Theorem domnmuln0 14525

Description: In a domain, a product of nonzero elements is nonzero. (Contributed by Mario Carneiro, 6-May-2015.)

Hypotheses

Ref	Expression
domneq0.b	|- B = ( Base ` R )
domneq0.t	|- .x. = ( .r ` R )
domneq0.z	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
domnmuln0	|- ( ( R e. Domn /\ ( X e. B /\ X =/= .0. ) /\ ( Y e. B /\ Y =/= .0. ) ) -> ( X .x. Y ) =/= .0. )

Proof of Theorem domnmuln0

Step	Hyp	Ref	Expression
1		an4 588	. . 3 |- ( ( ( X e. B /\ X =/= .0. ) /\ ( Y e. B /\ Y =/= .0. ) ) <-> ( ( X e. B /\ Y e. B ) /\ ( X =/= .0. /\ Y =/= .0. ) ) )
2		neanior 2501	. . . . . 6 |- ( ( X =/= .0. /\ Y =/= .0. ) <-> -. ( X = .0. \/ Y = .0. ) )
3		domneq0.b	. . . . . . . . 9 |- B = ( Base ` R )
4		domneq0.t	. . . . . . . . 9 |- .x. = ( .r ` R )
5		domneq0.z	. . . . . . . . 9 |- .0. = ( 0g ` R )
6	3, 4, 5	domneq0 14524	. . . . . . . 8 |- ( ( R e. Domn /\ X e. B /\ Y e. B ) -> ( ( X .x. Y ) = .0. <-> ( X = .0. \/ Y = .0. ) ) )
7	6	3expb 1231	. . . . . . 7 |- ( ( R e. Domn /\ ( X e. B /\ Y e. B ) ) -> ( ( X .x. Y ) = .0. <-> ( X = .0. \/ Y = .0. ) ) )
8	7	necon3abid 2453	. . . . . 6 |- ( ( R e. Domn /\ ( X e. B /\ Y e. B ) ) -> ( ( X .x. Y ) =/= .0. <-> -. ( X = .0. \/ Y = .0. ) ) )
9	2, 8	bitr4id 199	. . . . 5 |- ( ( R e. Domn /\ ( X e. B /\ Y e. B ) ) -> ( ( X =/= .0. /\ Y =/= .0. ) <-> ( X .x. Y ) =/= .0. ) )
10	9	biimpd 144	. . . 4 |- ( ( R e. Domn /\ ( X e. B /\ Y e. B ) ) -> ( ( X =/= .0. /\ Y =/= .0. ) -> ( X .x. Y ) =/= .0. ) )
11	10	expimpd 363	. . 3 |- ( R e. Domn -> ( ( ( X e. B /\ Y e. B ) /\ ( X =/= .0. /\ Y =/= .0. ) ) -> ( X .x. Y ) =/= .0. ) )
12	1, 11	biimtrid 152	. 2 |- ( R e. Domn -> ( ( ( X e. B /\ X =/= .0. ) /\ ( Y e. B /\ Y =/= .0. ) ) -> ( X .x. Y ) =/= .0. ) )
13	12	3impib 1228	1 |- ( ( R e. Domn /\ ( X e. B /\ X =/= .0. ) /\ ( Y e. B /\ Y =/= .0. ) ) -> ( X .x. Y ) =/= .0. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   ` cfv 5358  (class class class)co 6059   Basecbs 13301   .rcmulr 13380   0gc0g 13558  Domncdomn 14507

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4231  ax-sep 4234  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4665  ax-cnex 8235  ax-resscn 8236  ax-1cn 8237  ax-1re 8238  ax-icn 8239  ax-addcl 8240  ax-addrcl 8241  ax-mulcl 8242  ax-addcom 8244  ax-addass 8246  ax-i2m1 8249  ax-0lt1 8250  ax-0id 8252  ax-rnegex 8253  ax-pre-ltirr 8256  ax-pre-ltadd 8260

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-id 4420  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-rn 4766  df-res 4767  df-ima 4768  df-iota 5318  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-riota 6012  df-ov 6062  df-oprab 6063  df-mpo 6064  df-pnf 8327  df-mnf 8328  df-ltxr 8330  df-inn 9259  df-2 9317  df-3 9318  df-ndx 13304  df-slot 13305  df-base 13307  df-sets 13308  df-plusg 13392  df-mulr 13393  df-0g 13560  df-mgm 13624  df-sgrp 13670  df-mnd 13683  df-grp 13763  df-minusg 13764  df-mgp 14165  df-ring 14246  df-nzr 14430  df-domn 14510

This theorem is referenced by: (None)