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Mirrors > Home > MPE Home > Th. List > angneg | Structured version Visualization version GIF version |
Description: Cancel a negative sign in the angle function. (Contributed by Mario Carneiro, 23-Sep-2014.) |
Ref | Expression |
---|---|
ang.1 | ⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥)))) |
Ref | Expression |
---|---|
angneg | ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (-𝐴𝐹-𝐵) = (𝐴𝐹𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulm1 11639 | . . . 4 ⊢ (𝐴 ∈ ℂ → (-1 · 𝐴) = -𝐴) | |
2 | 1 | ad2antrr 724 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (-1 · 𝐴) = -𝐴) |
3 | mulm1 11639 | . . . 4 ⊢ (𝐵 ∈ ℂ → (-1 · 𝐵) = -𝐵) | |
4 | 3 | ad2antrl 726 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (-1 · 𝐵) = -𝐵) |
5 | 2, 4 | oveq12d 7412 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((-1 · 𝐴)𝐹(-1 · 𝐵)) = (-𝐴𝐹-𝐵)) |
6 | neg1cn 12310 | . . . 4 ⊢ -1 ∈ ℂ | |
7 | neg1ne0 12312 | . . . 4 ⊢ -1 ≠ 0 | |
8 | 6, 7 | pm3.2i 471 | . . 3 ⊢ (-1 ∈ ℂ ∧ -1 ≠ 0) |
9 | ang.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥)))) | |
10 | 9 | angcan 26236 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ (-1 ∈ ℂ ∧ -1 ≠ 0)) → ((-1 · 𝐴)𝐹(-1 · 𝐵)) = (𝐴𝐹𝐵)) |
11 | 8, 10 | mp3an3 1450 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((-1 · 𝐴)𝐹(-1 · 𝐵)) = (𝐴𝐹𝐵)) |
12 | 5, 11 | eqtr3d 2774 | 1 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (-𝐴𝐹-𝐵) = (𝐴𝐹𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∖ cdif 3942 {csn 4623 ‘cfv 6533 (class class class)co 7394 ∈ cmpo 7396 ℂcc 11092 0cc0 11094 1c1 11095 · cmul 11099 -cneg 11429 / cdiv 11855 ℑcim 15029 logclog 25994 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-resscn 11151 ax-1cn 11152 ax-icn 11153 ax-addcl 11154 ax-addrcl 11155 ax-mulcl 11156 ax-mulrcl 11157 ax-mulcom 11158 ax-addass 11159 ax-mulass 11160 ax-distr 11161 ax-i2m1 11162 ax-1ne0 11163 ax-1rid 11164 ax-rnegex 11165 ax-rrecex 11166 ax-cnre 11167 ax-pre-lttri 11168 ax-pre-lttrn 11169 ax-pre-ltadd 11170 ax-pre-mulgt0 11171 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5568 df-po 5582 df-so 5583 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-er 8688 df-en 8925 df-dom 8926 df-sdom 8927 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 df-sub 11430 df-neg 11431 df-div 11856 |
This theorem is referenced by: ang180 26248 isosctrlem3 26254 |
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