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Mirrors > Home > MPE Home > Th. List > asinlem | Structured version Visualization version GIF version |
Description: The argument to the logarithm in df-asin 26791 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 11192 | . . . 4 ⊢ i ∈ ℂ | |
2 | mulcl 11217 | . . . 4 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
3 | 1, 2 | mpan 689 | . . 3 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ) |
4 | ax-1cn 11191 | . . . . 5 ⊢ 1 ∈ ℂ | |
5 | sqcl 14109 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ) | |
6 | subcl 11484 | . . . . 5 ⊢ ((1 ∈ ℂ ∧ (𝐴↑2) ∈ ℂ) → (1 − (𝐴↑2)) ∈ ℂ) | |
7 | 4, 5, 6 | sylancr 586 | . . . 4 ⊢ (𝐴 ∈ ℂ → (1 − (𝐴↑2)) ∈ ℂ) |
8 | 7 | sqrtcld 15411 | . . 3 ⊢ (𝐴 ∈ ℂ → (√‘(1 − (𝐴↑2))) ∈ ℂ) |
9 | 3, 8 | subnegd 11603 | . 2 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) − -(√‘(1 − (𝐴↑2)))) = ((i · 𝐴) + (√‘(1 − (𝐴↑2))))) |
10 | 8 | negcld 11583 | . . 3 ⊢ (𝐴 ∈ ℂ → -(√‘(1 − (𝐴↑2))) ∈ ℂ) |
11 | 0ne1 12308 | . . . . . 6 ⊢ 0 ≠ 1 | |
12 | 0cnd 11232 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℂ) | |
13 | 1cnd 11234 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ) | |
14 | subcan2 11510 | . . . . . . . 8 ⊢ ((0 ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝐴↑2) ∈ ℂ) → ((0 − (𝐴↑2)) = (1 − (𝐴↑2)) ↔ 0 = 1)) | |
15 | 14 | necon3bid 2981 | . . . . . . 7 ⊢ ((0 ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝐴↑2) ∈ ℂ) → ((0 − (𝐴↑2)) ≠ (1 − (𝐴↑2)) ↔ 0 ≠ 1)) |
16 | 12, 13, 5, 15 | syl3anc 1369 | . . . . . 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 14110 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((i · 𝐴)↑2) = ((i↑2) · (𝐴↑2))) | |
19 | 1, 18 | mpan 689 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) = ((i↑2) · (𝐴↑2))) |
20 | i2 14192 | . . . . . . . . 9 ⊢ (i↑2) = -1 | |
21 | 20 | oveq1i 7425 | . . . . . . . 8 ⊢ ((i↑2) · (𝐴↑2)) = (-1 · (𝐴↑2)) |
22 | 5 | mulm1d 11691 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (-1 · (𝐴↑2)) = -(𝐴↑2)) |
23 | 21, 22 | eqtrid 2780 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((i↑2) · (𝐴↑2)) = -(𝐴↑2)) |
24 | 19, 23 | eqtrd 2768 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) = -(𝐴↑2)) |
25 | df-neg 11472 | . . . . . 6 ⊢ -(𝐴↑2) = (0 − (𝐴↑2)) | |
26 | 24, 25 | eqtrdi 2784 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) = (0 − (𝐴↑2))) |
27 | sqneg 14107 | . . . . . . 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 15413 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((√‘(1 − (𝐴↑2)))↑2) = (1 − (𝐴↑2))) |
30 | 28, 29 | eqtrd 2768 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (-(√‘(1 − (𝐴↑2)))↑2) = (1 − (𝐴↑2))) |
31 | 17, 26, 30 | 3netr4d 3014 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) ≠ (-(√‘(1 − (𝐴↑2)))↑2)) |
32 | oveq1 7422 | . . . . 5 ⊢ ((i · 𝐴) = -(√‘(1 − (𝐴↑2))) → ((i · 𝐴)↑2) = (-(√‘(1 − (𝐴↑2)))↑2)) | |
33 | 32 | necon3i 2969 | . . . 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 11605 | . 2 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) − -(√‘(1 − (𝐴↑2)))) ≠ 0) |
36 | 9, 35 | eqnetrrd 3005 | 1 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) + (√‘(1 − (𝐴↑2)))) ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 ‘cfv 6543 (class class class)co 7415 ℂcc 11131 0cc0 11133 1c1 11134 ici 11135 + caddc 11136 · cmul 11138 − cmin 11469 -cneg 11470 2c2 12292 ↑cexp 14053 √csqrt 15207 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-sup 9460 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-n0 12498 df-z 12584 df-uz 12848 df-rp 13002 df-seq 13994 df-exp 14054 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 |
This theorem is referenced by: asinlem3 26797 asinf 26798 asinneg 26812 efiasin 26814 asinbnd 26825 dvasin 37172 |
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