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Mirrors > Home > ILE Home > Th. List > sincosq1eq | GIF version |
Description: Complementarity of the sine and cosine functions in the first quadrant. (Contributed by Paul Chapman, 25-Jan-2008.) |
Ref | Expression |
---|---|
sincosq1eq | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐴 + 𝐵) = 1) → (sin‘(𝐴 · (π / 2))) = (cos‘(𝐵 · (π / 2)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | halfpire 13311 | . . . . . 6 ⊢ (π / 2) ∈ ℝ | |
2 | 1 | recni 7905 | . . . . 5 ⊢ (π / 2) ∈ ℂ |
3 | mulcl 7874 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (π / 2) ∈ ℂ) → (𝐴 · (π / 2)) ∈ ℂ) | |
4 | 2, 3 | mpan2 422 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 · (π / 2)) ∈ ℂ) |
5 | coshalfpim 13342 | . . . 4 ⊢ ((𝐴 · (π / 2)) ∈ ℂ → (cos‘((π / 2) − (𝐴 · (π / 2)))) = (sin‘(𝐴 · (π / 2)))) | |
6 | 4, 5 | syl 14 | . . 3 ⊢ (𝐴 ∈ ℂ → (cos‘((π / 2) − (𝐴 · (π / 2)))) = (sin‘(𝐴 · (π / 2)))) |
7 | 6 | 3ad2ant1 1007 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐴 + 𝐵) = 1) → (cos‘((π / 2) − (𝐴 · (π / 2)))) = (sin‘(𝐴 · (π / 2)))) |
8 | adddir 7884 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (π / 2) ∈ ℂ) → ((𝐴 + 𝐵) · (π / 2)) = ((𝐴 · (π / 2)) + (𝐵 · (π / 2)))) | |
9 | 2, 8 | mp3an3 1315 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) · (π / 2)) = ((𝐴 · (π / 2)) + (𝐵 · (π / 2)))) |
10 | 9 | 3adant3 1006 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐴 + 𝐵) = 1) → ((𝐴 + 𝐵) · (π / 2)) = ((𝐴 · (π / 2)) + (𝐵 · (π / 2)))) |
11 | oveq1 5846 | . . . . . . 7 ⊢ ((𝐴 + 𝐵) = 1 → ((𝐴 + 𝐵) · (π / 2)) = (1 · (π / 2))) | |
12 | 2 | mulid2i 7896 | . . . . . . 7 ⊢ (1 · (π / 2)) = (π / 2) |
13 | 11, 12 | eqtrdi 2213 | . . . . . 6 ⊢ ((𝐴 + 𝐵) = 1 → ((𝐴 + 𝐵) · (π / 2)) = (π / 2)) |
14 | 13 | 3ad2ant3 1009 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐴 + 𝐵) = 1) → ((𝐴 + 𝐵) · (π / 2)) = (π / 2)) |
15 | 10, 14 | eqtr3d 2199 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐴 + 𝐵) = 1) → ((𝐴 · (π / 2)) + (𝐵 · (π / 2))) = (π / 2)) |
16 | mulcl 7874 | . . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ (π / 2) ∈ ℂ) → (𝐵 · (π / 2)) ∈ ℂ) | |
17 | 2, 16 | mpan2 422 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → (𝐵 · (π / 2)) ∈ ℂ) |
18 | subadd 8095 | . . . . . 6 ⊢ (((π / 2) ∈ ℂ ∧ (𝐴 · (π / 2)) ∈ ℂ ∧ (𝐵 · (π / 2)) ∈ ℂ) → (((π / 2) − (𝐴 · (π / 2))) = (𝐵 · (π / 2)) ↔ ((𝐴 · (π / 2)) + (𝐵 · (π / 2))) = (π / 2))) | |
19 | 2, 4, 17, 18 | mp3an3an 1332 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((π / 2) − (𝐴 · (π / 2))) = (𝐵 · (π / 2)) ↔ ((𝐴 · (π / 2)) + (𝐵 · (π / 2))) = (π / 2))) |
20 | 19 | 3adant3 1006 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐴 + 𝐵) = 1) → (((π / 2) − (𝐴 · (π / 2))) = (𝐵 · (π / 2)) ↔ ((𝐴 · (π / 2)) + (𝐵 · (π / 2))) = (π / 2))) |
21 | 15, 20 | mpbird 166 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐴 + 𝐵) = 1) → ((π / 2) − (𝐴 · (π / 2))) = (𝐵 · (π / 2))) |
