Theorem mulnzcnopr 10088

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 9468	. . . . 5 |- x. : ( CC X. CC ) --> CC
2		ffnov 6299	. . . . 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 208	. . . 4 |- ( x. Fn ( CC X. CC ) /\ A. x e. CC A. y e. CC ( x x. y ) e. CC )
4	3	simpli 458	. . 3 |- x. Fn ( CC X. CC )
5		difss 3586	. . . 4 |- ( CC \ { 0 } ) C_ CC
6		xpss12 5048	. . . 4 |- ( ( ( CC \ { 0 } ) C_ CC /\ ( CC \ { 0 } ) C_ CC ) -> ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) C_ ( CC X. CC ) )
7	5, 5, 6	mp2an 672	. . 3 |- ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) C_ ( CC X. CC )
8		fnssres 5627	. . 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 672	. 2 |- ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) Fn ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) )
10		ovres 6335	. . . 4 |- ( ( x e. ( CC \ { 0 } ) /\ y e. ( CC \ { 0 } ) ) -> ( x ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) y ) = ( x x. y ) )
11		eldifsn 4103	. . . . . 6 |- ( x e. ( CC \ { 0 } ) <-> ( x e. CC /\ x =/= 0 ) )
12		eldifsn 4103	. . . . . 6 |- ( y e. ( CC \ { 0 } ) <-> ( y e. CC /\ y =/= 0 ) )
13		mulcl 9472	. . . . . . . 8 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
14	13	ad2ant2r 746	. . . . . . 7 |- ( ( ( x e. CC /\ x =/= 0 ) /\ ( y e. CC /\ y =/= 0 ) ) -> ( x x. y ) e. CC )
15		mulne0 10084	. . . . . . 7 |- ( ( ( x e. CC /\ x =/= 0 ) /\ ( y e. CC /\ y =/= 0 ) ) -> ( x x. y ) =/= 0 )
16	14, 15	jca 532	. . . . . 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 479	. . . . 5 |- ( ( x e. ( CC \ { 0 } ) /\ y e. ( CC \ { 0 } ) ) -> ( ( x x. y ) e. CC /\ ( x x. y ) =/= 0 ) )
18		eldifsn 4103	. . . . 5 |- ( ( x x. y ) e. ( CC \ { 0 } ) <-> ( ( x x. y ) e. CC /\ ( x x. y ) =/= 0 ) )
19	17, 18	sylibr 212	. . . 4 |- ( ( x e. ( CC \ { 0 } ) /\ y e. ( CC \ { 0 } ) ) -> ( x x. y ) e. ( CC \ { 0 } ) )
20	10, 19	eqeltrd 2540	. . 3 |- ( ( x e. ( CC \ { 0 } ) /\ y e. ( CC \ { 0 } ) ) -> ( x ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) y ) e. ( CC \ { 0 } ) )
21	20	rgen2a 2894	. 2 |- A. x e. ( CC \ { 0 } ) A. y e. ( CC \ { 0 } ) ( x ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) y ) e. ( CC \ { 0 } )
22		ffnov 6299	. 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 911	1 |- ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) : ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) --> ( CC \ { 0 } )

Syntax hints:   /\ wa 369   e. wcel 1758   =/= wne 2645   A.wral 2796   \ cdif 3428   C_ wss 3431   {csn 3980   X. cxp 4941   |` cres 4945   Fn wfn 5516   -->wf 5517  (class class class)co 6195   CCcc 9386   0cc0 9388   x. cmul 9393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-mulf 9468

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-po 4744  df-so 4745  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704

This theorem is referenced by:  ablomul  23989  mulid  23990