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Theorem ofoacom 42111
Description: Component-wise addition of natural numnber-yielding functions commutes. (Contributed by RP, 5-Jan-2025.)
Assertion
Ref Expression
ofoacom ((𝐴𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐹f +o 𝐺) = (𝐺f +o 𝐹))

Proof of Theorem ofoacom
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 elmapfn 8859 . . . 4 (𝐹 ∈ (ω ↑m 𝐴) → 𝐹 Fn 𝐴)
21ad2antrl 727 . . 3 ((𝐴𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → 𝐹 Fn 𝐴)
3 elmapfn 8859 . . . 4 (𝐺 ∈ (ω ↑m 𝐴) → 𝐺 Fn 𝐴)
43ad2antll 728 . . 3 ((𝐴𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → 𝐺 Fn 𝐴)
5 simpl 484 . . 3 ((𝐴𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → 𝐴𝑉)
6 inidm 4219 . . 3 (𝐴𝐴) = 𝐴
72, 4, 5, 5, 6offn 7683 . 2 ((𝐴𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐹f +o 𝐺) Fn 𝐴)
84, 2, 5, 5, 6offn 7683 . 2 ((𝐴𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐺f +o 𝐹) Fn 𝐴)
9 elmapi 8843 . . . . . 6 (𝐹 ∈ (ω ↑m 𝐴) → 𝐹:𝐴⟶ω)
109ad2antrl 727 . . . . 5 ((𝐴𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → 𝐹:𝐴⟶ω)
1110ffvelcdmda 7087 . . . 4 (((𝐴𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) ∧ 𝑎𝐴) → (𝐹𝑎) ∈ ω)
12 elmapi 8843 . . . . . 6 (𝐺 ∈ (ω ↑m 𝐴) → 𝐺:𝐴⟶ω)
1312ad2antll 728 . . . . 5 ((𝐴𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → 𝐺:𝐴⟶ω)
1413ffvelcdmda 7087 . . . 4 (((𝐴𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) ∧ 𝑎𝐴) → (𝐺𝑎) ∈ ω)
15 nnacom 8617 . . . 4 (((𝐹𝑎) ∈ ω ∧ (𝐺𝑎) ∈ ω) → ((𝐹𝑎) +o (𝐺𝑎)) = ((𝐺𝑎) +o (𝐹𝑎)))
1611, 14, 15syl2anc 585 . . 3 (((𝐴𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) ∧ 𝑎𝐴) → ((𝐹𝑎) +o (𝐺𝑎)) = ((𝐺𝑎) +o (𝐹𝑎)))
172, 4jca 513 . . . 4 ((𝐴𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐹 Fn 𝐴𝐺 Fn 𝐴))
185anim1i 616 . . . 4 (((𝐴𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) ∧ 𝑎𝐴) → (𝐴𝑉𝑎𝐴))
19 fnfvof 7687 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐴𝑉𝑎𝐴)) → ((𝐹f +o 𝐺)‘𝑎) = ((𝐹𝑎) +o (𝐺𝑎)))
2017, 18, 19syl2an2r 684 . . 3 (((𝐴𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) ∧ 𝑎𝐴) → ((𝐹f +o 𝐺)‘𝑎) = ((𝐹𝑎) +o (𝐺𝑎)))
214, 2jca 513 . . . 4 ((𝐴𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐺 Fn 𝐴𝐹 Fn 𝐴))
22 fnfvof 7687 . . . 4 (((𝐺 Fn 𝐴𝐹 Fn 𝐴) ∧ (𝐴𝑉𝑎𝐴)) → ((𝐺f +o 𝐹)‘𝑎) = ((𝐺𝑎) +o (𝐹𝑎)))
2321, 18, 22syl2an2r 684 . . 3 (((𝐴𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) ∧ 𝑎𝐴) → ((𝐺f +o 𝐹)‘𝑎) = ((𝐺𝑎) +o (𝐹𝑎)))
2416, 20, 233eqtr4d 2783 . 2 (((𝐴𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) ∧ 𝑎𝐴) → ((𝐹f +o 𝐺)‘𝑎) = ((𝐺f +o 𝐹)‘𝑎))
257, 8, 24eqfnfvd 7036 1 ((𝐴𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐹f +o 𝐺) = (𝐺f +o 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107   Fn wfn 6539  wf 6540  cfv 6544  (class class class)co 7409  f cof 7668  ωcom 7855   +o coa 8463  m cmap 8820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-oadd 8470  df-map 8822
This theorem is referenced by: (None)
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