Theorem axcnre 9985

Description: A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom 17 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-cnre 10009. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
axcnre	|- ( A e. CC -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )

Distinct variable group:   x, y, A

Proof of Theorem axcnre

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-c 9942	. 2 |- CC = ( R. X. R. )
2		eqeq1 2626	. . 3 |- ( <. z , w >. = A -> ( <. z , w >. = ( x + ( _i x. y ) ) <-> A = ( x + ( _i x. y ) ) ) )
3	2	2rexbidv 3057	. 2 |- ( <. z , w >. = A -> ( E. x e. RR E. y e. RR <. z , w >. = ( x + ( _i x. y ) ) <-> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) ) )
4		opelreal 9951	. . . . . 6 |- ( <. z , 0R >. e. RR <-> z e. R. )
5		opelreal 9951	. . . . . 6 |- ( <. w , 0R >. e. RR <-> w e. R. )
6	4, 5	anbi12i 733	. . . . 5 |- ( ( <. z , 0R >. e. RR /\ <. w , 0R >. e. RR ) <-> ( z e. R. /\ w e. R. ) )
7	6	biimpri 218	. . . 4 |- ( ( z e. R. /\ w e. R. ) -> ( <. z , 0R >. e. RR /\ <. w , 0R >. e. RR ) )
8		df-i 9945	. . . . . . . . 9 |- _i = <. 0R , 1R >.
9	8	oveq1i 6660	. . . . . . . 8 |- ( _i x. <. w , 0R >. ) = ( <. 0R , 1R >. x. <. w , 0R >. )
10		0r 9901	. . . . . . . . . 10 |- 0R e. R.
11		1sr 9902	. . . . . . . . . . 11 |- 1R e. R.
12		mulcnsr 9957	. . . . . . . . . . 11 |- ( ( ( 0R e. R. /\ 1R e. R. ) /\ ( w e. R. /\ 0R e. R. ) ) -> ( <. 0R , 1R >. x. <. w , 0R >. ) = <. ( ( 0R .R w ) +R ( -1R .R ( 1R .R 0R ) ) ) , ( ( 1R .R w ) +R ( 0R .R 0R ) ) >. )
13	10, 11, 12	mpanl12 718	. . . . . . . . . 10 |- ( ( w e. R. /\ 0R e. R. ) -> ( <. 0R , 1R >. x. <. w , 0R >. ) = <. ( ( 0R .R w ) +R ( -1R .R ( 1R .R 0R ) ) ) , ( ( 1R .R w ) +R ( 0R .R 0R ) ) >. )
14	10, 13	mpan2 707	. . . . . . . . 9 |- ( w e. R. -> ( <. 0R , 1R >. x. <. w , 0R >. ) = <. ( ( 0R .R w ) +R ( -1R .R ( 1R .R 0R ) ) ) , ( ( 1R .R w ) +R ( 0R .R 0R ) ) >. )
15		mulcomsr 9910	. . . . . . . . . . . . 13 |- ( 0R .R w ) = ( w .R 0R )
16		00sr 9920	. . . . . . . . . . . . 13 |- ( w e. R. -> ( w .R 0R ) = 0R )
17	15, 16	syl5eq 2668	. . . . . . . . . . . 12 |- ( w e. R. -> ( 0R .R w ) = 0R )
18	17	oveq1d 6665	. . . . . . . . . . 11 |- ( w e. R. -> ( ( 0R .R w ) +R ( -1R .R ( 1R .R 0R ) ) ) = ( 0R +R ( -1R .R ( 1R .R 0R ) ) ) )
19		00sr 9920	. . . . . . . . . . . . . . . 16 |- ( 1R e. R. -> ( 1R .R 0R ) = 0R )
20	11, 19	ax-mp 5	. . . . . . . . . . . . . . 15 |- ( 1R .R 0R ) = 0R
21	20	oveq2i 6661	. . . . . . . . . . . . . 14 |- ( -1R .R ( 1R .R 0R ) ) = ( -1R .R 0R )
22		m1r 9903	. . . . . . . . . . . . . . 15 |- -1R e. R.
23		00sr 9920	. . . . . . . . . . . . . . 15 |- ( -1R e. R. -> ( -1R .R 0R ) = 0R )
24	22, 23	ax-mp 5	. . . . . . . . . . . . . 14 |- ( -1R .R 0R ) = 0R
25	21, 24	eqtri 2644	. . . . . . . . . . . . 13 |- ( -1R .R ( 1R .R 0R ) ) = 0R
