Theorem drngmul0or 18768

Description: A product is zero iff one of its factors is zero. (Contributed by NM, 8-Oct-2014.)

Hypotheses

Ref	Expression
drngmuleq0.b	|- B = ( Base ` R )
drngmuleq0.o	|- .0. = ( 0g ` R )
drngmuleq0.t	|- .x. = ( .r ` R )
drngmuleq0.r	|- ( ph -> R e. DivRing )
drngmuleq0.x	|- ( ph -> X e. B )
drngmuleq0.y	|- ( ph -> Y e. B )

Assertion

Ref	Expression
drngmul0or	|- ( ph -> ( ( X .x. Y ) = .0. <-> ( X = .0. \/ Y = .0. ) ) )

Proof of Theorem drngmul0or

Step	Hyp	Ref	Expression
1		df-ne 2795	. . . . 5 |- ( X =/= .0. <-> -. X = .0. )
2		oveq2 6658	. . . . . . . 8 |- ( ( X .x. Y ) = .0. -> ( ( ( invr ` R ) ` X ) .x. ( X .x. Y ) ) = ( ( ( invr ` R ) ` X ) .x. .0. ) )
3	2	ad2antlr 763	. . . . . . 7 |- ( ( ( ph /\ ( X .x. Y ) = .0. ) /\ X =/= .0. ) -> ( ( ( invr ` R ) ` X ) .x. ( X .x. Y ) ) = ( ( ( invr ` R ) ` X ) .x. .0. ) )
4		drngmuleq0.r	. . . . . . . . . . . 12 |- ( ph -> R e. DivRing )
5	4	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ X =/= .0. ) -> R e. DivRing )
6		drngmuleq0.x	. . . . . . . . . . . 12 |- ( ph -> X e. B )
7	6	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ X =/= .0. ) -> X e. B )
8		simpr 477	. . . . . . . . . . 11 |- ( ( ph /\ X =/= .0. ) -> X =/= .0. )
9		drngmuleq0.b	. . . . . . . . . . . 12 |- B = ( Base ` R )
10		drngmuleq0.o	. . . . . . . . . . . 12 |- .0. = ( 0g ` R )
11		drngmuleq0.t	. . . . . . . . . . . 12 |- .x. = ( .r ` R )
12		eqid 2622	. . . . . . . . . . . 12 |- ( 1r ` R ) = ( 1r ` R )
13		eqid 2622	. . . . . . . . . . . 12 |- ( invr ` R ) = ( invr ` R )
14	9, 10, 11, 12, 13	drnginvrl 18766	. . . . . . . . . . 11 |- ( ( R e. DivRing /\ X e. B /\ X =/= .0. ) -> ( ( ( invr ` R ) ` X ) .x. X ) = ( 1r ` R ) )
15	5, 7, 8, 14	syl3anc 1326	. . . . . . . . . 10 |- ( ( ph /\ X =/= .0. ) -> ( ( ( invr ` R ) ` X ) .x. X ) = ( 1r ` R ) )
16	15	oveq1d 6665	. . . . . . . . 9 |- ( ( ph /\ X =/= .0. ) -> ( ( ( ( invr ` R ) ` X ) .x. X ) .x. Y ) = ( ( 1r ` R ) .x. Y ) )
17		drngring 18754	. . . . . . . . . . . 12 |- ( R e. DivRing -> R e. Ring )
18	4, 17	syl 17	. . . . . . . . . . 11 |- ( ph -> R e. Ring )
19	18	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ X =/= .0. ) -> R e. Ring )
20	9, 10, 13	drnginvrcl 18764	. . . . . . . . . . 11 |- ( ( R e. DivRing /\ X e. B /\ X =/= .0. ) -> ( ( invr ` R ) ` X ) e. B )
21	5, 7, 8, 20	syl3anc 1326	. . . . . . . . . 10 |- ( ( ph /\ X =/= .0. ) -> ( ( invr ` R ) ` X ) e. B )
22		drngmuleq0.y	. . . . . . . . . . 11 |- ( ph -> Y e. B )
23	22	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ X =/= .0. ) -> Y e. B )
24	9, 11	ringass 18564	. . . . . . . . . 10 |- ( ( R e. Ring /\ ( ( ( invr ` R ) ` X ) e. B /\ X e. B /\ Y e. B ) ) -> ( ( ( ( invr ` R ) ` X ) .x. X ) .x. Y ) = ( ( ( invr ` R ) ` X ) .x. ( X .x. Y ) ) )
25	19, 21, 7, 23, 24	syl13anc 1328	. . . . . . . . 9 |- ( ( ph /\ X =/= .0. ) -> ( ( ( ( invr ` R ) ` X ) .x. X ) .x. Y ) = ( ( ( invr ` R ) ` X ) .x. ( X .x. Y ) ) )
26	9, 11, 12	ringlidm 18571	. . . . . . . . . . 11 |- ( ( R e. Ring /\ Y e. B ) -> ( ( 1r ` R ) .x. Y ) = Y )
