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Mirrors > Home > ILE Home > Th. List > mulid2 | GIF version |
Description: Identity law for multiplication. Note: see mulid1 7731 for commuted version. (Contributed by NM, 8-Oct-1999.) |
Ref | Expression |
---|---|
mulid2 | ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 7681 | . . 3 ⊢ 1 ∈ ℂ | |
2 | mulcom 7717 | . . 3 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 · 𝐴) = (𝐴 · 1)) | |
3 | 1, 2 | mpan 420 | . 2 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = (𝐴 · 1)) |
4 | mulid1 7731 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴) | |
5 | 3, 4 | eqtrd 2150 | 1 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∈ wcel 1465 (class class class)co 5742 ℂcc 7586 1c1 7589 · cmul 7593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-resscn 7680 ax-1cn 7681 ax-icn 7683 ax-addcl 7684 ax-mulcl 7686 ax-mulcom 7689 ax-mulass 7691 ax-distr 7692 ax-1rid 7695 ax-cnre 7699 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-iota 5058 df-fv 5101 df-ov 5745 |
This theorem is referenced by: mulid2i 7737 mulid2d 7752 muladd11 7863 1p1times 7864 mulm1 8130 div1 8430 recdivap 8445 divdivap2 8451 conjmulap 8456 expp1 10255 recan 10836 arisum 11222 geo2sum 11238 demoivreALT 11394 gcdadd 11585 gcdid 11586 |
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