Theorem mulnzcnopr 10673

Description: Multiplication maps nonzero complex numbers to nonzero complex numbers. (Contributed by Steve Rodriguez, 23-Feb-2007.)

Assertion

Ref	Expression
mulnzcnopr	|- ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) : ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) --> ( CC \ { 0 } )

Proof of Theorem mulnzcnopr

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

Step	Hyp	Ref	Expression
1		ax-mulf 10016	. . . . 5 |- x. : ( CC X. CC ) --> CC
2		ffnov 6764	. . . . 5 |- ( x. : ( CC X. CC ) --> CC <-> ( x. Fn ( CC X. CC ) /\ A. x e. CC A. y e. CC ( x x. y ) e. CC ) )
3	1, 2	mpbi 220	. . . 4 |- ( x. Fn ( CC X. CC ) /\ A. x e. CC A. y e. CC ( x x. y ) e. CC )
4	3	simpli 474	. . 3 |- x. Fn ( CC X. CC )
5		difss 3737	. . . 4 |- ( CC \ { 0 } ) C_ CC
6		xpss12 5225	. . . 4 |- ( ( ( CC \ { 0 } ) C_ CC /\ ( CC \ { 0 } ) C_ CC ) -> ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) C_ ( CC X. CC ) )
7	5, 5, 6	mp2an 708	. . 3 |- ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) C_ ( CC X. CC )
8		fnssres 6004	. . 3 |- ( ( x. Fn ( CC X. CC ) /\ ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) C_ ( CC X. CC ) ) -> ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) Fn ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) )
9	4, 7, 8	mp2an 708	. 2 |- ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) Fn ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) )
10		ovres 6800	. . . 4 |- ( ( x e. ( CC \ { 0 } ) /\ y e. ( CC \ { 0 } ) ) -> ( x ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) y ) = ( x x. y ) )
11		eldifsn 4317	. . . . . 6 |- ( x e. ( CC \ { 0 } ) <-> ( x e. CC /\ x =/= 0 ) )
12		eldifsn 4317	. . . . . 6 |- ( y e. ( CC \ { 0 } ) <-> ( y e. CC /\ y =/= 0 ) )
13		mulcl 10020	. . . . . . . 8 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
14	13	ad2ant2r 783	. . . . . . 7 |- ( ( ( x e. CC /\ x =/= 0 ) /\ ( y e. CC /\ y =/= 0 ) ) -> ( x x. y ) e. CC )
15		mulne0 10669	. . . . . . 7 |- ( ( ( x e. CC /\ x =/= 0 ) /\ ( y e. CC /\ y =/= 0 ) ) -> ( x x. y ) =/= 0 )
16	14, 15	jca 554	. . . . . 6 |- ( ( ( x e. CC /\ x =/= 0 ) /\ ( y e. CC /\ y =/= 0 ) ) -> ( ( x x. y ) e. CC /\ ( x x. y ) =/= 0 ) )
17	11, 12, 16	syl2anb 496	. . . . 5 |- ( ( x e. ( CC \ { 0 } ) /\ y e. ( CC \ { 0 } ) ) -> ( ( x x. y ) e. CC /\ ( x x. y ) =/= 0 ) )
18		eldifsn 4317	. . . . 5 |- ( ( x x. y ) e. ( CC \ { 0 } ) <-> ( ( x x. y ) e. CC /\ ( x x. y ) =/= 0 ) )
19	17, 18	sylibr 224	. . . 4 |- ( ( x e. ( CC \ { 0 } ) /\ y e. ( CC \ { 0 } ) ) -> ( x x. y ) e. ( CC \ { 0 } ) )
20	10, 19	eqeltrd 2701	. . 3 |- ( ( x e. ( CC \ { 0 } ) /\ y e. ( CC \ { 0 } ) ) -> ( x ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) y ) e. ( CC \ { 0 } ) )
21	20	rgen2a 2977	. 2 |- A. x e. ( CC \ { 0 } ) A. y e. ( CC \ { 0 } ) ( x ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) y ) e. ( CC \ { 0 } )
22		ffnov 6764	. 2 |- ( ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) : ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) --> ( CC \ { 0 } ) <-> ( ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) Fn ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) /\ A. x e. ( CC \ { 0 } ) A. y e. ( CC \ { 0 } ) ( x ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) y ) e. ( CC \ { 0 } ) ) )
23	9, 21, 22	mpbir2an 955	1 |- ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) : ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) --> ( CC \ { 0 } )

Syntax hints:   /\ wa 384   e. wcel 1990   =/= wne 2794   A.wral 2912   \ cdif 3571   C_ wss 3574   {csn 4177   X. cxp 5112   |` cres 5116   Fn wfn 5883   -->wf 5884  (class class class)co 6650   CCcc 9934   0cc0 9936   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-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-mulf 10016

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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269

This theorem is referenced by: (None)