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Mirrors > Home > MPE Home > Th. List > sinmul | Structured version Visualization version GIF version |
Description: Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 16113 and cossub 16117. (Contributed by David A. Wheeler, 26-May-2015.) |
Ref | Expression |
---|---|
sinmul | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((sin‘𝐴) · (sin‘𝐵)) = (((cos‘(𝐴 − 𝐵)) − (cos‘(𝐴 + 𝐵))) / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cossub 16117 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (cos‘(𝐴 − 𝐵)) = (((cos‘𝐴) · (cos‘𝐵)) + ((sin‘𝐴) · (sin‘𝐵)))) | |
2 | cosadd 16113 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (cos‘(𝐴 + 𝐵)) = (((cos‘𝐴) · (cos‘𝐵)) − ((sin‘𝐴) · (sin‘𝐵)))) | |
3 | 1, 2 | oveq12d 7422 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((cos‘(𝐴 − 𝐵)) − (cos‘(𝐴 + 𝐵))) = ((((cos‘𝐴) · (cos‘𝐵)) + ((sin‘𝐴) · (sin‘𝐵))) − (((cos‘𝐴) · (cos‘𝐵)) − ((sin‘𝐴) · (sin‘𝐵))))) |
4 | coscl 16075 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) ∈ ℂ) | |
5 | coscl 16075 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → (cos‘𝐵) ∈ ℂ) | |
6 | mulcl 11193 | . . . . . 6 ⊢ (((cos‘𝐴) ∈ ℂ ∧ (cos‘𝐵) ∈ ℂ) → ((cos‘𝐴) · (cos‘𝐵)) ∈ ℂ) | |
7 | 4, 5, 6 | syl2an 595 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((cos‘𝐴) · (cos‘𝐵)) ∈ ℂ) |
8 | sincl 16074 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) ∈ ℂ) | |
9 | sincl 16074 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → (sin‘𝐵) ∈ ℂ) | |
10 | mulcl 11193 | . . . . . 6 ⊢ (((sin‘𝐴) ∈ ℂ ∧ (sin‘𝐵) ∈ ℂ) → ((sin‘𝐴) · (sin‘𝐵)) ∈ ℂ) | |
11 | 8, 9, 10 | syl2an 595 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((sin‘𝐴) · (sin‘𝐵)) ∈ ℂ) |
12 | pnncan 11502 | . . . . . . 7 ⊢ ((((cos‘𝐴) · (cos‘𝐵)) ∈ ℂ ∧ ((sin‘𝐴) · (sin‘𝐵)) ∈ ℂ ∧ ((sin‘𝐴) · (sin‘𝐵)) ∈ ℂ) → ((((cos‘𝐴) · (cos‘𝐵)) + ((sin‘𝐴) · (sin‘𝐵))) − (((cos‘𝐴) · (cos‘𝐵)) − ((sin‘𝐴) · (sin‘𝐵)))) = (((sin‘𝐴) · (sin‘𝐵)) + ((sin‘𝐴) · (sin‘𝐵)))) | |
13 | 12 | 3anidm23 1418 | . . . . . 6 ⊢ ((((cos‘𝐴) · (cos‘𝐵)) ∈ ℂ ∧ ((sin‘𝐴) · (sin‘𝐵)) ∈ ℂ) → ((((cos‘𝐴) · (cos‘𝐵)) + ((sin‘𝐴) · (sin‘𝐵))) − (((cos‘𝐴) · (cos‘𝐵)) − ((sin‘𝐴) · (sin‘𝐵)))) = (((sin‘𝐴) · (sin‘𝐵)) + ((sin‘𝐴) · (sin‘𝐵)))) |
14 | 2times 12349 | . . . . . . 7 ⊢ (((sin‘𝐴) · (sin‘𝐵)) ∈ ℂ → (2 · ((sin‘𝐴) · (sin‘𝐵))) = (((sin‘𝐴) · (sin‘𝐵)) + ((sin‘𝐴) · (sin‘𝐵)))) | |