26	25	oveq2i 6661	. . . . . . . . . . . 12 |- ( 0R +R ( -1R .R ( 1R .R 0R ) ) ) = ( 0R +R 0R )
27		0idsr 9918	. . . . . . . . . . . . 13 |- ( 0R e. R. -> ( 0R +R 0R ) = 0R )
28	10, 27	ax-mp 5	. . . . . . . . . . . 12 |- ( 0R +R 0R ) = 0R
29	26, 28	eqtri 2644	. . . . . . . . . . 11 |- ( 0R +R ( -1R .R ( 1R .R 0R ) ) ) = 0R
30	18, 29	syl6eq 2672	. . . . . . . . . 10 |- ( w e. R. -> ( ( 0R .R w ) +R ( -1R .R ( 1R .R 0R ) ) ) = 0R )
31		mulcomsr 9910	. . . . . . . . . . . . 13 |- ( 1R .R w ) = ( w .R 1R )
32		1idsr 9919	. . . . . . . . . . . . 13 |- ( w e. R. -> ( w .R 1R ) = w )
33	31, 32	syl5eq 2668	. . . . . . . . . . . 12 |- ( w e. R. -> ( 1R .R w ) = w )
34	33	oveq1d 6665	. . . . . . . . . . 11 |- ( w e. R. -> ( ( 1R .R w ) +R ( 0R .R 0R ) ) = ( w +R ( 0R .R 0R ) ) )
35		00sr 9920	. . . . . . . . . . . . . 14 |- ( 0R e. R. -> ( 0R .R 0R ) = 0R )
36	10, 35	ax-mp 5	. . . . . . . . . . . . 13 |- ( 0R .R 0R ) = 0R
37	36	oveq2i 6661	. . . . . . . . . . . 12 |- ( w +R ( 0R .R 0R ) ) = ( w +R 0R )
38		0idsr 9918	. . . . . . . . . . . 12 |- ( w e. R. -> ( w +R 0R ) = w )
39	37, 38	syl5eq 2668	. . . . . . . . . . 11 |- ( w e. R. -> ( w +R ( 0R .R 0R ) ) = w )
40	34, 39	eqtrd 2656	. . . . . . . . . 10 |- ( w e. R. -> ( ( 1R .R w ) +R ( 0R .R 0R ) ) = w )
41	30, 40	opeq12d 4410	. . . . . . . . 9 |- ( w e. R. -> <. ( ( 0R .R w ) +R ( -1R .R ( 1R .R 0R ) ) ) , ( ( 1R .R w ) +R ( 0R .R 0R ) ) >. = <. 0R , w >. )
42	14, 41	eqtrd 2656	. . . . . . . 8 |- ( w e. R. -> ( <. 0R , 1R >. x. <. w , 0R >. ) = <. 0R , w >. )
43	9, 42	syl5eq 2668	. . . . . . 7 |- ( w e. R. -> ( _i x. <. w , 0R >. ) = <. 0R , w >. )
44	43	oveq2d 6666	. . . . . 6 |- ( w e. R. -> ( <. z , 0R >. + ( _i x. <. w , 0R >. ) ) = ( <. z , 0R >. + <. 0R , w >. ) )
45	44	adantl 482	. . . . 5 |- ( ( z e. R. /\ w e. R. ) -> ( <. z , 0R >. + ( _i x. <. w , 0R >. ) ) = ( <. z , 0R >. + <. 0R , w >. ) )
46		addcnsr 9956	. . . . . . 7 |- ( ( ( z e. R. /\ 0R e. R. ) /\ ( 0R e. R. /\ w e. R. ) ) -> ( <. z , 0R >. + <. 0R , w >. ) = <. ( z +R 0R ) , ( 0R +R w ) >. )
47	10, 46	mpanl2 717	. . . . . 6 |- ( ( z e. R. /\ ( 0R e. R. /\ w e. R. ) ) -> ( <. z , 0R >. + <. 0R , w >. ) = <. ( z +R 0R ) , ( 0R +R w ) >. )
48	10, 47	mpanr1 719	. . . . 5 |- ( ( z e. R. /\ w e. R. ) -> ( <. z , 0R >. + <. 0R , w >. ) = <. ( z +R 0R ) , ( 0R +R w ) >. )
49		0idsr 9918	. . . . . 6 |- ( z e. R. -> ( z +R 0R ) = z )
50		addcomsr 9908	. . . . . . 7 |- ( 0R +R w ) = ( w +R 0R )
51	50, 38	syl5eq 2668	. . . . . 6 |- ( w e. R. -> ( 0R +R w ) = w )
52		opeq12 4404	. . . . . 6 |- ( ( ( z +R 0R ) = z /\ ( 0R +R w ) = w ) -> <. ( z +R 0R ) , ( 0R +R w ) >. = <. z , w >. )
53	49, 51, 52	syl2an 494	. . . . 5 |- ( ( z e. R. /\ w e. R. ) -> <. ( z +R 0R ) , ( 0R +R w ) >. = <. z , w >. )
