Theorem mulcnsr 9534

Description: Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
mulcnsr

Proof of Theorem mulcnsr

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

Step	Hyp	Ref	Expression
1		opex 4716	. 2
2		oveq1 6303	. . . . 5
3		oveq1 6303	. . . . . 6
4	3	oveq2d 6312	. . . . 5
5	2, 4	oveqan12d 6315	. . . 4
6		oveq1 6303	. . . . 5
7		oveq1 6303	. . . . 5
8	6, 7	oveqan12rd 6316	. . . 4
9	5, 8	opeq12d 4225	. . 3
10		oveq2 6304	. . . . 5
11		oveq2 6304	. . . . . 6
12	11	oveq2d 6312	. . . . 5
13	10, 12	oveqan12d 6315	. . . 4
14		oveq2 6304	. . . . 5
15		oveq2 6304	. . . . 5
16	14, 15	oveqan12d 6315	. . . 4
17	13, 16	opeq12d 4225	. . 3
18	9, 17	sylan9eq 2518	. 2
19		df-mul 9525	. . 3
20		df-c 9519	. . . . . . 7
21	20	eleq2i 2535	. . . . . 6
22	20	eleq2i 2535	. . . . . 6
23	21, 22	anbi12i 697	. . . . 5
24	23	anbi1i 695	. . . 4
25	24	oprabbii 6352	. . 3
26	19, 25	eqtri 2486	. 2
27	1, 18, 26	ov3 6439	1

Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  <.cop 4035  X.cxp 5002  (class class class)co 6296  {coprab 6297  cnr 9264  cm1r 9267  cplr 9268  cmr 9269  cc 9511  cmul 9518

This theorem is referenced by:  mulresr  9537  mulcnsrec  9542  axmulf  9544  axi2m1  9557  axcnre  9562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-iota 5556  df-fun 5595  df-fv 5601  df-ov 6299  df-oprab 6300  df-c 9519  df-mul 9525