Theorem mulnzcnopr 10286

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 9645	. . . . 5 |- x. : ( CC X. CC ) --> CC
2		ffnov 6427	. . . . 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 213	. . . 4 |- ( x. Fn ( CC X. CC ) /\ A. x e. CC A. y e. CC ( x x. y ) e. CC )
4	3	simpli 464	. . 3 |- x. Fn ( CC X. CC )
5		difss 3572	. . . 4 |- ( CC \ { 0 } ) C_ CC
6		xpss12 4959	. . . 4 |- ( ( ( CC \ { 0 } ) C_ CC /\ ( CC \ { 0 } ) C_ CC ) -> ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) C_ ( CC X. CC ) )
7	5, 5, 6	mp2an 683	. . 3 |- ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) C_ ( CC X. CC )
8		fnssres 5711	. . 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 683	. 2 |- ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) Fn ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) )
10		ovres 6463	. . . 4 |- ( ( x e. ( CC \ { 0 } ) /\ y e. ( CC \ { 0 } ) ) -> ( x ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) y ) = ( x x. y ) )
11		eldifsn 4110	. . . . . 6 |- ( x e. ( CC \ { 0 } ) <-> ( x e. CC /\ x =/= 0 ) )
12		eldifsn 4110	. . . . . 6 |- ( y e. ( CC \ { 0 } ) <-> ( y e. CC /\ y =/= 0 ) )
13		mulcl 9649	. . . . . . . 8 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
14	13	ad2ant2r 758	. . . . . . 7 |- ( ( ( x e. CC /\ x =/= 0 ) /\ ( y e. CC /\ y =/= 0 ) ) -> ( x x. y ) e. CC )
15		mulne0 10282	. . . . . . 7 |- ( ( ( x e. CC /\ x =/= 0 ) /\ ( y e. CC /\ y =/= 0 ) ) -> ( x x. y ) =/= 0 )
16	14, 15	jca 539	. . . . . 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 486	. . . . 5 |- ( ( x e. ( CC \ { 0 } ) /\ y e. ( CC \ { 0 } ) ) -> ( ( x x. y ) e. CC /\ ( x x. y ) =/= 0 ) )
18		eldifsn 4110	. . . . 5 |- ( ( x x. y ) e. ( CC \ { 0 } ) <-> ( ( x x. y ) e. CC /\ ( x x. y ) =/= 0 ) )
19	17, 18	sylibr 217	. . . 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 2827	. 2 |- A. x e. ( CC \ { 0 } ) A. y e. ( CC \ { 0 } ) ( x ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) y ) e. ( CC \ { 0 } )
22		ffnov 6427	. 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 936	1 |- ( x. |` ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) ) : ( ( CC \ { 0 } ) X. ( CC \ { 0 } ) ) --> ( CC \ { 0 } )

Syntax hints:   /\ wa 375   e. wcel 1898   =/= wne 2633   A.wral 2749   \ cdif 3413   C_ wss 3416   {csn 3980   X. cxp 4851   |` cres 4855   Fn wfn 5596   -->wf 5597  (class class class)co 6315   CCcc 9563   0cc0 9565   x. cmul 9570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642  ax-mulf 9645

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-po 4774  df-so 4775  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-er 7389  df-en 7596  df-dom 7597  df-sdom 7598  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889

This theorem is referenced by:  ablomul  26132  mulid  26133