Theorem asinlem 23842

Description: The argument to the logarithm in df-asin 23839 is always nonzero. (Contributed by Mario Carneiro, 31-Mar-2015.)

Assertion

Ref	Expression
asinlem	|- ( A e. CC -> ( ( _i x. A ) + ( sqr ` ( 1 - ( A ^ 2 ) ) ) ) =/= 0 )

Proof of Theorem asinlem

Step	Hyp	Ref	Expression
1		ax-icn 9623	. . . 4 |- _i e. CC
2		mulcl 9648	. . . 4 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
3	1, 2	mpan 681	. . 3 |- ( A e. CC -> ( _i x. A ) e. CC )
4		ax-1cn 9622	. . . . 5 |- 1 e. CC
5		sqcl 12368	. . . . 5 |- ( A e. CC -> ( A ^ 2 ) e. CC )
6		subcl 9899	. . . . 5 |- ( ( 1 e. CC /\ ( A ^ 2 ) e. CC ) -> ( 1 - ( A ^ 2 ) ) e. CC )
7	4, 5, 6	sylancr 674	. . . 4 |- ( A e. CC -> ( 1 - ( A ^ 2 ) ) e. CC )
8	7	sqrtcld 13547	. . 3 |- ( A e. CC -> ( sqr ` ( 1 - ( A ^ 2 ) ) ) e. CC )
9	3, 8	subnegd 10018	. 2 |- ( A e. CC -> ( ( _i x. A ) - -u ( sqr ` ( 1 - ( A ^ 2 ) ) ) ) = ( ( _i x. A ) + ( sqr ` ( 1 - ( A ^ 2 ) ) ) ) )
10	8	negcld 9998	. . 3 |- ( A e. CC -> -u ( sqr ` ( 1 - ( A ^ 2 ) ) ) e. CC )
11		0ne1 10704	. . . . . 6 |- 0 =/= 1
12		0cnd 9661	. . . . . . 7 |- ( A e. CC -> 0 e. CC )
13		1cnd 9684	. . . . . . 7 |- ( A e. CC -> 1 e. CC )
14		subcan2 9924	. . . . . . . 8 |- ( ( 0 e. CC /\ 1 e. CC /\ ( A ^ 2 ) e. CC ) -> ( ( 0 - ( A ^ 2 ) ) = ( 1 - ( A ^ 2 ) ) <-> 0 = 1 ) )
15	14	necon3bid 2679	. . . . . . 7 |- ( ( 0 e. CC /\ 1 e. CC /\ ( A ^ 2 ) e. CC ) -> ( ( 0 - ( A ^ 2 ) ) =/= ( 1 - ( A ^ 2 ) ) <-> 0 =/= 1 ) )
16	12, 13, 5, 15	syl3anc 1276	. . . . . 6 |- ( A e. CC -> ( ( 0 - ( A ^ 2 ) ) =/= ( 1 - ( A ^ 2 ) ) <-> 0 =/= 1 ) )
17	11, 16	mpbiri 241	. . . . 5 |- ( A e. CC -> ( 0 - ( A ^ 2 ) ) =/= ( 1 - ( A ^ 2 ) ) )
18		sqmul 12369	. . . . . . . 8 |- ( ( _i e. CC /\ A e. CC ) -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) )
19	1, 18	mpan 681	. . . . . . 7 |- ( A e. CC -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) )
20		i2 12406	. . . . . . . . 9 |- ( _i ^ 2 ) = -u 1
21	20	oveq1i 6324	. . . . . . . 8 |- ( ( _i ^ 2 ) x. ( A ^ 2 ) ) = ( -u 1 x. ( A ^ 2 ) )
22	5	mulm1d 10097	. . . . . . . 8 |- ( A e. CC -> ( -u 1 x. ( A ^ 2 ) ) = -u ( A ^ 2 ) )
23	21, 22	syl5eq 2507	. . . . . . 7 |- ( A e. CC -> ( ( _i ^ 2 ) x. ( A ^ 2 ) ) = -u ( A ^ 2 ) )
24	19, 23	eqtrd 2495	. . . . . 6 |- ( A e. CC -> ( ( _i x. A ) ^ 2 ) = -u ( A ^ 2 ) )
25		df-neg 9888	. . . . . 6 |- -u ( A ^ 2 ) = ( 0 - ( A ^ 2 ) )
26	24, 25	syl6eq 2511	. . . . 5 |- ( A e. CC -> ( ( _i x. A ) ^ 2 ) = ( 0 - ( A ^ 2 ) ) )
27		sqneg 12366	. . . . . . 7 |- ( ( sqr ` ( 1 - ( A ^ 2 ) ) ) e. CC -> ( -u ( sqr ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) = ( ( sqr ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) )
28	8, 27	syl 17	. . . . . 6 |- ( A e. CC -> ( -u ( sqr ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) = ( ( sqr ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) )
29	7	sqsqrtd 13549	. . . . . 6 |- ( A e. CC -> ( ( sqr ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) = ( 1 - ( A ^ 2 ) ) )
30	28, 29	eqtrd 2495	. . . . 5 |- ( A e. CC -> ( -u ( sqr ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) = ( 1 - ( A ^ 2 ) ) )
31	17, 26, 30	3netr4d 2712	. . . 4 |- ( A e. CC -> ( ( _i x. A ) ^ 2 ) =/= ( -u ( sqr ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) )
32		oveq1 6321	. . . . 5 |- ( ( _i x. A ) = -u ( sqr ` ( 1 - ( A ^ 2 ) ) ) -> ( ( _i x. A ) ^ 2 ) = ( -u ( sqr ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) )
33	32	necon3i 2667	. . . 4 |- ( ( ( _i x. A ) ^ 2 ) =/= ( -u ( sqr ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) -> ( _i x. A ) =/= -u ( sqr ` ( 1 - ( A ^ 2 ) ) ) )
34	31, 33	syl 17	. . 3 |- ( A e. CC -> ( _i x. A ) =/= -u ( sqr ` ( 1 - ( A ^ 2 ) ) ) )
35	3, 10, 34	subne0d 10020	. 2 |- ( A e. CC -> ( ( _i x. A ) - -u ( sqr ` ( 1 - ( A ^ 2 ) ) ) ) =/= 0 )
36	9, 35	eqnetrrd 2703	1 |- ( A e. CC -> ( ( _i x. A ) + ( sqr ` ( 1 - ( A ^ 2 ) ) ) ) =/= 0 )

Syntax hints:   -> wi 4   <-> wb 189   /\ w3a 991   = wceq 1454   e. wcel 1897   =/= wne 2632   ` cfv 5600  (class class class)co 6314   CCcc 9562   0cc0 9564   1c1 9565   _ici 9566   + caddc 9567   x. cmul 9569   - cmin 9885   -ucneg 9886   2c2 10686   ^cexp 12303   sqrcsqrt 13344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641  ax-pre-sup 9642

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-2nd 6820  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-sup 7981  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-div 10297  df-nn 10637  df-2 10695  df-3 10696  df-n0 10898  df-z 10966  df-uz 11188  df-rp 11331  df-seq 12245  df-exp 12304  df-cj 13210  df-re 13211  df-im 13212  df-sqrt 13346  df-abs 13347

This theorem is referenced by:  asinlem3  23845  asinf  23846  asinneg  23860  efiasin  23862  asinbnd  23873  dvasin  32072