Theorem mulnzcnopr 10209

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 9570	. . . . 5 |- x. : ( CC X. CC ) --> CC
2		ffnov 6358	. . . . 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 211	. . . 4 |- ( x. Fn ( CC X. CC ) /\ A. x e. CC A. y e. CC ( x x. y ) e. CC )
4	3	simpli 459	. . 3 |- x. Fn ( CC X. CC )
5		difss 3535	. . . 4 |- ( CC \ { 0 } ) C_ CC
6		xpss12 4902	. . . 4 |- ( ( ( CC \ { 0 } ) C_ CC /\ ( CC \ { 0 } ) C_ CC ) -> ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) C_ ( CC X. CC ) )
7	5, 5, 6	mp2an 676	. . 3 |- ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) C_ ( CC X. CC )
8		fnssres 5650	. . 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 676	. 2 |- ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) Fn ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) )
10		ovres 6394	. . . 4 |- ( ( x e. ( CC \ { 0 } ) /\ y e. ( CC \ { 0 } ) ) -> ( x ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) y ) = ( x x. y ) )
11		eldifsn 4068	. . . . . 6 |- ( x e. ( CC \ { 0 } ) <-> ( x e. CC /\ x =/= 0 ) )
12		eldifsn 4068	. . . . . 6 |- ( y e. ( CC \ { 0 } ) <-> ( y e. CC /\ y =/= 0 ) )
13		mulcl 9574	. . . . . . . 8 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
14	13	ad2ant2r 751	. . . . . . 7 |- ( ( ( x e. CC /\ x =/= 0 ) /\ ( y e. CC /\ y =/= 0 ) ) -> ( x x. y ) e. CC )
15		mulne0 10205	. . . . . . 7 |- ( ( ( x e. CC /\ x =/= 0 ) /\ ( y e. CC /\ y =/= 0 ) ) -> ( x x. y ) =/= 0 )
16	14, 15	jca 534	. . . . . 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 481	. . . . 5 |- ( ( x e. ( CC \ { 0 } ) /\ y e. ( CC \ { 0 } ) ) -> ( ( x x. y ) e. CC /\ ( x x. y ) =/= 0 ) )
18		eldifsn 4068	. . . . 5 |- ( ( x x. y ) e. ( CC \ { 0 } ) <-> ( ( x x. y ) e. CC /\ ( x x. y ) =/= 0 ) )
19	17, 18	sylibr 215	. . . 4 |- ( ( x e. ( CC \ { 0 } ) /\ y e. ( CC \ { 0 } ) ) -> ( x x. y ) e. ( CC \ { 0 } ) )
20	10, 19	eqeltrd 2506	. . 3 |- ( ( x e. ( CC \ { 0 } ) /\ y e. ( CC \ { 0 } ) ) -> ( x ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) y ) e. ( CC \ { 0 } ) )
21	20	rgen2a 2792	. 2 |- A. x e. ( CC \ { 0 } ) A. y e. ( CC \ { 0 } ) ( x ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) y ) e. ( CC \ { 0 } )
22		ffnov 6358	. 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 928	1 |- ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) : ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) --> ( CC \ { 0 } )

Syntax hints:   /\ wa 370   e. wcel 1872   =/= wne 2599   A.wral 2714   \ cdif 3376   C_ wss 3379   {csn 3941   X. cxp 4794   |` cres 4798   Fn wfn 5539   -->wf 5540  (class class class)co 6249   CCcc 9488   0cc0 9490   x. cmul 9495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-mulf 9570

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-po 4717  df-so 4718  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814

This theorem is referenced by:  ablomul  26025  mulid  26026