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| Description: The argument to the logarithm in df-asin 26908 is always nonzero. (Contributed by Mario Carneiro, 31-Mar-2015.) | 
| Ref | Expression | 
|---|---|
| asinlem | ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) + (√‘(1 − (𝐴↑2)))) ≠ 0) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ax-icn 11214 | . . . 4 ⊢ i ∈ ℂ | |
| 2 | mulcl 11239 | . . . 4 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
| 3 | 1, 2 | mpan 690 | . . 3 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ) | 
| 4 | ax-1cn 11213 | . . . . 5 ⊢ 1 ∈ ℂ | |
| 5 | sqcl 14158 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ) | |
| 6 | subcl 11507 | . . . . 5 ⊢ ((1 ∈ ℂ ∧ (𝐴↑2) ∈ ℂ) → (1 − (𝐴↑2)) ∈ ℂ) | |
| 7 | 4, 5, 6 | sylancr 587 | . . . 4 ⊢ (𝐴 ∈ ℂ → (1 − (𝐴↑2)) ∈ ℂ) | 
| 8 | 7 | sqrtcld 15476 | . . 3 ⊢ (𝐴 ∈ ℂ → (√‘(1 − (𝐴↑2))) ∈ ℂ) | 
| 9 | 3, 8 | subnegd 11627 | . 2 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) − -(√‘(1 − (𝐴↑2)))) = ((i · 𝐴) + (√‘(1 − (𝐴↑2))))) | 
| 10 | 8 | negcld 11607 | . . 3 ⊢ (𝐴 ∈ ℂ → -(√‘(1 − (𝐴↑2))) ∈ ℂ) | 
| 11 | 0ne1 12337 | . . . . . 6 ⊢ 0 ≠ 1 | |
| 12 | 0cnd 11254 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℂ) | |
| 13 | 1cnd 11256 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ) | |
| 14 | subcan2 11534 | . . . . . . . 8 ⊢ ((0 ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝐴↑2) ∈ ℂ) → ((0 − (𝐴↑2)) = (1 − (𝐴↑2)) ↔ 0 = 1)) | |
| 15 | 14 | necon3bid 2985 | . . . . . . 7 ⊢ ((0 ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝐴↑2) ∈ ℂ) → ((0 − (𝐴↑2)) ≠ (1 − (𝐴↑2)) ↔ 0 ≠ 1)) | 
| 16 | 12, 13, 5, 15 | syl3anc 1373 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((0 − (𝐴↑2)) ≠ (1 − (𝐴↑2)) ↔ 0 ≠ 1)) | 
| 17 | 11, 16 | mpbiri 258 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (0 − (𝐴↑2)) ≠ (1 − (𝐴↑2))) | 
| 18 | sqmul 14159 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((i · 𝐴)↑2) = ((i↑2) · (𝐴↑2))) | |
| 19 | 1, 18 | mpan 690 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) = ((i↑2) · (𝐴↑2))) | 
| 20 | i2 14241 | . . . . . . . . 9 ⊢ (i↑2) = -1 | |
| 21 | 20 | oveq1i 7441 | . . . . . . . 8 ⊢ ((i↑2) · (𝐴↑2)) = (-1 · (𝐴↑2)) | 
| 22 | 5 | mulm1d 11715 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (-1 · (𝐴↑2)) = -(𝐴↑2)) | 
| 23 | 21, 22 | eqtrid 2789 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((i↑2) · (𝐴↑2)) = -(𝐴↑2)) | 
| 24 | 19, 23 | eqtrd 2777 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) = -(𝐴↑2)) | 
| 25 | df-neg 11495 | . . . . . 6 ⊢ -(𝐴↑2) = (0 − (𝐴↑2)) | |
| 26 | 24, 25 | eqtrdi 2793 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) = (0 − (𝐴↑2))) | 
| 27 | sqneg 14156 | . . . . . . 7 ⊢ ((√‘(1 − (𝐴↑2))) ∈ ℂ → (-(√‘(1 − (𝐴↑2)))↑2) = ((√‘(1 − (𝐴↑2)))↑2)) | |
| 28 | 8, 27 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (-(√‘(1 − (𝐴↑2)))↑2) = ((√‘(1 − (𝐴↑2)))↑2)) | 
| 29 | 7 | sqsqrtd 15478 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((√‘(1 − (𝐴↑2)))↑2) = (1 − (𝐴↑2))) | 
| 30 | 28, 29 | eqtrd 2777 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (-(√‘(1 − (𝐴↑2)))↑2) = (1 − (𝐴↑2))) | 
| 31 | 17, 26, 30 | 3netr4d 3018 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) ≠ (-(√‘(1 − (𝐴↑2)))↑2)) | 
| 32 | oveq1 7438 | . . . . 5 ⊢ ((i · 𝐴) = -(√‘(1 − (𝐴↑2))) → ((i · 𝐴)↑2) = (-(√‘(1 − (𝐴↑2)))↑2)) | |
| 33 | 32 | necon3i 2973 | . . . 4 ⊢ (((i · 𝐴)↑2) ≠ (-(√‘(1 − (𝐴↑2)))↑2) → (i · 𝐴) ≠ -(√‘(1 − (𝐴↑2)))) | 
| 34 | 31, 33 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ≠ -(√‘(1 − (𝐴↑2)))) | 
| 35 | 3, 10, 34 | subne0d 11629 | . 2 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) − -(√‘(1 − (𝐴↑2)))) ≠ 0) | 
| 36 | 9, 35 | eqnetrrd 3009 | 1 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) + (√‘(1 − (𝐴↑2)))) ≠ 0) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 ici 11157 + caddc 11158 · cmul 11160 − cmin 11492 -cneg 11493 2c2 12321 ↑cexp 14102 √csqrt 15272 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 | 
| This theorem is referenced by: asinlem3 26914 asinf 26915 asinneg 26929 efiasin 26931 asinbnd 26942 dvasin 37711 | 
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