Mathbox for David A. Wheeler |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > recsccl | Structured version Visualization version GIF version |
Description: The closure of the cosecant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.) |
Ref | Expression |
---|---|
recsccl | ⊢ ((𝐴 ∈ ℝ ∧ (sin‘𝐴) ≠ 0) → (csc‘𝐴) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recn 10945 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
2 | cscval 46402 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (csc‘𝐴) = (1 / (sin‘𝐴))) | |
3 | 1, 2 | sylan 579 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ (sin‘𝐴) ≠ 0) → (csc‘𝐴) = (1 / (sin‘𝐴))) |
4 | resincl 15830 | . . . 4 ⊢ (𝐴 ∈ ℝ → (sin‘𝐴) ∈ ℝ) | |
5 | 1red 10960 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 1 ∈ ℝ) | |
6 | redivcl 11677 | . . . . 5 ⊢ ((1 ∈ ℝ ∧ (sin‘𝐴) ∈ ℝ ∧ (sin‘𝐴) ≠ 0) → (1 / (sin‘𝐴)) ∈ ℝ) | |
7 | 5, 6 | syl3an1 1161 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (sin‘𝐴) ∈ ℝ ∧ (sin‘𝐴) ≠ 0) → (1 / (sin‘𝐴)) ∈ ℝ) |
8 | 4, 7 | syl3an2 1162 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (sin‘𝐴) ≠ 0) → (1 / (sin‘𝐴)) ∈ ℝ) |
9 | 8 | 3anidm12 1417 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ (sin‘𝐴) ≠ 0) → (1 / (sin‘𝐴)) ∈ ℝ) |
10 | 3, 9 | eqeltrd 2840 | 1 ⊢ ((𝐴 ∈ ℝ ∧ (sin‘𝐴) ≠ 0) → (csc‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2109 ≠ wne 2944 ‘cfv 6430 (class class class)co 7268 ℂcc 10853 ℝcr 10854 0cc0 10855 1c1 10856 / cdiv 11615 sincsin 15754 cscccsc 46396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-inf2 9360 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-se 5544 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-isom 6439 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-pm 8592 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-sup 9162 df-inf 9163 df-oi 9230 df-card 9681 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-n0 12217 df-z 12303 df-uz 12565 df-rp 12713 df-ico 13067 df-fz 13222 df-fzo 13365 df-fl 13493 df-seq 13703 df-exp 13764 df-fac 13969 df-hash 14026 df-shft 14759 df-cj 14791 df-re 14792 df-im 14793 df-sqrt 14927 df-abs 14928 df-limsup 15161 df-clim 15178 df-rlim 15179 df-sum 15379 df-ef 15758 df-sin 15760 df-csc 46399 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |