Theorem recexap 8927

Description: Existence of reciprocal of nonzero complex number. (Contributed by Jim Kingdon, 20-Feb-2020.)

Assertion

Ref	Expression
recexap	|- ( ( A e. CC /\ A # 0 ) -> E. x e. CC ( A x. x ) = 1 )

Distinct variable group:   x, A

Proof of Theorem recexap

Dummy variables y a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnre 8270	. . 3 |- ( A e. CC -> E. a e. RR E. b e. RR A = ( a + ( _i x. b ) ) )
2		recexaplem2 8926	. . . . . . . . 9 |- ( ( a e. RR /\ b e. RR /\ ( a + ( _i x. b ) ) # 0 ) -> ( ( a x. a ) + ( b x. b ) ) # 0 )
3	2	3expia 1232	. . . . . . . 8 |- ( ( a e. RR /\ b e. RR ) -> ( ( a + ( _i x. b ) ) # 0 -> ( ( a x. a ) + ( b x. b ) ) # 0 ) )
4		remulcl 8255	. . . . . . . . . . . 12 |- ( ( a e. RR /\ a e. RR ) -> ( a x. a ) e. RR )
5	4	anidms 397	. . . . . . . . . . 11 |- ( a e. RR -> ( a x. a ) e. RR )
6		remulcl 8255	. . . . . . . . . . . 12 |- ( ( b e. RR /\ b e. RR ) -> ( b x. b ) e. RR )
7	6	anidms 397	. . . . . . . . . . 11 |- ( b e. RR -> ( b x. b ) e. RR )
8		readdcl 8253	. . . . . . . . . . 11 |- ( ( ( a x. a ) e. RR /\ ( b x. b ) e. RR ) -> ( ( a x. a ) + ( b x. b ) ) e. RR )
9	5, 7, 8	syl2an 289	. . . . . . . . . 10 |- ( ( a e. RR /\ b e. RR ) -> ( ( a x. a ) + ( b x. b ) ) e. RR )
10		0re 8274	. . . . . . . . . 10 |- 0 e. RR
11		apreap 8861	. . . . . . . . . 10 |- ( ( ( ( a x. a ) + ( b x. b ) ) e. RR /\ 0 e. RR ) -> ( ( ( a x. a ) + ( b x. b ) ) # 0 <-> ( ( a x. a ) + ( b x. b ) ) #ℝ 0 ) )
12	9, 10, 11	sylancl 413	. . . . . . . . 9 |- ( ( a e. RR /\ b e. RR ) -> ( ( ( a x. a ) + ( b x. b ) ) # 0 <-> ( ( a x. a ) + ( b x. b ) ) #ℝ 0 ) )
13		recexre 8852	. . . . . . . . . . . 12 |- ( ( ( ( a x. a ) + ( b x. b ) ) e. RR /\ ( ( a x. a ) + ( b x. b ) ) #ℝ 0 ) -> E. y e. RR ( ( ( a x. a ) + ( b x. b ) ) x. y ) = 1 )
14	9, 13	sylan 283	. . . . . . . . . . 11 |- ( ( ( a e. RR /\ b e. RR ) /\ ( ( a x. a ) + ( b x. b ) ) #ℝ 0 ) -> E. y e. RR ( ( ( a x. a ) + ( b x. b ) ) x. y ) = 1 )
15		recn 8260	. . . . . . . . . . . . 13 |- ( a e. RR -> a e. CC )
16		recn 8260	. . . . . . . . . . . . 13 |- ( b e. RR -> b e. CC )
17		recn 8260	. . . . . . . . . . . . . . 15 |- ( y e. RR -> y e. CC )
18		ax-icn 8222	. . . . . . . . . . . . . . . . . . . . 21 |- _i e. CC
19		mulcl 8254	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( _i e. CC /\ b e. CC ) -> ( _i x. b ) e. CC )
20	18, 19	mpan 424	. . . . . . . . . . . . . . . . . . . 20 |- ( b e. CC -> ( _i x. b ) e. CC )
21		subcl 8472	. . . . . . . . . . . . . . . . . . . 20 |- ( ( a e. CC /\ ( _i x. b ) e. CC ) -> ( a - ( _i x. b ) ) e. CC )
22	20, 21	sylan2 286	. . . . . . . . . . . . . . . . . . 19 |- ( ( a e. CC /\ b e. CC ) -> ( a - ( _i x. b ) ) e. CC )
23		mulcl 8254	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( a - ( _i x. b ) ) e. CC /\ y e. CC ) -> ( ( a - ( _i x. b ) ) x. y ) e. CC )
24	22, 23	sylan 283	. . . . . . . . . . . . . . . . . 18 |- ( ( ( a e. CC /\ b e. CC ) /\ y e. CC ) -> ( ( a - ( _i x. b ) ) x. y ) e. CC )
25	24	adantr 276	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( a e. CC /\ b e. CC ) /\ y e. CC ) /\ ( ( ( a x. a ) + ( b x. b ) ) x. y ) = 1 ) -> ( ( a - ( _i x. b ) ) x. y ) e. CC )