54	45, 48, 53	3eqtrrd 2661	. . . 4 |- ( ( z e. R. /\ w e. R. ) -> <. z , w >. = ( <. z , 0R >. + ( _i x. <. w , 0R >. ) ) )
55		opex 4932	. . . . 5 |- <. z , 0R >. e. _V
56		opex 4932	. . . . 5 |- <. w , 0R >. e. _V
57		eleq1 2689	. . . . . . 7 |- ( x = <. z , 0R >. -> ( x e. RR <-> <. z , 0R >. e. RR ) )
58		eleq1 2689	. . . . . . 7 |- ( y = <. w , 0R >. -> ( y e. RR <-> <. w , 0R >. e. RR ) )
59	57, 58	bi2anan9 917	. . . . . 6 |- ( ( x = <. z , 0R >. /\ y = <. w , 0R >. ) -> ( ( x e. RR /\ y e. RR ) <-> ( <. z , 0R >. e. RR /\ <. w , 0R >. e. RR ) ) )
60		oveq1 6657	. . . . . . . 8 |- ( x = <. z , 0R >. -> ( x + ( _i x. y ) ) = ( <. z , 0R >. + ( _i x. y ) ) )
61		oveq2 6658	. . . . . . . . 9 |- ( y = <. w , 0R >. -> ( _i x. y ) = ( _i x. <. w , 0R >. ) )
62	61	oveq2d 6666	. . . . . . . 8 |- ( y = <. w , 0R >. -> ( <. z , 0R >. + ( _i x. y ) ) = ( <. z , 0R >. + ( _i x. <. w , 0R >. ) ) )
63	60, 62	sylan9eq 2676	. . . . . . 7 |- ( ( x = <. z , 0R >. /\ y = <. w , 0R >. ) -> ( x + ( _i x. y ) ) = ( <. z , 0R >. + ( _i x. <. w , 0R >. ) ) )
64	63	eqeq2d 2632	. . . . . 6 |- ( ( x = <. z , 0R >. /\ y = <. w , 0R >. ) -> ( <. z , w >. = ( x + ( _i x. y ) ) <-> <. z , w >. = ( <. z , 0R >. + ( _i x. <. w , 0R >. ) ) ) )
65	59, 64	anbi12d 747	. . . . 5 |- ( ( x = <. z , 0R >. /\ y = <. w , 0R >. ) -> ( ( ( x e. RR /\ y e. RR ) /\ <. z , w >. = ( x + ( _i x. y ) ) ) <-> ( ( <. z , 0R >. e. RR /\ <. w , 0R >. e. RR ) /\ <. z , w >. = ( <. z , 0R >. + ( _i x. <. w , 0R >. ) ) ) ) )
66	55, 56, 65	spc2ev 3301	. . . 4 |- ( ( ( <. z , 0R >. e. RR /\ <. w , 0R >. e. RR ) /\ <. z , w >. = ( <. z , 0R >. + ( _i x. <. w , 0R >. ) ) ) -> E. x E. y ( ( x e. RR /\ y e. RR ) /\ <. z , w >. = ( x + ( _i x. y ) ) ) )
67	7, 54, 66	syl2anc 693	. . 3 |- ( ( z e. R. /\ w e. R. ) -> E. x E. y ( ( x e. RR /\ y e. RR ) /\ <. z , w >. = ( x + ( _i x. y ) ) ) )
68		r2ex 3061	. . 3 |- ( E. x e. RR E. y e. RR <. z , w >. = ( x + ( _i x. y ) ) <-> E. x E. y ( ( x e. RR /\ y e. RR ) /\ <. z , w >. = ( x + ( _i x. y ) ) ) )
69	67, 68	sylibr 224	. 2 |- ( ( z e. R. /\ w e. R. ) -> E. x e. RR E. y e. RR <. z , w >. = ( x + ( _i x. y ) ) )
70	1, 3, 69	optocl 5195	1 |- ( A e. CC -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   <.cop 4183  (class class class)co 6650   R.cnr 9687   0Rc0r 9688   1Rc1r 9689   -1Rcm1r 9690   +R cplr 9691   .R cmr 9692   CCcc 9934   RRcr 9935   _ici 9938   + caddc 9939   x. cmul 9941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-rq 9739  df-ltnq 9740  df-np 9803  df-1p 9804  df-plp 9805  df-mp 9806  df-ltp 9807  df-enr 9877  df-nr 9878  df-plr 9879  df-mr 9880  df-0r 9882  df-1r 9883  df-m1r 9884  df-c 9942  df-i 9945  df-r 9946  df-add 9947  df-mul 9948

This theorem is referenced by: (None)