15 | 14 | adantl 481 | . . . . . 6 ⊢ ((((cos‘𝐴) · (cos‘𝐵)) ∈ ℂ ∧ ((sin‘𝐴) · (sin‘𝐵)) ∈ ℂ) → (2 · ((sin‘𝐴) · (sin‘𝐵))) = (((sin‘𝐴) · (sin‘𝐵)) + ((sin‘𝐴) · (sin‘𝐵)))) |
16 | 13, 15 | eqtr4d 2769 | . . . . 5 ⊢ ((((cos‘𝐴) · (cos‘𝐵)) ∈ ℂ ∧ ((sin‘𝐴) · (sin‘𝐵)) ∈ ℂ) → ((((cos‘𝐴) · (cos‘𝐵)) + ((sin‘𝐴) · (sin‘𝐵))) − (((cos‘𝐴) · (cos‘𝐵)) − ((sin‘𝐴) · (sin‘𝐵)))) = (2 · ((sin‘𝐴) · (sin‘𝐵)))) |
17 | 7, 11, 16 | syl2anc 583 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((((cos‘𝐴) · (cos‘𝐵)) + ((sin‘𝐴) · (sin‘𝐵))) − (((cos‘𝐴) · (cos‘𝐵)) − ((sin‘𝐴) · (sin‘𝐵)))) = (2 · ((sin‘𝐴) · (sin‘𝐵)))) |
18 | 2cn 12288 | . . . . 5 ⊢ 2 ∈ ℂ | |
19 | mulcom 11195 | . . . . 5 ⊢ ((2 ∈ ℂ ∧ ((sin‘𝐴) · (sin‘𝐵)) ∈ ℂ) → (2 · ((sin‘𝐴) · (sin‘𝐵))) = (((sin‘𝐴) · (sin‘𝐵)) · 2)) | |
20 | 18, 11, 19 | sylancr 586 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (2 · ((sin‘𝐴) · (sin‘𝐵))) = (((sin‘𝐴) · (sin‘𝐵)) · 2)) |
21 | 3, 17, 20 | 3eqtrd 2770 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((cos‘(𝐴 − 𝐵)) − (cos‘(𝐴 + 𝐵))) = (((sin‘𝐴) · (sin‘𝐵)) · 2)) |
22 | 21 | oveq1d 7419 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((cos‘(𝐴 − 𝐵)) − (cos‘(𝐴 + 𝐵))) / 2) = ((((sin‘𝐴) · (sin‘𝐵)) · 2) / 2)) |
23 | 2ne0 12317 | . . . 4 ⊢ 2 ≠ 0 | |
24 | divcan4 11900 | . . . 4 ⊢ ((((sin‘𝐴) · (sin‘𝐵)) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((((sin‘𝐴) · (sin‘𝐵)) · 2) / 2) = ((sin‘𝐴) · (sin‘𝐵))) | |
25 | 18, 23, 24 | mp3an23 1449 | . . 3 ⊢ (((sin‘𝐴) · (sin‘𝐵)) ∈ ℂ → ((((sin‘𝐴) · (sin‘𝐵)) · 2) / 2) = ((sin‘𝐴) · (sin‘𝐵))) |
26 | 11, 25 | syl 17 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((((sin‘𝐴) · (sin‘𝐵)) · 2) / 2) = ((sin‘𝐴) · (sin‘𝐵))) |
27 | 22, 26 | eqtr2d 2767 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((sin‘𝐴) · (sin‘𝐵)) = (((cos‘(𝐴 − 𝐵)) − (cos‘(𝐴 + 𝐵))) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ‘cfv 6536 (class class class)co 7404 ℂcc 11107 0cc0 11109 + caddc 11112 · cmul 11114 − cmin 11445 / cdiv 11872 2c2 12268 sincsin 16011 cosccos 16012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-rp 12978 df-ico 13333 df-fz 13488 df-fzo 13631 df-fl 13760 df-seq 13970 df-exp 14031 df-fac 14237 df-bc 14266 df-hash 14294 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-ef 16015 df-sin 16017 df-cos 16018 |
This theorem is referenced by: ptolemy 26382 |
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