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Mirrors > Home > MPE Home > Th. List > addcompi | Structured version Visualization version GIF version |
Description: Addition of positive integers is commutative. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
addcompi | ⊢ (𝐴 +N 𝐵) = (𝐵 +N 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pinn 10735 | . . . 4 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
2 | pinn 10735 | . . . 4 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω) | |
3 | nnacom 8519 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) = (𝐵 +o 𝐴)) | |
4 | 1, 2, 3 | syl2an 596 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +o 𝐵) = (𝐵 +o 𝐴)) |
5 | addpiord 10741 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐴 +o 𝐵)) | |
6 | addpiord 10741 | . . . 4 ⊢ ((𝐵 ∈ N ∧ 𝐴 ∈ N) → (𝐵 +N 𝐴) = (𝐵 +o 𝐴)) | |
7 | 6 | ancoms 459 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐵 +N 𝐴) = (𝐵 +o 𝐴)) |
8 | 4, 5, 7 | 3eqtr4d 2786 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐵 +N 𝐴)) |
9 | dmaddpi 10747 | . . 3 ⊢ dom +N = (N × N) | |
10 | 9 | ndmovcom 7521 | . 2 ⊢ (¬ (𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐵 +N 𝐴)) |
11 | 8, 10 | pm2.61i 182 | 1 ⊢ (𝐴 +N 𝐵) = (𝐵 +N 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1540 ∈ wcel 2105 (class class class)co 7337 ωcom 7780 +o coa 8364 Ncnpi 10701 +N cpli 10702 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-1st 7899 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-oadd 8371 df-ni 10729 df-pli 10730 |
This theorem is referenced by: addcompq 10807 adderpqlem 10811 |
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