Theorem ofoacom 41252

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 8684	. . . 4 ⊢ (𝐹 ∈ (ω ↑m 𝐴) → 𝐹 Fn 𝐴)
2	1	ad2antrl 726	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → 𝐹 Fn 𝐴)
3		elmapfn 8684	. . . 4 ⊢ (𝐺 ∈ (ω ↑m 𝐴) → 𝐺 Fn 𝐴)
4	3	ad2antll 727	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → 𝐺 Fn 𝐴)
5		simpl 484	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → 𝐴 ∈ 𝑉)
6		inidm 4158	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
7	2, 4, 5, 5, 6	offn 7578	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐹 ∘f +o 𝐺) Fn 𝐴)
8	4, 2, 5, 5, 6	offn 7578	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐺 ∘f +o 𝐹) Fn 𝐴)
9		elmapi 8668	. . . . . 6 ⊢ (𝐹 ∈ (ω ↑m 𝐴) → 𝐹:𝐴⟶ω)
10	9	ad2antrl 726	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → 𝐹:𝐴⟶ω)
11	10	ffvelcdmda 6993	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ ω)
12		elmapi 8668	. . . . . 6 ⊢ (𝐺 ∈ (ω ↑m 𝐴) → 𝐺:𝐴⟶ω)
13	12	ad2antll 727	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → 𝐺:𝐴⟶ω)
14	13	ffvelcdmda 6993	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ ω)
15		nnacom 8479	. . . 4 ⊢ (((𝐹‘𝑎) ∈ ω ∧ (𝐺‘𝑎) ∈ ω) → ((𝐹‘𝑎) +o (𝐺‘𝑎)) = ((𝐺‘𝑎) +o (𝐹‘𝑎)))
16	11, 14, 15	syl2anc 585	. . 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 513	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴))
18	5	anim1i 616	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴))
19		fnfvof 7582	. . . 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 683	. . 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 513	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐺 Fn 𝐴 ∧ 𝐹 Fn 𝐴))
22		fnfvof 7582	. . . 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 683	. . 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 2786	. 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 6944	1 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐹 ∘f +o 𝐺) = (𝐺 ∘f +o 𝐹))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1539   ∈ wcel 2104   Fn wfn 6453  ⟶wf 6454  ‘cfv 6458  (class class class)co 7307   ∘_f cof 7563  ωcom 7744   +_o coa 8325   ↑_m cmap 8646

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

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-1st 7863  df-2nd 7864  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-rdg 8272  df-oadd 8332  df-map 8648

This theorem is referenced by: (None)