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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 10022 | . . . 4 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
2 | pinn 10022 | . . . 4 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω) | |
3 | nnacom 7969 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) = (𝐵 +o 𝐴)) | |
4 | 1, 2, 3 | syl2an 589 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +o 𝐵) = (𝐵 +o 𝐴)) |
5 | addpiord 10028 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐴 +o 𝐵)) | |
6 | addpiord 10028 | . . . 4 ⊢ ((𝐵 ∈ N ∧ 𝐴 ∈ N) → (𝐵 +N 𝐴) = (𝐵 +o 𝐴)) | |
7 | 6 | ancoms 452 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐵 +N 𝐴) = (𝐵 +o 𝐴)) |
8 | 4, 5, 7 | 3eqtr4d 2871 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐵 +N 𝐴)) |
9 | dmaddpi 10034 | . . 3 ⊢ dom +N = (N × N) | |
10 | 9 | ndmovcom 7086 | . 2 ⊢ (¬ (𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐵 +N 𝐴)) |
11 | 8, 10 | pm2.61i 177 | 1 ⊢ (𝐴 +N 𝐵) = (𝐵 +N 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 = wceq 1656 ∈ wcel 2164 (class class class)co 6910 ωcom 7331 +o coa 7828 Ncnpi 9988 +N cpli 9989 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-oadd 7835 df-ni 10016 df-pli 10017 |
This theorem is referenced by: addcompq 10094 adderpqlem 10098 |
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