Theorem ofoacom 43950

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 8850	. . . 4 ⊢ (𝐹 ∈ (ω ↑m 𝐴) → 𝐹 Fn 𝐴)
2	1	ad2antrl 740	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → 𝐹 Fn 𝐴)
3		elmapfn 8850	. . . 4 ⊢ (𝐺 ∈ (ω ↑m 𝐴) → 𝐺 Fn 𝐴)
4	3	ad2antll 741	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → 𝐺 Fn 𝐴)
5		simpl 487	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → 𝐴 ∈ 𝑉)
6		inidm 4181	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
7	2, 4, 5, 5, 6	offn 7677	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐹 ∘f +o 𝐺) Fn 𝐴)
8	4, 2, 5, 5, 6	offn 7677	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐺 ∘f +o 𝐹) Fn 𝐴)
9		elmapi 8834	. . . . . 6 ⊢ (𝐹 ∈ (ω ↑m 𝐴) → 𝐹:𝐴⟶ω)
10	9	ad2antrl 740	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → 𝐹:𝐴⟶ω)
11	10	ffvelcdmda 7069	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ ω)
12		elmapi 8834	. . . . . 6 ⊢ (𝐺 ∈ (ω ↑m 𝐴) → 𝐺:𝐴⟶ω)
13	12	ad2antll 741	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → 𝐺:𝐴⟶ω)
14	13	ffvelcdmda 7069	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ ω)
15		nnacom 8591	. . . 4 ⊢ (((𝐹‘𝑎) ∈ ω ∧ (𝐺‘𝑎) ∈ ω) → ((𝐹‘𝑎) +o (𝐺‘𝑎)) = ((𝐺‘𝑎) +o (𝐹‘𝑎)))
16	11, 14, 15	syl2anc 595	. . 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 520	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴))
18	5	anim1i 626	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) ∧ 𝑎 ∈ 𝐴) → (𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴))
19		fnfvof 7681	. . . 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 697	. . 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 520	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐺 Fn 𝐴 ∧ 𝐹 Fn 𝐴))
22		fnfvof 7681	. . . 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 697	. . 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 2810	. 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 7018	1 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐹 ∘f +o 𝐺) = (𝐺 ∘f +o 𝐹))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145   Fn wfn 6520  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400   ∘_f cof 7662  ωcom 7850   +_o coa 8438   ↑_m cmap 8812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-of 7664  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-oadd 8445  df-map 8814

This theorem is referenced by: (None)