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Mirrors > Home > ILE Home > Th. List > mulcompig | GIF version |
Description: Multiplication of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.) |
Ref | Expression |
---|---|
mulcompig | ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐵 ·N 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pinn 6789 | . . 3 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
2 | pinn 6789 | . . 3 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω) | |
3 | nnmcom 6185 | . . 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 | mulpiord 6797 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·𝑜 𝐵)) | |
6 | mulpiord 6797 | . . 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 2127 | 1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐵 ·N 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1287 ∈ wcel 1436 ωcom 4371 (class class class)co 5594 ·𝑜 comu 6114 Ncnpi 6752 ·N cmi 6754 |
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 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-coll 3922 ax-sep 3925 ax-nul 3933 ax-pow 3977 ax-pr 4003 ax-un 4227 ax-setind 4319 ax-iinf 4369 |
This theorem depends on definitions: df-bi 115 df-3an 924 df-tru 1290 df-fal 1293 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ne 2252 df-ral 2360 df-rex 2361 df-reu 2362 df-rab 2364 df-v 2616 df-sbc 2829 df-csb 2922 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-nul 3273 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-uni 3631 df-int 3666 df-iun 3709 df-br 3815 df-opab 3869 df-mpt 3870 df-tr 3905 df-id 4087 df-iord 4160 df-on 4162 df-suc 4165 df-iom 4372 df-xp 4410 df-rel 4411 df-cnv 4412 df-co 4413 df-dm 4414 df-rn 4415 df-res 4416 df-ima 4417 df-iota 4937 df-fun 4974 df-fn 4975 df-f 4976 df-f1 4977 df-fo 4978 df-f1o 4979 df-fv 4980 df-ov 5597 df-oprab 5598 df-mpt2 5599 df-1st 5849 df-2nd 5850 df-recs 6005 df-irdg 6070 df-oadd 6120 df-omul 6121 df-ni 6784 df-mi 6786 |
This theorem is referenced by: dfplpq2 6834 enqbreq2 6837 enqer 6838 addcmpblnq 6847 mulcmpblnq 6848 ordpipqqs 6854 addcomnqg 6861 addassnqg 6862 mulcomnqg 6863 mulcanenq 6865 distrnqg 6867 mulidnq 6869 recexnq 6870 nqtri3or 6876 ltsonq 6878 ltanqg 6880 ltmnqg 6881 ltexnqq 6888 archnqq 6897 prarloclemarch2 6899 ltnnnq 6903 prarloclemlt 6973 |
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