22 | 21 | fveq2d 5487 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐴 + 𝐵) = 1) → (cos‘((π / 2) − (𝐴 · (π / 2)))) = (cos‘(𝐵 · (π / 2)))) |
23 | 7, 22 | eqtr3d 2199 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐴 + 𝐵) = 1) → (sin‘(𝐴 · (π / 2))) = (cos‘(𝐵 · (π / 2)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∧ w3a 967 = wceq 1342 ∈ wcel 2135 ‘cfv 5185 (class class class)co 5839 ℂcc 7745 1c1 7748 + caddc 7750 · cmul 7752 − cmin 8063 / cdiv 8562 2c2 8902 sincsin 11579 cosccos 11580 πcpi 11582 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4094 ax-sep 4097 ax-nul 4105 ax-pow 4150 ax-pr 4184 ax-un 4408 ax-setind 4511 ax-iinf 4562 ax-cnex 7838 ax-resscn 7839 ax-1cn 7840 ax-1re 7841 ax-icn 7842 ax-addcl 7843 ax-addrcl 7844 ax-mulcl 7845 ax-mulrcl 7846 ax-addcom 7847 ax-mulcom 7848 ax-addass 7849 ax-mulass 7850 ax-distr 7851 ax-i2m1 7852 ax-0lt1 7853 ax-1rid 7854 ax-0id 7855 ax-rnegex 7856 ax-precex 7857 ax-cnre 7858 ax-pre-ltirr 7859 ax-pre-ltwlin 7860 ax-pre-lttrn 7861 ax-pre-apti 7862 ax-pre-ltadd 7863 ax-pre-mulgt0 7864 ax-pre-mulext 7865 ax-arch 7866 ax-caucvg 7867 ax-pre-suploc 7868 ax-addf 7869 ax-mulf 7870 |
This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2726 df-sbc 2950 df-csb 3044 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-nul 3408 df-if 3519 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-int 3822 df-iun 3865 df-disj 3957 df-br 3980 df-opab 4041 df-mpt 4042 df-tr 4078 df-id 4268 df-po 4271 df-iso 4272 df-iord 4341 df-on 4343 df-ilim 4344 df-suc 4346 df-iom 4565 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-dm 4611 df-rn 4612 df-res 4613 df-ima 4614 df-iota 5150 df-fun 5187 df-fn 5188 df-f 5189 df-f1 5190 df-fo 5191 df-f1o 5192 df-fv 5193 df-isom 5194 df-riota 5795 df-ov 5842 df-oprab 5843 df-mpo 5844 df-of 6047 df-1st 6103 df-2nd 6104 df-recs 6267 df-irdg 6332 df-frec 6353 df-1o 6378 df-oadd 6382 df-er 6495 df-map 6610 df-pm 6611 df-en 6701 df-dom 6702 df-fin 6703 df-sup 6943 df-inf 6944 df-pnf 7929 df-mnf 7930 df-xr 7931 df-ltxr 7932 df-le 7933 df-sub 8065 df-neg 8066 df-reap 8467 df-ap 8474 df-div 8563 df-inn 8852 df-2 8910 df-3 8911 df-4 8912 df-5 8913 df-6 8914 df-7 8915 df-8 8916 df-9 8917 df-n0 9109 df-z 9186 df-uz 9461 df-q 9552 df-rp 9584 df-xneg 9702 df-xadd 9703 df-ioo 9822 df-ioc 9823 df-ico 9824 df-icc 9825 df-fz 9939 df-fzo 10072 df-seqfrec 10375 df-exp 10449 df-fac 10633 df-bc 10655 df-ihash 10683 df-shft 10751 df-cj 10778 df-re 10779 df-im 10780 df-rsqrt 10934 df-abs 10935 df-clim 11214 df-sumdc 11289 df-ef 11583 df-sin 11585 df-cos 11586 df-pi 11588 df-rest 12551 df-topgen 12570 df-psmet 12585 df-xmet 12586 df-met 12587 df-bl 12588 df-mopn 12589 df-top 12594 df-topon 12607 df-bases 12639 df-ntr 12694 df-cn 12786 df-cnp 12787 df-tx 12851 df-cncf 13156 df-limced 13223 df-dvap 13224 |
This theorem is referenced by: sincos4thpi 13359 sincos6thpi 13361 |
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