![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 479 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ) | |
2 | 1 | adantr 474 | . . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐵 ≠ 0) → 𝐵 ∈ ℂ) |
3 | 2 | mul02d 10553 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐵 ≠ 0) → (0 · 𝐵) = 0) |
4 | 3 | eqeq2d 2835 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐵 ≠ 0) → ((𝐴 · 𝐵) = (0 · 𝐵) ↔ (𝐴 · 𝐵) = 0)) |
5 | simpl 476 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℂ) | |
6 | 5 | adantr 474 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐵 ≠ 0) → 𝐴 ∈ ℂ) |
7 | 0cnd 10349 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐵 ≠ 0) → 0 ∈ ℂ) | |
8 | simpr 479 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐵 ≠ 0) → 𝐵 ≠ 0) | |
9 | 6, 7, 2, 8 | mulcan2d 10986 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐵 ≠ 0) → ((𝐴 · 𝐵) = (0 · 𝐵) ↔ 𝐴 = 0)) |
10 | 4, 9 | bitr3d 273 | . . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐵 ≠ 0) → ((𝐴 · 𝐵) = 0 ↔ 𝐴 = 0)) |
11 | 10 | biimpd 221 | . . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐵 ≠ 0) → ((𝐴 · 𝐵) = 0 → 𝐴 = 0)) |
12 | 11 | impancom 445 | . . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 · 𝐵) = 0) → (𝐵 ≠ 0 → 𝐴 = 0)) |
13 | 12 | necon1bd 3017 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 · 𝐵) = 0) → (¬ 𝐴 = 0 → 𝐵 = 0)) |
14 | 13 | orrd 896 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 · 𝐵) = 0) → (𝐴 = 0 ∨ 𝐵 = 0)) |
15 | 14 | ex 403 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵) = 0 → (𝐴 = 0 ∨ 𝐵 = 0))) |
16 | 1 | mul02d 10553 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (0 · 𝐵) = 0) |
17 | oveq1 6912 | . . . . 5 ⊢ (𝐴 = 0 → (𝐴 · 𝐵) = (0 · 𝐵)) | |
18 | 17 | eqeq1d 2827 | . . . 4 ⊢ (𝐴 = 0 → ((𝐴 · 𝐵) = 0 ↔ (0 · 𝐵) = 0)) |
19 | 16, 18 | syl5ibrcom 239 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 = 0 → (𝐴 · 𝐵) = 0)) |
20 | 5 | mul01d 10554 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 0) = 0) |
21 | oveq2 6913 | . . . . 5 ⊢ (𝐵 = 0 → (𝐴 · 𝐵) = (𝐴 · 0)) | |
22 | 21 | eqeq1d 2827 | . . . 4 ⊢ (𝐵 = 0 → ((𝐴 · 𝐵) = 0 ↔ (𝐴 · 0) = 0)) |
23 | 20, 22 | syl5ibrcom 239 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 = 0 → (𝐴 · 𝐵) = 0)) |
24 | 19, 23 | jaod 892 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 = 0 ∨ 𝐵 = 0) → (𝐴 · 𝐵) = 0)) |
25 | 15, 24 | impbid 204 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵) = 0 ↔ (𝐴 = 0 ∨ 𝐵 = 0))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∨ wo 880 = wceq 1658 ∈ wcel 2166 ≠ wne 2999 (class class class)co 6905 ℂcc 10250 0cc0 10252 · cmul 10257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-po 5263 df-so 5264 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 |
This theorem is referenced by: mulne0b 10993 msq0i 10999 mul0ori 11000 msq0d 11001 mul0ord 11002 coseq1 24674 efrlim 25109 |
Copyright terms: Public domain | W3C validator |