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Mirrors > Home > ILE Home > Th. List > mulid2 | GIF version |
Description: Identity law for multiplication. Note: see mulid1 7485 for commuted version. (Contributed by NM, 8-Oct-1999.) |
Ref | Expression |
---|---|
mulid2 | ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 7438 | . . 3 ⊢ 1 ∈ ℂ | |
2 | mulcom 7471 | . . 3 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 · 𝐴) = (𝐴 · 1)) | |
3 | 1, 2 | mpan 415 | . 2 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = (𝐴 · 1)) |
4 | mulid1 7485 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴) | |
5 | 3, 4 | eqtrd 2120 | 1 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1289 ∈ wcel 1438 (class class class)co 5652 ℂcc 7348 1c1 7351 · cmul 7355 |
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-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-resscn 7437 ax-1cn 7438 ax-icn 7440 ax-addcl 7441 ax-mulcl 7443 ax-mulcom 7446 ax-mulass 7448 ax-distr 7449 ax-1rid 7452 ax-cnre 7456 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-iota 4980 df-fv 5023 df-ov 5655 |
This theorem is referenced by: mulid2i 7491 mulid2d 7506 muladd11 7615 1p1times 7616 mulm1 7878 div1 8170 recdivap 8185 divdivap2 8191 conjmulap 8196 expp1 9962 recan 10542 arisum 10892 geo2sum 10908 demoivreALT 11063 gcdadd 11254 gcdid 11255 |
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