑥))
38	36, 37	syl 17	. . . 4 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) +o ∅) = (𝐹‘𝑥))
39	31, 34, 38	3eqtrd 2770	. . 3 ⊢ (((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) ∧ 𝑥 ∈ 𝑋) → ((𝐹 ∘f +o (𝑋 × {∅}))‘𝑥) = (𝐹‘𝑥))
40	25, 20, 39	eqfnfvd 6967	. 2 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) ∧ (𝑋 × {∅}) ∈ 𝑆) → (𝐹 ∘f +o (𝑋 × {∅})) = 𝐹)
41	14, 40	mpidan 689	1 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → (𝐹 ∘f +o (𝑋 × {∅})) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∅c0 4283  {csn 4576   class class class wbr 5091   × cxp 5614  dom cdm 5616  Oncon0 6306   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   ∘_f cof 7608  ωcom 7796   +_o coa 8382   finSupp cfsupp 9245   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-pow 5303  ax-pr 5370  ax-un 7668  ax-inf2 9531

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-pss 3922  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-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  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-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  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-of 7610  df-om 7797  df-2nd 7922  df-supp 8091  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-seqom 8367  df-oadd 8389  df-map 8752  df-en 8870  df-fin 8873  df-fsupp 9246  df-cnf 9552

This theorem is referenced by:  naddcnfid2  43400