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Mirrors > Home > ILE Home > Th. List > mul0eqap | GIF version |
Description: If two numbers are apart from each other and their product is zero, one of them must be zero. (Contributed by Jim Kingdon, 31-Jul-2023.) |
Ref | Expression |
---|---|
mul0eqap.a | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
mul0eqap.b | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
mul0eqap.ab | ⊢ (𝜑 → 𝐴 # 𝐵) |
mul0eqap.0 | ⊢ (𝜑 → (𝐴 · 𝐵) = 0) |
Ref | Expression |
---|---|
mul0eqap | ⊢ (𝜑 → (𝐴 = 0 ∨ 𝐵 = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul0eqap.ab | . . . 4 ⊢ (𝜑 → 𝐴 # 𝐵) | |
2 | mul0eqap.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
3 | mul0eqap.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
4 | 0cnd 7928 | . . . . 5 ⊢ (𝜑 → 0 ∈ ℂ) | |
5 | apcotr 8541 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐴 # 𝐵 → (𝐴 # 0 ∨ 𝐵 # 0))) | |
6 | 2, 3, 4, 5 | syl3anc 1238 | . . . 4 ⊢ (𝜑 → (𝐴 # 𝐵 → (𝐴 # 0 ∨ 𝐵 # 0))) |
7 | 1, 6 | mpd 13 | . . 3 ⊢ (𝜑 → (𝐴 # 0 ∨ 𝐵 # 0)) |
8 | mul0eqap.0 | . . . . . . 7 ⊢ (𝜑 → (𝐴 · 𝐵) = 0) | |
9 | 8 | adantr 276 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 # 0) → (𝐴 · 𝐵) = 0) |
10 | 3 | adantr 276 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 # 0) → 𝐵 ∈ ℂ) |
11 | 0cnd 7928 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 # 0) → 0 ∈ ℂ) | |
12 | 2, 3 | mulcld 7955 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℂ) |
13 | 12 | adantr 276 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 # 0) → (𝐴 · 𝐵) ∈ ℂ) |
14 | ibar 301 | . . . . . . . 8 ⊢ (𝐴 # 0 → (𝐵 # 0 ↔ (𝐴 # 0 ∧ 𝐵 # 0))) | |
15 | 2, 3 | mulap0bd 8590 | . . . . . . . 8 ⊢ (𝜑 → ((𝐴 # 0 ∧ 𝐵 # 0) ↔ (𝐴 · 𝐵) # 0)) |
16 | 14, 15 | sylan9bbr 463 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 # 0) → (𝐵 # 0 ↔ (𝐴 · 𝐵) # 0)) |
17 | 10, 11, 13, 11, 16 | apcon4bid 8558 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 # 0) → (𝐵 = 0 ↔ (𝐴 · 𝐵) = 0)) |
18 | 9, 17 | mpbird 167 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 # 0) → 𝐵 = 0) |
19 | 18 | ex 115 | . . . 4 ⊢ (𝜑 → (𝐴 # 0 → 𝐵 = 0)) |
20 | 8 | adantr 276 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 # 0) → (𝐴 · 𝐵) = 0) |
21 | 2 | adantr 276 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 # 0) → 𝐴 ∈ ℂ) |
22 | 0cnd 7928 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 # 0) → 0 ∈ ℂ) | |
23 | 12 | adantr 276 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 # 0) → (𝐴 · 𝐵) ∈ ℂ) |
24 | iba 300 | . . . . . . . 8 ⊢ (𝐵 # 0 → (𝐴 # 0 ↔ (𝐴 # 0 ∧ 𝐵 # 0))) | |
25 | 24, 15 | sylan9bbr 463 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 # 0) → (𝐴 # 0 ↔ (𝐴 · 𝐵) # 0)) |
26 | 21, 22, 23, 22, 25 | apcon4bid 8558 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 # 0) → (𝐴 = 0 ↔ (𝐴 · 𝐵) = 0)) |
27 | 20, 26 | mpbird 167 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 # 0) → 𝐴 = 0) |
28 | 27 | ex 115 | . . . 4 ⊢ (𝜑 → (𝐵 # 0 → 𝐴 = 0)) |
29 | 19, 28 | orim12d 786 | . . 3 ⊢ (𝜑 → ((𝐴 # 0 ∨ 𝐵 # 0) → (𝐵 = 0 ∨ 𝐴 = 0))) |
30 | 7, 29 | mpd 13 | . 2 ⊢ (𝜑 → (𝐵 = 0 ∨ 𝐴 = 0)) |
31 | 30 | orcomd 729 | 1 ⊢ (𝜑 → (𝐴 = 0 ∨ 𝐵 = 0)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∨ wo 708 = wceq 1353 ∈ wcel 2148 class class class wbr 4000 (class class class)co 5868 ℂcc 7787 0cc0 7789 · cmul 7794 # cap 8515 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-mulrcl 7888 ax-addcom 7889 ax-mulcom 7890 ax-addass 7891 ax-mulass 7892 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-1rid 7896 ax-0id 7897 ax-rnegex 7898 ax-precex 7899 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-apti 7904 ax-pre-ltadd 7905 ax-pre-mulgt0 7906 ax-pre-mulext 7907 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4289 df-po 4292 df-iso 4293 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-iota 5173 df-fun 5213 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-reap 8509 df-ap 8516 |
This theorem is referenced by: (None) |
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