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Mirrors > Home > MPE Home > Th. List > resinhcl | Structured version Visualization version GIF version |
Description: The hyperbolic sine of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.) |
Ref | Expression |
---|---|
resinhcl | ⊢ (𝐴 ∈ ℝ → ((sin‘(i · 𝐴)) / i) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recn 10343 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
2 | sinhval 15257 | . . 3 ⊢ (𝐴 ∈ ℂ → ((sin‘(i · 𝐴)) / i) = (((exp‘𝐴) − (exp‘-𝐴)) / 2)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ ℝ → ((sin‘(i · 𝐴)) / i) = (((exp‘𝐴) − (exp‘-𝐴)) / 2)) |
4 | reefcl 15190 | . . . 4 ⊢ (𝐴 ∈ ℝ → (exp‘𝐴) ∈ ℝ) | |
5 | renegcl 10666 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
6 | 5 | reefcld 15191 | . . . 4 ⊢ (𝐴 ∈ ℝ → (exp‘-𝐴) ∈ ℝ) |
7 | 4, 6 | resubcld 10783 | . . 3 ⊢ (𝐴 ∈ ℝ → ((exp‘𝐴) − (exp‘-𝐴)) ∈ ℝ) |
8 | 7 | rehalfcld 11606 | . 2 ⊢ (𝐴 ∈ ℝ → (((exp‘𝐴) − (exp‘-𝐴)) / 2) ∈ ℝ) |
9 | 3, 8 | eqeltrd 2907 | 1 ⊢ (𝐴 ∈ ℝ → ((sin‘(i · 𝐴)) / i) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ‘cfv 6124 (class class class)co 6906 ℂcc 10251 ℝcr 10252 ici 10255 · cmul 10258 − cmin 10586 -cneg 10587 / cdiv 11010 2c2 11407 expce 15165 sincsin 15167 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-inf2 8816 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-pre-sup 10331 ax-addf 10332 ax-mulf 10333 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-se 5303 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-isom 6133 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-er 8010 df-pm 8126 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-sup 8618 df-inf 8619 df-oi 8685 df-card 9079 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-div 11011 df-nn 11352 df-2 11415 df-3 11416 df-n0 11620 df-z 11706 df-uz 11970 df-rp 12114 df-ico 12470 df-fz 12621 df-fzo 12762 df-fl 12889 df-seq 13097 df-exp 13156 df-fac 13355 df-hash 13412 df-shft 14185 df-cj 14217 df-re 14218 df-im 14219 df-sqrt 14353 df-abs 14354 df-limsup 14580 df-clim 14597 df-rlim 14598 df-sum 14795 df-ef 15171 df-sin 15173 |
This theorem is referenced by: retanhcl 15262 |
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