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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 26923 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 11212 | . . . 4 ⊢ i ∈ ℂ | |
2 | mulcl 11237 | . . . 4 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
3 | 1, 2 | mpan 690 | . . 3 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ) |
4 | ax-1cn 11211 | . . . . 5 ⊢ 1 ∈ ℂ | |
5 | sqcl 14155 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ) | |
6 | subcl 11505 | . . . . 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 15473 | . . 3 ⊢ (𝐴 ∈ ℂ → (√‘(1 − (𝐴↑2))) ∈ ℂ) |
9 | 3, 8 | subnegd 11625 | . 2 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) − -(√‘(1 − (𝐴↑2)))) = ((i · 𝐴) + (√‘(1 − (𝐴↑2))))) |
10 | 8 | negcld 11605 | . . 3 ⊢ (𝐴 ∈ ℂ → -(√‘(1 − (𝐴↑2))) ∈ ℂ) |
11 | 0ne1 12335 | . . . . . 6 ⊢ 0 ≠ 1 | |
12 | 0cnd 11252 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℂ) | |
13 | 1cnd 11254 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ) | |
14 | subcan2 11532 | . . . . . . . 8 ⊢ ((0 ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝐴↑2) ∈ ℂ) → ((0 − (𝐴↑2)) = (1 − (𝐴↑2)) ↔ 0 = 1)) | |
15 | 14 | necon3bid 2983 | . . . . . . 7 ⊢ ((0 ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝐴↑2) ∈ ℂ) → ((0 − (𝐴↑2)) ≠ (1 − (𝐴↑2)) ↔ 0 ≠ 1)) |
16 | 12, 13, 5, 15 | syl3anc 1370 | . . . . . 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 14156 | . . . . . . . 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 14238 | . . . . . . . . 9 ⊢ (i↑2) = -1 | |
21 | 20 | oveq1i 7441 | . . . . . . . 8 ⊢ ((i↑2) · (𝐴↑2)) = (-1 · (𝐴↑2)) |
22 | 5 | mulm1d 11713 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (-1 · (𝐴↑2)) = -(𝐴↑2)) |
23 | 21, 22 | eqtrid 2787 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((i↑2) · (𝐴↑2)) = -(𝐴↑2)) |
24 | 19, 23 | eqtrd 2775 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) = -(𝐴↑2)) |
25 | df-neg 11493 | . . . . . 6 ⊢ -(𝐴↑2) = (0 − (𝐴↑2)) | |
26 | 24, 25 | eqtrdi 2791 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) = (0 − (𝐴↑2))) |
27 | sqneg 14153 | . . . . . . 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 15475 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((√‘(1 − (𝐴↑2)))↑2) = (1 − (𝐴↑2))) |
30 | 28, 29 | eqtrd 2775 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (-(√‘(1 − (𝐴↑2)))↑2) = (1 − (𝐴↑2))) |
31 | 17, 26, 30 | 3netr4d 3016 | . . . 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 2971 | . . . 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 11627 | . 2 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) − -(√‘(1 − (𝐴↑2)))) ≠ 0) |
36 | 9, 35 | eqnetrrd 3007 | 1 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) + (√‘(1 − (𝐴↑2)))) ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 ici 11155 + caddc 11156 · cmul 11158 − cmin 11490 -cneg 11491 2c2 12319 ↑cexp 14099 √csqrt 15269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 |
This theorem is referenced by: asinlem3 26929 asinf 26930 asinneg 26944 efiasin 26946 asinbnd 26957 dvasin 37691 |
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