Theorem mulnzcnopr 10112

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 9483	. . . . 5 |- x. : ( CC X. CC ) --> CC
2		ffnov 6305	. . . . 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 456	. . 3 |- x. Fn ( CC X. CC )
5		difss 3545	. . . 4 |- ( CC \ { 0 } ) C_ CC
6		xpss12 5021	. . . 4 |- ( ( ( CC \ { 0 } ) C_ CC /\ ( CC \ { 0 } ) C_ CC ) -> ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) C_ ( CC X. CC ) )
7	5, 5, 6	mp2an 670	. . 3 |- ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) C_ ( CC X. CC )
8		fnssres 5602	. . 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 670	. 2 |- ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) Fn ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) )
10		ovres 6341	. . . 4 |- ( ( x e. ( CC \ { 0 } ) /\ y e. ( CC \ { 0 } ) ) -> ( x ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) y ) = ( x x. y ) )
11		eldifsn 4069	. . . . . 6 |- ( x e. ( CC \ { 0 } ) <-> ( x e. CC /\ x =/= 0 ) )
12		eldifsn 4069	. . . . . 6 |- ( y e. ( CC \ { 0 } ) <-> ( y e. CC /\ y =/= 0 ) )
13		mulcl 9487	. . . . . . . 8 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
14	13	ad2ant2r 744	. . . . . . 7 |- ( ( ( x e. CC /\ x =/= 0 ) /\ ( y e. CC /\ y =/= 0 ) ) -> ( x x. y ) e. CC )
15		mulne0 10108	. . . . . . 7 |- ( ( ( x e. CC /\ x =/= 0 ) /\ ( y e. CC /\ y =/= 0 ) ) -> ( x x. y ) =/= 0 )
16	14, 15	jca 530	. . . . . 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 477	. . . . 5 |- ( ( x e. ( CC \ { 0 } ) /\ y e. ( CC \ { 0 } ) ) -> ( ( x x. y ) e. CC /\ ( x x. y ) =/= 0 ) )
18		eldifsn 4069	. . . . 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 2470	. . 3 |- ( ( x e. ( CC \ { 0 } ) /\ y e. ( CC \ { 0 } ) ) -> ( x ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) y ) e. ( CC \ { 0 } ) )
21	20	rgen2a 2809	. 2 |- A. x e. ( CC \ { 0 } ) A. y e. ( CC \ { 0 } ) ( x ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) y ) e. ( CC \ { 0 } )
22		ffnov 6305	. 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 918	1 |- ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) : ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) --> ( CC \ { 0 } )

Syntax hints:   /\ wa 367   e. wcel 1826   =/= wne 2577   A.wral 2732   \ cdif 3386   C_ wss 3389   {csn 3944   X. cxp 4911   |` cres 4915   Fn wfn 5491   -->wf 5492  (class class class)co 6196   CCcc 9401   0cc0 9403   x. cmul 9408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-mulf 9483

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-po 4714  df-so 4715  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721

This theorem is referenced by:  ablomul  25474  mulid  25475