26		addcl 8252	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( a e. CC /\ ( _i x. b ) e. CC ) -> ( a + ( _i x. b ) ) e. CC )
27	20, 26	sylan2 286	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( a e. CC /\ b e. CC ) -> ( a + ( _i x. b ) ) e. CC )
28	27	adantr 276	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( a e. CC /\ b e. CC ) /\ y e. CC ) -> ( a + ( _i x. b ) ) e. CC )
29	22	adantr 276	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( a e. CC /\ b e. CC ) /\ y e. CC ) -> ( a - ( _i x. b ) ) e. CC )
30		simpr 110	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( a e. CC /\ b e. CC ) /\ y e. CC ) -> y e. CC )
31	28, 29, 30	mulassd 8297	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( a e. CC /\ b e. CC ) /\ y e. CC ) -> ( ( ( a + ( _i x. b ) ) x. ( a - ( _i x. b ) ) ) x. y ) = ( ( a + ( _i x. b ) ) x. ( ( a - ( _i x. b ) ) x. y ) ) )
32		recextlem1 8925	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( a e. CC /\ b e. CC ) -> ( ( a + ( _i x. b ) ) x. ( a - ( _i x. b ) ) ) = ( ( a x. a ) + ( b x. b ) ) )
33	32	adantr 276	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( a e. CC /\ b e. CC ) /\ y e. CC ) -> ( ( a + ( _i x. b ) ) x. ( a - ( _i x. b ) ) ) = ( ( a x. a ) + ( b x. b ) ) )
34	33	oveq1d 6065	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( a e. CC /\ b e. CC ) /\ y e. CC ) -> ( ( ( a + ( _i x. b ) ) x. ( a - ( _i x. b ) ) ) x. y ) = ( ( ( a x. a ) + ( b x. b ) ) x. y ) )
35	31, 34	eqtr3d 2267	. . . . . . . . . . . . . . . . . 18 |- ( ( ( a e. CC /\ b e. CC ) /\ y e. CC ) -> ( ( a + ( _i x. b ) ) x. ( ( a - ( _i x. b ) ) x. y ) ) = ( ( ( a x. a ) + ( b x. b ) ) x. y ) )
36		id 19	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( a x. a ) + ( b x. b ) ) x. y ) = 1 -> ( ( ( a x. a ) + ( b x. b ) ) x. y ) = 1 )
37	35, 36	sylan9eq 2285	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( a e. CC /\ b e. CC ) /\ y e. CC ) /\ ( ( ( a x. a ) + ( b x. b ) ) x. y ) = 1 ) -> ( ( a + ( _i x. b ) ) x. ( ( a - ( _i x. b ) ) x. y ) ) = 1 )
38		oveq2 6058	. . . . . . . . . . . . . . . . . . 19 |- ( x = ( ( a - ( _i x. b ) ) x. y ) -> ( ( a + ( _i x. b ) ) x. x ) = ( ( a + ( _i x. b ) ) x. ( ( a - ( _i x. b ) ) x. y ) ) )
39	38	eqeq1d 2241	. . . . . . . . . . . . . . . . . 18 |- ( x = ( ( a - ( _i x. b ) ) x. y ) -> ( ( ( a + ( _i x. b ) ) x. x ) = 1 <-> ( ( a + ( _i x. b ) ) x. ( ( a - ( _i x. b ) ) x. y ) ) = 1 ) )
40	39	rspcev 2921	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( a - ( _i x. b ) ) x. y ) e. CC /\ ( ( a + ( _i x. b ) ) x. ( ( a - ( _i x. b ) ) x. y ) ) = 1 ) -> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 )
41	25, 37, 40	syl2anc 411	. . . . . . . . . . . . . . . 16 |- ( ( ( ( a e. CC /\ b e. CC ) /\ y e. CC ) /\ ( ( ( a x. a ) + ( b x. b ) ) x. y ) = 1 ) -> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 )
42	41	exp31 364	. . . . . . . . . . . . . . 15 |- ( ( a e. CC /\ b e. CC ) -> ( y e. CC -> ( ( ( ( a x. a ) + ( b x. b ) ) x. y ) = 1 -> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 ) ) )
43	17, 42	syl5 32	. . . . . . . . . . . . . 14 |- ( ( a e. CC /\ b e. CC ) -> ( y e. RR -> ( ( ( ( a x. a ) + ( b x. b ) ) x. y ) = 1 -> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 ) ) )
