Theorem ofoacom 43350

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 8838	. . . 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 8838	. . . 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 4190	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
7	2, 4, 5, 5, 6	offn 7666	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐹 ∘f +o 𝐺) Fn 𝐴)
8	4, 2, 5, 5, 6	offn 7666	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐺 ∘f +o 𝐹) Fn 𝐴)
9		elmapi 8822	. . . . . 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 7056	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ ω)
12		elmapi 8822	. . . . . 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 7056	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ ω)
15		nnacom 8581	. . . 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 7670	. . . 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 7670	. . . 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 2774	. 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 7006	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 1540   ∈ wcel 2109   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   ∘_f cof 7651  ωcom 7842   +_o coa 8431   ↑_m cmap 8799

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-oadd 8438  df-map 8801

This theorem is referenced by: (None)