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Mirrors > Home > ILE Home > Th. List > apmul1 | GIF version |
Description: Multiplication of both sides of complex apartness by a complex number apart from zero. (Contributed by Jim Kingdon, 20-Mar-2020.) |
Ref | Expression |
---|---|
apmul1 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐴 # 𝐵 ↔ (𝐴 · 𝐶) # (𝐵 · 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 999 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → 𝐴 ∈ ℂ) | |
2 | simp3l 1027 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → 𝐶 ∈ ℂ) | |
3 | simp3r 1028 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → 𝐶 # 0) | |
4 | 2, 3 | recclapd 8800 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (1 / 𝐶) ∈ ℂ) |
5 | 1, 2, 4 | mulassd 8043 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 · 𝐶) · (1 / 𝐶)) = (𝐴 · (𝐶 · (1 / 𝐶)))) |
6 | 2, 3 | recidapd 8802 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐶 · (1 / 𝐶)) = 1) |
7 | 6 | oveq2d 5934 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐴 · (𝐶 · (1 / 𝐶))) = (𝐴 · 1)) |
8 | 1 | mulridd 8036 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐴 · 1) = 𝐴) |
9 | 5, 7, 8 | 3eqtrd 2230 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 · 𝐶) · (1 / 𝐶)) = 𝐴) |
10 | simp2 1000 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → 𝐵 ∈ ℂ) | |
11 | 10, 2, 4 | mulassd 8043 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐵 · 𝐶) · (1 / 𝐶)) = (𝐵 · (𝐶 · (1 / 𝐶)))) |
12 | 6 | oveq2d 5934 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐵 · (𝐶 · (1 / 𝐶))) = (𝐵 · 1)) |
13 | 10 | mulridd 8036 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐵 · 1) = 𝐵) |
14 | 11, 12, 13 | 3eqtrd 2230 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐵 · 𝐶) · (1 / 𝐶)) = 𝐵) |
15 | 9, 14 | breq12d 4042 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (((𝐴 · 𝐶) · (1 / 𝐶)) # ((𝐵 · 𝐶) · (1 / 𝐶)) ↔ 𝐴 # 𝐵)) |
16 | 1, 2 | mulcld 8040 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐴 · 𝐶) ∈ ℂ) |
17 | 10, 2 | mulcld 8040 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐵 · 𝐶) ∈ ℂ) |
18 | mulext1 8631 | . . . 4 ⊢ (((𝐴 · 𝐶) ∈ ℂ ∧ (𝐵 · 𝐶) ∈ ℂ ∧ (1 / 𝐶) ∈ ℂ) → (((𝐴 · 𝐶) · (1 / 𝐶)) # ((𝐵 · 𝐶) · (1 / 𝐶)) → (𝐴 · 𝐶) # (𝐵 · 𝐶))) | |
19 | 16, 17, 4, 18 | syl3anc 1249 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (((𝐴 · 𝐶) · (1 / 𝐶)) # ((𝐵 · 𝐶) · (1 / 𝐶)) → (𝐴 · 𝐶) # (𝐵 · 𝐶))) |
20 | 15, 19 | sylbird 170 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐴 # 𝐵 → (𝐴 · 𝐶) # (𝐵 · 𝐶))) |
21 | mulext1 8631 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐶) # (𝐵 · 𝐶) → 𝐴 # 𝐵)) | |
22 | 21 | 3adant3r 1237 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 · 𝐶) # (𝐵 · 𝐶) → 𝐴 # 𝐵)) |
23 | 20, 22 | impbid 129 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐴 # 𝐵 ↔ (𝐴 · 𝐶) # (𝐵 · 𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 980 ∈ wcel 2164 class class class wbr 4029 (class class class)co 5918 ℂcc 7870 0cc0 7872 1c1 7873 · cmul 7877 # cap 8600 / cdiv 8691 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-id 4324 df-po 4327 df-iso 4328 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-iota 5215 df-fun 5256 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 |
This theorem is referenced by: apmul2 8808 divap1d 8820 apdivmuld 8832 qapne 9704 apcxp2 15072 |
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