44	43	rexlimdv 2659	. . . . . . . . . . . . 13 |- ( ( a e. CC /\ b e. CC ) -> ( E. y e. RR ( ( ( a x. a ) + ( b x. b ) ) x. y ) = 1 -> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 ) )
45	15, 16, 44	syl2an 289	. . . . . . . . . . . 12 |- ( ( a e. RR /\ b e. RR ) -> ( E. y e. RR ( ( ( a x. a ) + ( b x. b ) ) x. y ) = 1 -> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 ) )
46	45	adantr 276	. . . . . . . . . . 11 |- ( ( ( a e. RR /\ b e. RR ) /\ ( ( a x. a ) + ( b x. b ) ) #ℝ 0 ) -> ( E. y e. RR ( ( ( a x. a ) + ( b x. b ) ) x. y ) = 1 -> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 ) )
47	14, 46	mpd 13	. . . . . . . . . 10 |- ( ( ( a e. RR /\ b e. RR ) /\ ( ( a x. a ) + ( b x. b ) ) #ℝ 0 ) -> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 )
48	47	ex 115	. . . . . . . . 9 |- ( ( a e. RR /\ b e. RR ) -> ( ( ( a x. a ) + ( b x. b ) ) #ℝ 0 -> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 ) )
49	12, 48	sylbid 150	. . . . . . . 8 |- ( ( a e. RR /\ b e. RR ) -> ( ( ( a x. a ) + ( b x. b ) ) # 0 -> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 ) )
50	3, 49	syld 45	. . . . . . 7 |- ( ( a e. RR /\ b e. RR ) -> ( ( a + ( _i x. b ) ) # 0 -> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 ) )
51	50	adantr 276	. . . . . 6 |- ( ( ( a e. RR /\ b e. RR ) /\ A = ( a + ( _i x. b ) ) ) -> ( ( a + ( _i x. b ) ) # 0 -> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 ) )
52		breq1 4112	. . . . . . 7 |- ( A = ( a + ( _i x. b ) ) -> ( A # 0 <-> ( a + ( _i x. b ) ) # 0 ) )
53	52	adantl 277	. . . . . 6 |- ( ( ( a e. RR /\ b e. RR ) /\ A = ( a + ( _i x. b ) ) ) -> ( A # 0 <-> ( a + ( _i x. b ) ) # 0 ) )
54		oveq1 6057	. . . . . . . . 9 |- ( A = ( a + ( _i x. b ) ) -> ( A x. x ) = ( ( a + ( _i x. b ) ) x. x ) )
55	54	eqeq1d 2241	. . . . . . . 8 |- ( A = ( a + ( _i x. b ) ) -> ( ( A x. x ) = 1 <-> ( ( a + ( _i x. b ) ) x. x ) = 1 ) )
56	55	rexbidv 2543	. . . . . . 7 |- ( A = ( a + ( _i x. b ) ) -> ( E. x e. CC ( A x. x ) = 1 <-> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 ) )
57	56	adantl 277	. . . . . 6 |- ( ( ( a e. RR /\ b e. RR ) /\ A = ( a + ( _i x. b ) ) ) -> ( E. x e. CC ( A x. x ) = 1 <-> E. x e. CC ( ( a + ( _i x. b ) ) x. x ) = 1 ) )
58	51, 53, 57	3imtr4d 203	. . . . 5 |- ( ( ( a e. RR /\ b e. RR ) /\ A = ( a + ( _i x. b ) ) ) -> ( A # 0 -> E. x e. CC ( A x. x ) = 1 ) )
59	58	ex 115	. . . 4 |- ( ( a e. RR /\ b e. RR ) -> ( A = ( a + ( _i x. b ) ) -> ( A # 0 -> E. x e. CC ( A x. x ) = 1 ) ) )
60	59	rexlimivv 2666	. . 3 |- ( E. a e. RR E. b e. RR A = ( a + ( _i x. b ) ) -> ( A # 0 -> E. x e. CC ( A x. x ) = 1 ) )
61	1, 60	syl 14	. 2 |- ( A e. CC -> ( A # 0 -> E. x e. CC ( A x. x ) = 1 ) )
62	61	imp 124	1 |- ( ( A e. CC /\ A # 0 ) -> E. x e. CC ( A x. x ) = 1 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   E.wrex 2521   class class class wbr 4109  (class class class)co 6050   CCcc 8125   RRcr 8126   0cc0 8127   1c1 8128   _ici 8129   + caddc 8130   x. cmul 8132   - cmin 8444   #_ℝ creap 8848   # cap 8855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-po 4417  df-iso 4418  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856

This theorem is referenced by:  mulap0  8928  mulcanapd  8935  receuap  8943  recapb  8945