Theorem naddcnfcom 41257

Description: Component-wise ordinal addition of Cantor normal forms commutes. (Contributed by RP, 2-Jan-2025.)

Assertion

Ref	Expression
naddcnfcom	⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) → (𝐹 ∘f +o 𝐺) = (𝐺 ∘f +o 𝐹))

Proof of Theorem naddcnfcom

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 486	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑆 = dom (ω CNF 𝑋))
2	1	eleq2d 2822	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ 𝑆 ↔ 𝐹 ∈ dom (ω CNF 𝑋)))
3		eqid 2736	. . . . . . . 8 ⊢ dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
4		omelon 9452	. . . . . . . . 9 ⊢ ω ∈ On
5	4	a1i 11	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ω ∈ On)
6		simpl 484	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑋 ∈ On)
7	3, 5, 6	cantnfs 9472	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ dom (ω CNF 𝑋) ↔ (𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅)))
8	2, 7	bitrd 279	. . . . . 6 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ 𝑆 ↔ (𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅)))
9		simpl 484	. . . . . 6 ⊢ ((𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅) → 𝐹:𝑋⟶ω)
10	8, 9	syl6bi 253	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ 𝑆 → 𝐹:𝑋⟶ω))
11		simpl 484	. . . . 5 ⊢ ((𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆) → 𝐹 ∈ 𝑆)
12	10, 11	impel 507	. . . 4 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) → 𝐹:𝑋⟶ω)
13	12	ffnd 6631	. . 3 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) → 𝐹 Fn 𝑋)
14	1	eleq2d 2822	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺 ∈ 𝑆 ↔ 𝐺 ∈ dom (ω CNF 𝑋)))
15	3, 5, 6	cantnfs 9472	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺 ∈ dom (ω CNF 𝑋) ↔ (𝐺:𝑋⟶ω ∧ 𝐺 finSupp ∅)))
16	14, 15	bitrd 279	. . . . . 6 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺 ∈ 𝑆 ↔ (𝐺:𝑋⟶ω ∧ 𝐺 finSupp ∅)))
17		simpl 484	. . . . . 6 ⊢ ((𝐺:𝑋⟶ω ∧ 𝐺 finSupp ∅) → 𝐺:𝑋⟶ω)
18	16, 17	syl6bi 253	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺 ∈ 𝑆 → 𝐺:𝑋⟶ω))
19		simpr 486	. . . . 5 ⊢ ((𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆) → 𝐺 ∈ 𝑆)
20	18, 19	impel 507	. . . 4 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) → 𝐺:𝑋⟶ω)
21	20	ffnd 6631	. . 3 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) → 𝐺 Fn 𝑋)
22		simpll 765	. . 3 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) → 𝑋 ∈ On)
23		inidm 4158	. . 3 ⊢ (𝑋 ∩ 𝑋) = 𝑋
24	13, 21, 22, 22, 23	offn 7578	. 2 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) → (𝐹 ∘f +o 𝐺) Fn 𝑋)
25	21, 13, 22, 22, 23	offn 7578	. 2 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) → (𝐺 ∘f +o 𝐹) Fn 𝑋)
26	12	ffvelcdmda 6993	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ω)
27	20	ffvelcdmda 6993	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) ∈ ω)
28		nnacom 8479	. . . 4 ⊢ (((𝐹‘𝑥) ∈ ω ∧ (𝐺‘𝑥) ∈ ω) → ((𝐹‘𝑥) +o (𝐺‘𝑥)) = ((𝐺‘𝑥) +o (𝐹‘𝑥)))
29	26, 27, 28	syl2anc 585	. . 3 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) +o (𝐺‘𝑥)) = ((𝐺‘𝑥) +o (𝐹‘𝑥)))
30	13	adantr 482	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → 𝐹 Fn 𝑋)
31	21	adantr 482	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → 𝐺 Fn 𝑋)
32		simplll 773	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → 𝑋 ∈ On)
33		simpr 486	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
34		fnfvof 7582	. . . 4 ⊢ (((𝐹 Fn 𝑋 ∧ 𝐺 Fn 𝑋) ∧ (𝑋 ∈ On ∧ 𝑥 ∈ 𝑋)) → ((𝐹 ∘f +o 𝐺)‘𝑥) = ((𝐹‘𝑥) +o (𝐺‘𝑥)))
35	30, 31, 32, 33, 34	syl22anc 837	. . 3 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → ((𝐹 ∘f +o 𝐺)‘𝑥) = ((𝐹‘𝑥) +o (𝐺‘𝑥)))
36		fnfvof 7582	. . . 4 ⊢ (((𝐺 Fn 𝑋 ∧ 𝐹 Fn 𝑋) ∧ (𝑋 ∈ On ∧ 𝑥 ∈ 𝑋)) → ((𝐺 ∘f +o 𝐹)‘𝑥) = ((𝐺‘𝑥) +o (𝐹‘𝑥)))
37	31, 30, 32, 33, 36	syl22anc 837	. . 3 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → ((𝐺 ∘f +o 𝐹)‘𝑥) = ((𝐺‘𝑥) +o (𝐹‘𝑥)))
38	29, 35, 37	3eqtr4d 2786	. 2 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → ((𝐹 ∘f +o 𝐺)‘𝑥) = ((𝐺 ∘f +o 𝐹)‘𝑥))
39	24, 25, 38	eqfnfvd 6944	1 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) → (𝐹 ∘f +o 𝐺) = (𝐺 ∘f +o 𝐹))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1539   ∈ wcel 2104  ∅c0 4262   class class class wbr 5081  dom cdm 5600  Oncon0 6281   Fn wfn 6453  ⟶wf 6454  ‘cfv 6458  (class class class)co 7307   ∘_f cof 7563  ωcom 7744   +_o coa 8325   finSupp cfsupp 9176   CNF ccnf 9467

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620  ax-inf2 9447

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3332  df-rab 3333  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-ov 7310  df-oprab 7311  df-mpo 7312  df-of 7565  df-om 7745  df-2nd 7864  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-rdg 8272  df-seqom 8310  df-oadd 8332  df-map 8648  df-cnf 9468

This theorem is referenced by:  naddcnfid2  41259