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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 24392 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 9874 | . . . 4 ⊢ i ∈ ℂ | |
2 | mulcl 9899 | . . . 4 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
3 | 1, 2 | mpan 702 | . . 3 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ) |
4 | ax-1cn 9873 | . . . . 5 ⊢ 1 ∈ ℂ | |
5 | sqcl 12787 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ) | |
6 | subcl 10159 | . . . . 5 ⊢ ((1 ∈ ℂ ∧ (𝐴↑2) ∈ ℂ) → (1 − (𝐴↑2)) ∈ ℂ) | |
7 | 4, 5, 6 | sylancr 694 | . . . 4 ⊢ (𝐴 ∈ ℂ → (1 − (𝐴↑2)) ∈ ℂ) |
8 | 7 | sqrtcld 14024 | . . 3 ⊢ (𝐴 ∈ ℂ → (√‘(1 − (𝐴↑2))) ∈ ℂ) |
9 | 3, 8 | subnegd 10278 | . 2 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) − -(√‘(1 − (𝐴↑2)))) = ((i · 𝐴) + (√‘(1 − (𝐴↑2))))) |
10 | 8 | negcld 10258 | . . 3 ⊢ (𝐴 ∈ ℂ → -(√‘(1 − (𝐴↑2))) ∈ ℂ) |
11 | 0ne1 10965 | . . . . . 6 ⊢ 0 ≠ 1 | |
12 | 0cnd 9912 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℂ) | |
13 | 1cnd 9935 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ) | |
14 | subcan2 10185 | . . . . . . . 8 ⊢ ((0 ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝐴↑2) ∈ ℂ) → ((0 − (𝐴↑2)) = (1 − (𝐴↑2)) ↔ 0 = 1)) | |
15 | 14 | necon3bid 2826 | . . . . . . 7 ⊢ ((0 ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝐴↑2) ∈ ℂ) → ((0 − (𝐴↑2)) ≠ (1 − (𝐴↑2)) ↔ 0 ≠ 1)) |
16 | 12, 13, 5, 15 | syl3anc 1318 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((0 − (𝐴↑2)) ≠ (1 − (𝐴↑2)) ↔ 0 ≠ 1)) |
17 | 11, 16 | mpbiri 247 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (0 − (𝐴↑2)) ≠ (1 − (𝐴↑2))) |
18 | sqmul 12788 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((i · 𝐴)↑2) = ((i↑2) · (𝐴↑2))) | |
19 | 1, 18 | mpan 702 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) = ((i↑2) · (𝐴↑2))) |
20 | i2 12827 | . . . . . . . . 9 ⊢ (i↑2) = -1 | |
21 | 20 | oveq1i 6559 | . . . . . . . 8 ⊢ ((i↑2) · (𝐴↑2)) = (-1 · (𝐴↑2)) |
22 | 5 | mulm1d 10361 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (-1 · (𝐴↑2)) = -(𝐴↑2)) |
23 | 21, 22 | syl5eq 2656 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((i↑2) · (𝐴↑2)) = -(𝐴↑2)) |
24 | 19, 23 | eqtrd 2644 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) = -(𝐴↑2)) |
25 | df-neg 10148 | . . . . . 6 ⊢ -(𝐴↑2) = (0 − (𝐴↑2)) | |
26 | 24, 25 | syl6eq 2660 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) = (0 − (𝐴↑2))) |
27 | sqneg 12785 | . . . . . . 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 14026 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((√‘(1 − (𝐴↑2)))↑2) = (1 − (𝐴↑2))) |
30 | 28, 29 | eqtrd 2644 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (-(√‘(1 − (𝐴↑2)))↑2) = (1 − (𝐴↑2))) |
31 | 17, 26, 30 | 3netr4d 2859 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) ≠ (-(√‘(1 − (𝐴↑2)))↑2)) |
32 | oveq1 6556 | . . . . 5 ⊢ ((i · 𝐴) = -(√‘(1 − (𝐴↑2))) → ((i · 𝐴)↑2) = (-(√‘(1 − (𝐴↑2)))↑2)) | |
33 | 32 | necon3i 2814 | . . . 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 10280 | . 2 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) − -(√‘(1 − (𝐴↑2)))) ≠ 0) |
36 | 9, 35 | eqnetrrd 2850 | 1 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) + (√‘(1 − (𝐴↑2)))) ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 ici 9817 + caddc 9818 · cmul 9820 − cmin 10145 -cneg 10146 2c2 10947 ↑cexp 12722 √csqrt 13821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 |
This theorem is referenced by: asinlem3 24398 asinf 24399 asinneg 24413 efiasin 24415 asinbnd 24426 dvasin 32666 |
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