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Mirrors > Home > ILE Home > Th. List > mul02 | GIF version |
Description: Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 10-Aug-1999.) |
Ref | Expression |
---|---|
mul02 | ⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 7758 | . . . 4 ⊢ 0 ∈ ℂ | |
2 | 1 | subidi 8033 | . . 3 ⊢ (0 − 0) = 0 |
3 | 2 | oveq1i 5784 | . 2 ⊢ ((0 − 0) · 𝐴) = (0 · 𝐴) |
4 | subdir 8148 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((0 − 0) · 𝐴) = ((0 · 𝐴) − (0 · 𝐴))) | |
5 | 1, 1, 4 | mp3an12 1305 | . . 3 ⊢ (𝐴 ∈ ℂ → ((0 − 0) · 𝐴) = ((0 · 𝐴) − (0 · 𝐴))) |
6 | mulcl 7747 | . . . . 5 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (0 · 𝐴) ∈ ℂ) | |
7 | 6 | subidd 8061 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((0 · 𝐴) − (0 · 𝐴)) = 0) |
8 | 1, 7 | mpan 420 | . . 3 ⊢ (𝐴 ∈ ℂ → ((0 · 𝐴) − (0 · 𝐴)) = 0) |
9 | 5, 8 | eqtrd 2172 | . 2 ⊢ (𝐴 ∈ ℂ → ((0 − 0) · 𝐴) = 0) |
10 | 3, 9 | syl5eqr 2186 | 1 ⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 (class class class)co 5774 ℂcc 7618 0cc0 7620 · cmul 7625 − cmin 7933 |
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-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 ax-resscn 7712 ax-1cn 7713 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sub 7935 |
This theorem is referenced by: mul02lem2 8150 mul01 8151 mul02i 8152 mul02d 8154 demoivreALT 11480 |
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