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Mirrors > Home > ILE Home > Th. List > addcompig | GIF version |
Description: Addition of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.) |
Ref | Expression |
---|---|
addcompig | ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐵 +N 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pinn 6561 | . . 3 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
2 | pinn 6561 | . . 3 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω) | |
3 | nnacom 6128 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 𝐵) = (𝐵 +𝑜 𝐴)) | |
4 | 1, 2, 3 | syl2an 283 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +𝑜 𝐵) = (𝐵 +𝑜 𝐴)) |
5 | addpiord 6568 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐴 +𝑜 𝐵)) | |
6 | addpiord 6568 | . . 3 ⊢ ((𝐵 ∈ N ∧ 𝐴 ∈ N) → (𝐵 +N 𝐴) = (𝐵 +𝑜 𝐴)) | |
7 | 6 | ancoms 264 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐵 +N 𝐴) = (𝐵 +𝑜 𝐴)) |
8 | 4, 5, 7 | 3eqtr4d 2124 | 1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐵 +N 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1285 ∈ wcel 1434 ωcom 4339 (class class class)co 5543 +𝑜 coa 6062 Ncnpi 6524 +N cpli 6525 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-coll 3901 ax-sep 3904 ax-nul 3912 ax-pow 3956 ax-pr 3972 ax-un 4196 ax-setind 4288 ax-iinf 4337 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ne 2247 df-ral 2354 df-rex 2355 df-reu 2356 df-rab 2358 df-v 2604 df-sbc 2817 df-csb 2910 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-nul 3259 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-uni 3610 df-int 3645 df-iun 3688 df-br 3794 df-opab 3848 df-mpt 3849 df-tr 3884 df-id 4056 df-iord 4129 df-on 4131 df-suc 4134 df-iom 4340 df-xp 4377 df-rel 4378 df-cnv 4379 df-co 4380 df-dm 4381 df-rn 4382 df-res 4383 df-ima 4384 df-iota 4897 df-fun 4934 df-fn 4935 df-f 4936 df-f1 4937 df-fo 4938 df-f1o 4939 df-fv 4940 df-ov 5546 df-oprab 5547 df-mpt2 5548 df-1st 5798 df-2nd 5799 df-recs 5954 df-irdg 6019 df-oadd 6069 df-ni 6556 df-pli 6557 |
This theorem is referenced by: addcomnqg 6633 |
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