Theorem asinlem 22222

Description: The argument to the logarithm in df-asin 22219 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 9337	. . . 4 |- _i e. CC
2		mulcl 9362	. . . 4 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
3	1, 2	mpan 665	. . 3 |- ( A e. CC -> ( _i x. A ) e. CC )
4		ax-1cn 9336	. . . . 5 |- 1 e. CC
5		sqcl 11924	. . . . 5 |- ( A e. CC -> ( A ^ 2 ) e. CC )
6		subcl 9605	. . . . 5 |- ( ( 1 e. CC /\ ( A ^ 2 ) e. CC ) -> ( 1 - ( A ^ 2 ) ) e. CC )
7	4, 5, 6	sylancr 658	. . . 4 |- ( A e. CC -> ( 1 - ( A ^ 2 ) ) e. CC )
8	7	sqrcld 12919	. . 3 |- ( A e. CC -> ( sqr ` ( 1 - ( A ^ 2 ) ) ) e. CC )
9	3, 8	subnegd 9722	. 2 |- ( A e. CC -> ( ( _i x. A ) - -u ( sqr ` ( 1 - ( A ^ 2 ) ) ) ) = ( ( _i x. A ) + ( sqr ` ( 1 - ( A ^ 2 ) ) ) ) )
10	8	negcld 9702	. . 3 |- ( A e. CC -> -u ( sqr ` ( 1 - ( A ^ 2 ) ) ) e. CC )
11		0ne1 10385	. . . . . 6 |- 0 =/= 1
12		0cnd 9375	. . . . . . 7 |- ( A e. CC -> 0 e. CC )
13	4	a1i 11	. . . . . . 7 |- ( A e. CC -> 1 e. CC )
14		subcan2 9630	. . . . . . . 8 |- ( ( 0 e. CC /\ 1 e. CC /\ ( A ^ 2 ) e. CC ) -> ( ( 0 - ( A ^ 2 ) ) = ( 1 - ( A ^ 2 ) ) <-> 0 = 1 ) )
15	14	necon3bid 2641	. . . . . . 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 1213	. . . . . 6 |- ( A e. CC -> ( ( 0 - ( A ^ 2 ) ) =/= ( 1 - ( A ^ 2 ) ) <-> 0 =/= 1 ) )
17	11, 16	mpbiri 233	. . . . 5 |- ( A e. CC -> ( 0 - ( A ^ 2 ) ) =/= ( 1 - ( A ^ 2 ) ) )
18		sqmul 11925	. . . . . . . 8 |- ( ( _i e. CC /\ A e. CC ) -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) )
19	1, 18	mpan 665	. . . . . . 7 |- ( A e. CC -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) )
20		i2 11962	. . . . . . . . 9 |- ( _i ^ 2 ) = -u 1
21	20	oveq1i 6100	. . . . . . . 8 |- ( ( _i ^ 2 ) x. ( A ^ 2 ) ) = ( -u 1 x. ( A ^ 2 ) )
22	5	mulm1d 9792	. . . . . . . 8 |- ( A e. CC -> ( -u 1 x. ( A ^ 2 ) ) = -u ( A ^ 2 ) )
23	21, 22	syl5eq 2485	. . . . . . 7 |- ( A e. CC -> ( ( _i ^ 2 ) x. ( A ^ 2 ) ) = -u ( A ^ 2 ) )
24	19, 23	eqtrd 2473	. . . . . 6 |- ( A e. CC -> ( ( _i x. A ) ^ 2 ) = -u ( A ^ 2 ) )
25		df-neg 9594	. . . . . 6 |- -u ( A ^ 2 ) = ( 0 - ( A ^ 2 ) )
26	24, 25	syl6eq 2489	. . . . 5 |- ( A e. CC -> ( ( _i x. A ) ^ 2 ) = ( 0 - ( A ^ 2 ) ) )
27		sqneg 11922	. . . . . . 7 |- ( ( sqr ` ( 1 - ( A ^ 2 ) ) ) e. CC -> ( -u ( sqr ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) = ( ( sqr ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) )
28	8, 27	syl 16	. . . . . 6 |- ( A e. CC -> ( -u ( sqr ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) = ( ( sqr ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) )
29	7	sqsqrd 12921	. . . . . 6 |- ( A e. CC -> ( ( sqr ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) = ( 1 - ( A ^ 2 ) ) )
30	28, 29	eqtrd 2473	. . . . 5 |- ( A e. CC -> ( -u ( sqr ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) = ( 1 - ( A ^ 2 ) ) )
31	17, 26, 30	3netr4d 2633	. . . 4 |- ( A e. CC -> ( ( _i x. A ) ^ 2 ) =/= ( -u ( sqr ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) )
32		oveq1 6097	. . . . 5 |- ( ( _i x. A ) = -u ( sqr ` ( 1 - ( A ^ 2 ) ) ) -> ( ( _i x. A ) ^ 2 ) = ( -u ( sqr ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) )
33	32	necon3i 2648	. . . 4 |- ( ( ( _i x. A ) ^ 2 ) =/= ( -u ( sqr ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) -> ( _i x. A ) =/= -u ( sqr ` ( 1 - ( A ^ 2 ) ) ) )
34	31, 33	syl 16	. . 3 |- ( A e. CC -> ( _i x. A ) =/= -u ( sqr ` ( 1 - ( A ^ 2 ) ) ) )
35	3, 10, 34	subne0d 9724	. 2 |- ( A e. CC -> ( ( _i x. A ) - -u ( sqr ` ( 1 - ( A ^ 2 ) ) ) ) =/= 0 )
36	9, 35	eqnetrrd 2626	1 |- ( A e. CC -> ( ( _i x. A ) + ( sqr ` ( 1 - ( A ^ 2 ) ) ) ) =/= 0 )

Syntax hints:   -> wi 4   <-> wb 184   /\ w3a 960   = wceq 1364   e. wcel 1761   =/= wne 2604   ` cfv 5415  (class class class)co 6090   CCcc 9276   0cc0 9278   1c1 9279   _ici 9280   + caddc 9281   x. cmul 9283   - cmin 9591   -ucneg 9592   2c2 10367   ^cexp 11861   sqrcsqr 12718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-seq 11803  df-exp 11862  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721

This theorem is referenced by:  asinlem3  22225  asinf  22226  asinneg  22240  efiasin  22242  asinbnd  22253  dvasin  28405