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Mirrors > Home > MPE Home > Th. List > mul0or | Structured version Visualization version GIF version |
Description: If a product is zero, one of its factors must be zero. Theorem I.11 of [Apostol] p. 18. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
mul0or | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵) = 0 ↔ (𝐴 = 0 ∨ 𝐵 = 0))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 484 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ) | |
2 | 1 | adantr 480 | . . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐵 ≠ 0) → 𝐵 ∈ ℂ) |
3 | 2 | mul02d 11457 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐵 ≠ 0) → (0 · 𝐵) = 0) |
4 | 3 | eqeq2d 2746 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐵 ≠ 0) → ((𝐴 · 𝐵) = (0 · 𝐵) ↔ (𝐴 · 𝐵) = 0)) |
5 | simpl 482 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℂ) | |
6 | 5 | adantr 480 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐵 ≠ 0) → 𝐴 ∈ ℂ) |
7 | 0cnd 11252 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐵 ≠ 0) → 0 ∈ ℂ) | |
8 | simpr 484 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐵 ≠ 0) → 𝐵 ≠ 0) | |
9 | 6, 7, 2, 8 | mulcan2d 11895 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐵 ≠ 0) → ((𝐴 · 𝐵) = (0 · 𝐵) ↔ 𝐴 = 0)) |
10 | 4, 9 | bitr3d 281 | . . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐵 ≠ 0) → ((𝐴 · 𝐵) = 0 ↔ 𝐴 = 0)) |
11 | 10 | biimpd 229 | . . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐵 ≠ 0) → ((𝐴 · 𝐵) = 0 → 𝐴 = 0)) |
12 | 11 | impancom 451 | . . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 · 𝐵) = 0) → (𝐵 ≠ 0 → 𝐴 = 0)) |
13 | 12 | necon1bd 2956 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 · 𝐵) = 0) → (¬ 𝐴 = 0 → 𝐵 = 0)) |
14 | 13 | orrd 863 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 · 𝐵) = 0) → (𝐴 = 0 ∨ 𝐵 = 0)) |
15 | 14 | ex 412 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵) = 0 → (𝐴 = 0 ∨ 𝐵 = 0))) |
16 | 1 | mul02d 11457 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (0 · 𝐵) = 0) |
17 | oveq1 7438 | . . . . 5 ⊢ (𝐴 = 0 → (𝐴 · 𝐵) = (0 · 𝐵)) | |
18 | 17 | eqeq1d 2737 | . . . 4 ⊢ (𝐴 = 0 → ((𝐴 · 𝐵) = 0 ↔ (0 · 𝐵) = 0)) |
19 | 16, 18 | syl5ibrcom 247 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 = 0 → (𝐴 · 𝐵) = 0)) |
20 | 5 | mul01d 11458 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 0) = 0) |
21 | oveq2 7439 | . . . . 5 ⊢ (𝐵 = 0 → (𝐴 · 𝐵) = (𝐴 · 0)) | |
22 | 21 | eqeq1d 2737 | . . . 4 ⊢ (𝐵 = 0 → ((𝐴 · 𝐵) = 0 ↔ (𝐴 · 0) = 0)) |
23 | 20, 22 | syl5ibrcom 247 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 = 0 → (𝐴 · 𝐵) = 0)) |
24 | 19, 23 | jaod 859 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 = 0 ∨ 𝐵 = 0) → (𝐴 · 𝐵) = 0)) |
25 | 15, 24 | impbid 212 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵) = 0 ↔ (𝐴 = 0 ∨ 𝐵 = 0))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 (class class class)co 7431 ℂcc 11151 0cc0 11153 · cmul 11158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 |
This theorem is referenced by: mulne0b 11902 msq0i 11908 mul0ori 11909 msq0d 11910 mul0ord 11911 coseq1 26582 efrlim 27027 efrlimOLD 27028 zringidom 33559 |
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