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| Description: The argument to the logarithm in df-asin 26909 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 11215 | . . . 4 ⊢ i ∈ ℂ | |
| 2 | mulcl 11240 | . . . 4 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
| 3 | 1, 2 | mpan 690 | . . 3 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ) | 
| 4 | ax-1cn 11214 | . . . . 5 ⊢ 1 ∈ ℂ | |
| 5 | sqcl 14159 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ) | |
| 6 | subcl 11508 | . . . . 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 15477 | . . 3 ⊢ (𝐴 ∈ ℂ → (√‘(1 − (𝐴↑2))) ∈ ℂ) | 
| 9 | 3, 8 | subnegd 11628 | . 2 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) − -(√‘(1 − (𝐴↑2)))) = ((i · 𝐴) + (√‘(1 − (𝐴↑2))))) | 
| 10 | 8 | negcld 11608 | . . 3 ⊢ (𝐴 ∈ ℂ → -(√‘(1 − (𝐴↑2))) ∈ ℂ) | 
| 11 | 0ne1 12338 | . . . . . 6 ⊢ 0 ≠ 1 | |
| 12 | 0cnd 11255 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℂ) | |
| 13 | 1cnd 11257 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ) | |
| 14 | subcan2 11535 | . . . . . . . 8 ⊢ ((0 ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝐴↑2) ∈ ℂ) → ((0 − (𝐴↑2)) = (1 − (𝐴↑2)) ↔ 0 = 1)) | |
| 15 | 14 | necon3bid 2984 | . . . . . . 7 ⊢ ((0 ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝐴↑2) ∈ ℂ) → ((0 − (𝐴↑2)) ≠ (1 − (𝐴↑2)) ↔ 0 ≠ 1)) | 
| 16 | 12, 13, 5, 15 | syl3anc 1372 | . . . . . 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 14160 | . . . . . . . 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 14242 | . . . . . . . . 9 ⊢ (i↑2) = -1 | |
| 21 | 20 | oveq1i 7442 | . . . . . . . 8 ⊢ ((i↑2) · (𝐴↑2)) = (-1 · (𝐴↑2)) | 
| 22 | 5 | mulm1d 11716 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (-1 · (𝐴↑2)) = -(𝐴↑2)) | 
| 23 | 21, 22 | eqtrid 2788 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((i↑2) · (𝐴↑2)) = -(𝐴↑2)) | 
| 24 | 19, 23 | eqtrd 2776 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) = -(𝐴↑2)) | 
| 25 | df-neg 11496 | . . . . . 6 ⊢ -(𝐴↑2) = (0 − (𝐴↑2)) | |
| 26 | 24, 25 | eqtrdi 2792 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) = (0 − (𝐴↑2))) | 
| 27 | sqneg 14157 | . . . . . . 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 15479 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((√‘(1 − (𝐴↑2)))↑2) = (1 − (𝐴↑2))) | 
| 30 | 28, 29 | eqtrd 2776 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (-(√‘(1 − (𝐴↑2)))↑2) = (1 − (𝐴↑2))) | 
| 31 | 17, 26, 30 | 3netr4d 3017 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) ≠ (-(√‘(1 − (𝐴↑2)))↑2)) | 
| 32 | oveq1 7439 | . . . . 5 ⊢ ((i · 𝐴) = -(√‘(1 − (𝐴↑2))) → ((i · 𝐴)↑2) = (-(√‘(1 − (𝐴↑2)))↑2)) | |
| 33 | 32 | necon3i 2972 | . . . 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 11630 | . 2 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) − -(√‘(1 − (𝐴↑2)))) ≠ 0) | 
| 36 | 9, 35 | eqnetrrd 3008 | 1 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) + (√‘(1 − (𝐴↑2)))) ≠ 0) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ‘cfv 6560 (class class class)co 7432 ℂcc 11154 0cc0 11156 1c1 11157 ici 11158 + caddc 11159 · cmul 11161 − cmin 11493 -cneg 11494 2c2 12322 ↑cexp 14103 √csqrt 15273 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-sup 9483 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-n0 12529 df-z 12616 df-uz 12880 df-rp 13036 df-seq 14044 df-exp 14104 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 | 
| This theorem is referenced by: asinlem3 26915 asinf 26916 asinneg 26930 efiasin 26932 asinbnd 26943 dvasin 37712 | 
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