Theorem ofoacom 43336

Description: Component-wise addition of natural numnber-yielding functions commutes. (Contributed by RP, 5-Jan-2025.)

Assertion

Ref	Expression
ofoacom	⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐹 ∘f +o 𝐺) = (𝐺 ∘f +o 𝐹))

Proof of Theorem ofoacom

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmapfn 8887	. . . 4 ⊢ (𝐹 ∈ (ω ↑m 𝐴) → 𝐹 Fn 𝐴)
2	1	ad2antrl 728	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → 𝐹 Fn 𝐴)
3		elmapfn 8887	. . . 4 ⊢ (𝐺 ∈ (ω ↑m 𝐴) → 𝐺 Fn 𝐴)
4	3	ad2antll 729	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → 𝐺 Fn 𝐴)
5		simpl 482	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → 𝐴 ∈ 𝑉)
6		inidm 4207	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
7	2, 4, 5, 5, 6	offn 7692	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐹 ∘f +o 𝐺) Fn 𝐴)
8	4, 2, 5, 5, 6	offn 7692	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐺 ∘f +o 𝐹) Fn 𝐴)
9		elmapi 8871	. . . . . 6 ⊢ (𝐹 ∈ (ω ↑m 𝐴) → 𝐹:𝐴⟶ω)
10	9	ad2antrl 728	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → 𝐹:𝐴⟶ω)
11	10	ffvelcdmda 7084	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ ω)
12		elmapi 8871	. . . . . 6 ⊢ (𝐺 ∈ (ω ↑m 𝐴) → 𝐺:𝐴⟶ω)
13	12	ad2antll 729	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → 𝐺:𝐴⟶ω)
14	13	ffvelcdmda 7084	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ ω)
15		nnacom 8637	. . . 4 ⊢ (((𝐹‘𝑎) ∈ ω ∧ (𝐺‘𝑎) ∈ ω) → ((𝐹‘𝑎) +o (𝐺‘𝑎)) = ((𝐺‘𝑎) +o (𝐹‘𝑎)))
16	11, 14, 15	syl2anc 584	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → ((𝐹‘𝑎) +o (𝐺‘𝑎)) = ((𝐺‘𝑎) +o (𝐹‘𝑎)))
17	2, 4	jca 511	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴))
18	5	anim1i 615	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴))
19		fnfvof 7696	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴)) → ((𝐹 ∘f +o 𝐺)‘𝑎) = ((𝐹‘𝑎) +o (𝐺‘𝑎)))
20	17, 18, 19	syl2an2r 685	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → ((𝐹 ∘f +o 𝐺)‘𝑎) = ((𝐹‘𝑎) +o (𝐺‘𝑎)))
21	4, 2	jca 511	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐺 Fn 𝐴 ∧ 𝐹 Fn 𝐴))
22		fnfvof 7696	. . . 4 ⊢ (((𝐺 Fn 𝐴 ∧ 𝐹 Fn 𝐴) ∧ (𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴)) → ((𝐺 ∘f +o 𝐹)‘𝑎) = ((𝐺‘𝑎) +o (𝐹‘𝑎)))
23	21, 18, 22	syl2an2r 685	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → ((𝐺 ∘f +o 𝐹)‘𝑎) = ((𝐺‘𝑎) +o (𝐹‘𝑎)))
24	16, 20, 23	3eqtr4d 2779	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → ((𝐹 ∘f +o 𝐺)‘𝑎) = ((𝐺 ∘f +o 𝐹)‘𝑎))
25	7, 8, 24	eqfnfvd 7034	1 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐹 ∘f +o 𝐺) = (𝐺 ∘f +o 𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107   Fn wfn 6536  ⟶wf 6537  ‘cfv 6541  (class class class)co 7413   ∘_f cof 7677  ωcom 7869   +_o coa 8485   ↑_m cmap 8848

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7679  df-om 7870  df-1st 7996  df-2nd 7997  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-oadd 8492  df-map 8850

This theorem is referenced by: (None)