Theorem mul0or 10667

Description: If a product is zero, one of its factors must be zero. Theorem I.11 of [Apostol] p. 18. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Assertion

Ref	Expression
mul0or	|- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) = 0 <-> ( A = 0 \/ B = 0 ) ) )

Proof of Theorem mul0or

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . . . . . . 11 |- ( ( A e. CC /\ B e. CC ) -> B e. CC )
2	1	adantr 481	. . . . . . . . . 10 |- ( ( ( A e. CC /\ B e. CC ) /\ B =/= 0 ) -> B e. CC )
3	2	mul02d 10234	. . . . . . . . 9 |- ( ( ( A e. CC /\ B e. CC ) /\ B =/= 0 ) -> ( 0 x. B ) = 0 )
4	3	eqeq2d 2632	. . . . . . . 8 |- ( ( ( A e. CC /\ B e. CC ) /\ B =/= 0 ) -> ( ( A x. B ) = ( 0 x. B ) <-> ( A x. B ) = 0 ) )
5		simpl 473	. . . . . . . . . 10 |- ( ( A e. CC /\ B e. CC ) -> A e. CC )
6	5	adantr 481	. . . . . . . . 9 |- ( ( ( A e. CC /\ B e. CC ) /\ B =/= 0 ) -> A e. CC )
7		0cnd 10033	. . . . . . . . 9 |- ( ( ( A e. CC /\ B e. CC ) /\ B =/= 0 ) -> 0 e. CC )
8		simpr 477	. . . . . . . . 9 |- ( ( ( A e. CC /\ B e. CC ) /\ B =/= 0 ) -> B =/= 0 )
9	6, 7, 2, 8	mulcan2d 10661	. . . . . . . 8 |- ( ( ( A e. CC /\ B e. CC ) /\ B =/= 0 ) -> ( ( A x. B ) = ( 0 x. B ) <-> A = 0 ) )
10	4, 9	bitr3d 270	. . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ B =/= 0 ) -> ( ( A x. B ) = 0 <-> A = 0 ) )
11	10	biimpd 219	. . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ B =/= 0 ) -> ( ( A x. B ) = 0 -> A = 0 ) )
12	11	impancom 456	. . . . 5 |- ( ( ( A e. CC /\ B e. CC ) /\ ( A x. B ) = 0 ) -> ( B =/= 0 -> A = 0 ) )
13	12	necon1bd 2812	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( A x. B ) = 0 ) -> ( -. A = 0 -> B = 0 ) )
14	13	orrd 393	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( A x. B ) = 0 ) -> ( A = 0 \/ B = 0 ) )
15	14	ex 450	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) = 0 -> ( A = 0 \/ B = 0 ) ) )
16	1	mul02d 10234	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( 0 x. B ) = 0 )
17		oveq1 6657	. . . . 5 |- ( A = 0 -> ( A x. B ) = ( 0 x. B ) )
18	17	eqeq1d 2624	. . . 4 |- ( A = 0 -> ( ( A x. B ) = 0 <-> ( 0 x. B ) = 0 ) )
19	16, 18	syl5ibrcom 237	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A = 0 -> ( A x. B ) = 0 ) )
20	5	mul01d 10235	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A x. 0 ) = 0 )
21		oveq2 6658	. . . . 5 |- ( B = 0 -> ( A x. B ) = ( A x. 0 ) )
22	21	eqeq1d 2624	. . . 4 |- ( B = 0 -> ( ( A x. B ) = 0 <-> ( A x. 0 ) = 0 ) )
23	20, 22	syl5ibrcom 237	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( B = 0 -> ( A x. B ) = 0 ) )
24	19, 23	jaod 395	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( A = 0 \/ B = 0 ) -> ( A x. B ) = 0 ) )
25	15, 24	impbid 202	1 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) = 0 <-> ( A = 0 \/ B = 0 ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794  (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

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-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:  mulne0b  10668  msq0i  10674  mul0ori  10675  msq0d  10676  mul0ord  10677  coseq1  24274  efrlim  24696