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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 10299 | . . . 4 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
2 | pinn 10299 | . . . 4 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω) | |
3 | nnacom 8242 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) = (𝐵 +o 𝐴)) | |
4 | 1, 2, 3 | syl2an 597 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +o 𝐵) = (𝐵 +o 𝐴)) |
5 | addpiord 10305 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐴 +o 𝐵)) | |
6 | addpiord 10305 | . . . 4 ⊢ ((𝐵 ∈ N ∧ 𝐴 ∈ N) → (𝐵 +N 𝐴) = (𝐵 +o 𝐴)) | |
7 | 6 | ancoms 461 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐵 +N 𝐴) = (𝐵 +o 𝐴)) |
8 | 4, 5, 7 | 3eqtr4d 2866 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐵 +N 𝐴)) |
9 | dmaddpi 10311 | . . 3 ⊢ dom +N = (N × N) | |
10 | 9 | ndmovcom 7334 | . 2 ⊢ (¬ (𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐵 +N 𝐴)) |
11 | 8, 10 | pm2.61i 184 | 1 ⊢ (𝐴 +N 𝐵) = (𝐵 +N 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∈ wcel 2110 (class class class)co 7155 ωcom 7579 +o coa 8098 Ncnpi 10265 +N cpli 10266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-oadd 8105 df-ni 10293 df-pli 10294 |
This theorem is referenced by: addcompq 10371 adderpqlem 10375 |
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