27	18, 22, 26	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( ( 1r ` R ) .x. Y ) = Y )
28	27	adantr 481	. . . . . . . . 9 |- ( ( ph /\ X =/= .0. ) -> ( ( 1r ` R ) .x. Y ) = Y )
29	16, 25, 28	3eqtr3d 2664	. . . . . . . 8 |- ( ( ph /\ X =/= .0. ) -> ( ( ( invr ` R ) ` X ) .x. ( X .x. Y ) ) = Y )
30	29	adantlr 751	. . . . . . 7 |- ( ( ( ph /\ ( X .x. Y ) = .0. ) /\ X =/= .0. ) -> ( ( ( invr ` R ) ` X ) .x. ( X .x. Y ) ) = Y )
31	18	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( X .x. Y ) = .0. ) -> R e. Ring )
32	31	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ ( X .x. Y ) = .0. ) /\ X =/= .0. ) -> R e. Ring )
33	21	adantlr 751	. . . . . . . 8 |- ( ( ( ph /\ ( X .x. Y ) = .0. ) /\ X =/= .0. ) -> ( ( invr ` R ) ` X ) e. B )
34	9, 11, 10	ringrz 18588	. . . . . . . 8 |- ( ( R e. Ring /\ ( ( invr ` R ) ` X ) e. B ) -> ( ( ( invr ` R ) ` X ) .x. .0. ) = .0. )
35	32, 33, 34	syl2anc 693	. . . . . . 7 |- ( ( ( ph /\ ( X .x. Y ) = .0. ) /\ X =/= .0. ) -> ( ( ( invr ` R ) ` X ) .x. .0. ) = .0. )
36	3, 30, 35	3eqtr3d 2664	. . . . . 6 |- ( ( ( ph /\ ( X .x. Y ) = .0. ) /\ X =/= .0. ) -> Y = .0. )
37	36	ex 450	. . . . 5 |- ( ( ph /\ ( X .x. Y ) = .0. ) -> ( X =/= .0. -> Y = .0. ) )
38	1, 37	syl5bir 233	. . . 4 |- ( ( ph /\ ( X .x. Y ) = .0. ) -> ( -. X = .0. -> Y = .0. ) )
39	38	orrd 393	. . 3 |- ( ( ph /\ ( X .x. Y ) = .0. ) -> ( X = .0. \/ Y = .0. ) )
40	39	ex 450	. 2 |- ( ph -> ( ( X .x. Y ) = .0. -> ( X = .0. \/ Y = .0. ) ) )
41	9, 11, 10	ringlz 18587	. . . . 5 |- ( ( R e. Ring /\ Y e. B ) -> ( .0. .x. Y ) = .0. )
42	18, 22, 41	syl2anc 693	. . . 4 |- ( ph -> ( .0. .x. Y ) = .0. )
43		oveq1 6657	. . . . 5 |- ( X = .0. -> ( X .x. Y ) = ( .0. .x. Y ) )
44	43	eqeq1d 2624	. . . 4 |- ( X = .0. -> ( ( X .x. Y ) = .0. <-> ( .0. .x. Y ) = .0. ) )
45	42, 44	syl5ibrcom 237	. . 3 |- ( ph -> ( X = .0. -> ( X .x. Y ) = .0. ) )
46	9, 11, 10	ringrz 18588	. . . . 5 |- ( ( R e. Ring /\ X e. B ) -> ( X .x. .0. ) = .0. )
47	18, 6, 46	syl2anc 693	. . . 4 |- ( ph -> ( X .x. .0. ) = .0. )
48		oveq2 6658	. . . . 5 |- ( Y = .0. -> ( X .x. Y ) = ( X .x. .0. ) )
49	48	eqeq1d 2624	. . . 4 |- ( Y = .0. -> ( ( X .x. Y ) = .0. <-> ( X .x. .0. ) = .0. ) )
50	47, 49	syl5ibrcom 237	. . 3 |- ( ph -> ( Y = .0. -> ( X .x. Y ) = .0. ) )
51	45, 50	jaod 395	. 2 |- ( ph -> ( ( X = .0. \/ Y = .0. ) -> ( X .x. Y ) = .0. ) )
52	40, 51	impbid 202	1 |- ( ph -> ( ( X .x. Y ) = .0. <-> ( X = .0. \/ Y = .0. ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   Basecbs 15857   .rcmulr 15942   0gc0g 16100   1rcur 18501   Ringcrg 18547   invrcinvr 18671   DivRingcdr 18747

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  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-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  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  df-nn 11021  df-2 11079  df-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-drng 18749

This theorem is referenced by:  drngmulne0  18769  drngmuleq0  18770