Theorem naddcnfcom 42937

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 483	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑆 = dom (ω CNF 𝑋))
2	1	eleq2d 2811	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ 𝑆 ↔ 𝐹 ∈ dom (ω CNF 𝑋)))
3		eqid 2725	. . . . . . . 8 ⊢ dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
4		omelon 9671	. . . . . . . . 9 ⊢ ω ∈ On
5	4	a1i 11	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ω ∈ On)
6		simpl 481	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → 𝑋 ∈ On)
7	3, 5, 6	cantnfs 9691	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ dom (ω CNF 𝑋) ↔ (𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅)))
8	2, 7	bitrd 278	. . . . . 6 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ 𝑆 ↔ (𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅)))
9		simpl 481	. . . . . 6 ⊢ ((𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅) → 𝐹:𝑋⟶ω)
10	8, 9	biimtrdi 252	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐹 ∈ 𝑆 → 𝐹:𝑋⟶ω))
11		simpl 481	. . . . 5 ⊢ ((𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆) → 𝐹 ∈ 𝑆)
12	10, 11	impel 504	. . . 4 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) → 𝐹:𝑋⟶ω)
13	12	ffnd 6724	. . 3 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) → 𝐹 Fn 𝑋)
14	1	eleq2d 2811	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺 ∈ 𝑆 ↔ 𝐺 ∈ dom (ω CNF 𝑋)))
15	3, 5, 6	cantnfs 9691	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺 ∈ dom (ω CNF 𝑋) ↔ (𝐺:𝑋⟶ω ∧ 𝐺 finSupp ∅)))
16	14, 15	bitrd 278	. . . . . 6 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺 ∈ 𝑆 ↔ (𝐺:𝑋⟶ω ∧ 𝐺 finSupp ∅)))
17		simpl 481	. . . . . 6 ⊢ ((𝐺:𝑋⟶ω ∧ 𝐺 finSupp ∅) → 𝐺:𝑋⟶ω)
18	16, 17	biimtrdi 252	. . . . 5 ⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → (𝐺 ∈ 𝑆 → 𝐺:𝑋⟶ω))
19		simpr 483	. . . . 5 ⊢ ((𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆) → 𝐺 ∈ 𝑆)
20	18, 19	impel 504	. . . 4 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) → 𝐺:𝑋⟶ω)
21	20	ffnd 6724	. . 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 4217	. . 3 ⊢ (𝑋 ∩ 𝑋) = 𝑋
24	13, 21, 22, 22, 23	offn 7698	. 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 7698	. 2 ⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) → (𝐺 ∘f +o 𝐹) Fn 𝑋)
26	12	ffvelcdmda 7093	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ω)
27	20	ffvelcdmda 7093	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) ∈ ω)
28		nnacom 8638	. . . 4 ⊢ (((𝐹‘𝑥) ∈ ω ∧ (𝐺‘𝑥) ∈ ω) → ((𝐹‘𝑥) +o (𝐺‘𝑥)) = ((𝐺‘𝑥) +o (𝐹‘𝑥)))
29	26, 27, 28	syl2anc 582	. . 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 479	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → 𝐹 Fn 𝑋)
31	21	adantr 479	. . . 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 483	. . . 4 ⊢ ((((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
34		fnfvof 7702	. . . 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 7702	. . . 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 2775	. 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 7042	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 394   = wceq 1533   ∈ wcel 2098  ∅c0 4322   class class class wbr 5149  dom cdm 5678  Oncon0 6371   Fn wfn 6544  ⟶wf 6545  ‘cfv 6549  (class class class)co 7419   ∘_f cof 7683  ωcom 7871   +_o coa 8484   finSupp cfsupp 9387   CNF ccnf 9686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-seqom 8469  df-oadd 8491  df-map 8847  df-cnf 9687

This theorem is referenced by:  naddcnfid2  42939