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Mirrors > Home > MPE Home > Th. List > Mathboxes > sinh-conventional | Structured version Visualization version GIF version |
Description: Conventional definition of sinh. Here we show that the sinh definition we're using has the same meaning as the conventional definition used in some other sources. We choose a slightly different definition of sinh because it has fewer operations, and thus is more convenient to manipulate using set.mm. (Contributed by David A. Wheeler, 10-May-2015.) |
Ref | Expression |
---|---|
sinh-conventional | β’ (π΄ β β β (sinhβπ΄) = (-i Β· (sinβ(i Β· π΄)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sinhval-named 47734 | . 2 β’ (π΄ β β β (sinhβπ΄) = ((sinβ(i Β· π΄)) / i)) | |
2 | ax-icn 11165 | . . . . 5 β’ i β β | |
3 | mulcl 11190 | . . . . 5 β’ ((i β β β§ π΄ β β) β (i Β· π΄) β β) | |
4 | 2, 3 | mpan 688 | . . . 4 β’ (π΄ β β β (i Β· π΄) β β) |
5 | 4 | sincld 16069 | . . 3 β’ (π΄ β β β (sinβ(i Β· π΄)) β β) |
6 | ine0 11645 | . . . 4 β’ i β 0 | |
7 | divrec2 11885 | . . . 4 β’ (((sinβ(i Β· π΄)) β β β§ i β β β§ i β 0) β ((sinβ(i Β· π΄)) / i) = ((1 / i) Β· (sinβ(i Β· π΄)))) | |
8 | 2, 6, 7 | mp3an23 1453 | . . 3 β’ ((sinβ(i Β· π΄)) β β β ((sinβ(i Β· π΄)) / i) = ((1 / i) Β· (sinβ(i Β· π΄)))) |
9 | 5, 8 | syl 17 | . 2 β’ (π΄ β β β ((sinβ(i Β· π΄)) / i) = ((1 / i) Β· (sinβ(i Β· π΄)))) |
10 | irec 14161 | . . . 4 β’ (1 / i) = -i | |
11 | 10 | oveq1i 7415 | . . 3 β’ ((1 / i) Β· (sinβ(i Β· π΄))) = (-i Β· (sinβ(i Β· π΄))) |
12 | 11 | a1i 11 | . 2 β’ (π΄ β β β ((1 / i) Β· (sinβ(i Β· π΄))) = (-i Β· (sinβ(i Β· π΄)))) |
13 | 1, 9, 12 | 3eqtrd 2776 | 1 β’ (π΄ β β β (sinhβπ΄) = (-i Β· (sinβ(i Β· π΄)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 β wne 2940 βcfv 6540 (class class class)co 7405 βcc 11104 0cc0 11106 1c1 11107 ici 11108 Β· cmul 11111 -cneg 11441 / cdiv 11867 sincsin 16003 sinhcsinh 47728 |
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-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-ico 13326 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-fac 14230 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-sinh 47731 |
This theorem is referenced by: (None) |
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