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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 7726 | . . . 4 ⊢ 0 ∈ ℂ | |
2 | 1 | subidi 8001 | . . 3 ⊢ (0 − 0) = 0 |
3 | 2 | oveq1i 5752 | . 2 ⊢ ((0 − 0) · 𝐴) = (0 · 𝐴) |
4 | subdir 8116 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((0 − 0) · 𝐴) = ((0 · 𝐴) − (0 · 𝐴))) | |
5 | 1, 1, 4 | mp3an12 1290 | . . 3 ⊢ (𝐴 ∈ ℂ → ((0 − 0) · 𝐴) = ((0 · 𝐴) − (0 · 𝐴))) |
6 | mulcl 7715 | . . . . 5 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (0 · 𝐴) ∈ ℂ) | |
7 | 6 | subidd 8029 | . . . 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 2150 | . 2 ⊢ (𝐴 ∈ ℂ → ((0 − 0) · 𝐴) = 0) |
10 | 3, 9 | syl5eqr 2164 | 1 ⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 (class class class)co 5742 ℂcc 7586 0cc0 7588 · cmul 7593 − cmin 7901 |
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 588 ax-in2 589 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-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-setind 4422 ax-resscn 7680 ax-1cn 7681 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-sub 7903 |
This theorem is referenced by: mul02lem2 8118 mul01 8119 mul02i 8120 mul02d 8122 demoivreALT 11407 |
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