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Mirrors > Home > ILE Home > Th. List > mulneg1 | GIF version |
Description: Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
mulneg1 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 · 𝐵) = -(𝐴 · 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 7019 | . . . 4 ⊢ 0 ∈ ℂ | |
2 | subdir 7383 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((0 − 𝐴) · 𝐵) = ((0 · 𝐵) − (𝐴 · 𝐵))) | |
3 | 1, 2 | mp3an1 1219 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((0 − 𝐴) · 𝐵) = ((0 · 𝐵) − (𝐴 · 𝐵))) |
4 | simpr 103 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ) | |
5 | 4 | mul02d 7389 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (0 · 𝐵) = 0) |
6 | 5 | oveq1d 5527 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((0 · 𝐵) − (𝐴 · 𝐵)) = (0 − (𝐴 · 𝐵))) |
7 | 3, 6 | eqtrd 2072 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((0 − 𝐴) · 𝐵) = (0 − (𝐴 · 𝐵))) |
8 | df-neg 7185 | . . 3 ⊢ -𝐴 = (0 − 𝐴) | |
9 | 8 | oveq1i 5522 | . 2 ⊢ (-𝐴 · 𝐵) = ((0 − 𝐴) · 𝐵) |
10 | df-neg 7185 | . 2 ⊢ -(𝐴 · 𝐵) = (0 − (𝐴 · 𝐵)) | |
11 | 7, 9, 10 | 3eqtr4g 2097 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 · 𝐵) = -(𝐴 · 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 (class class class)co 5512 ℂcc 6887 0cc0 6889 · cmul 6894 − cmin 7182 -cneg 7183 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-setind 4262 ax-resscn 6976 ax-1cn 6977 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-distr 6988 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-sub 7184 df-neg 7185 |
This theorem is referenced by: mulneg2 7393 mulneg12 7394 mulm1 7397 mulneg1i 7401 mulneg1d 7408 divnegap 7683 zmulcl 8297 cjreim 9503 |
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