Theorem abvdom 18838

Description: Any ring with an absolute value is a domain, which is to say that it contains no zero divisors. (Contributed by Mario Carneiro, 10-Sep-2014.)

Hypotheses

Ref	Expression
abv0.a	|- A = (AbsVal ` R )
abvneg.b	|- B = ( Base ` R )
abvrec.z	|- .0. = ( 0g ` R )
abvdom.t	|- .x. = ( .r ` R )

Assertion

Ref	Expression
abvdom	|- ( ( F e. A /\ ( X e. B /\ X =/= .0. ) /\ ( Y e. B /\ Y =/= .0. ) ) -> ( X .x. Y ) =/= .0. )

Proof of Theorem abvdom

Step	Hyp	Ref	Expression
1		simp1 1061	. . . 4 |- ( ( F e. A /\ ( X e. B /\ X =/= .0. ) /\ ( Y e. B /\ Y =/= .0. ) ) -> F e. A )
2		simp2l 1087	. . . 4 |- ( ( F e. A /\ ( X e. B /\ X =/= .0. ) /\ ( Y e. B /\ Y =/= .0. ) ) -> X e. B )
3		simp3l 1089	. . . 4 |- ( ( F e. A /\ ( X e. B /\ X =/= .0. ) /\ ( Y e. B /\ Y =/= .0. ) ) -> Y e. B )
4		abv0.a	. . . . 5 |- A = (AbsVal ` R )
5		abvneg.b	. . . . 5 |- B = ( Base ` R )
6		abvdom.t	. . . . 5 |- .x. = ( .r ` R )
7	4, 5, 6	abvmul 18829	. . . 4 |- ( ( F e. A /\ X e. B /\ Y e. B ) -> ( F ` ( X .x. Y ) ) = ( ( F ` X ) x. ( F ` Y ) ) )
8	1, 2, 3, 7	syl3anc 1326	. . 3 |- ( ( F e. A /\ ( X e. B /\ X =/= .0. ) /\ ( Y e. B /\ Y =/= .0. ) ) -> ( F ` ( X .x. Y ) ) = ( ( F ` X ) x. ( F ` Y ) ) )
9	4, 5	abvcl 18824	. . . . . 6 |- ( ( F e. A /\ X e. B ) -> ( F ` X ) e. RR )
10	1, 2, 9	syl2anc 693	. . . . 5 |- ( ( F e. A /\ ( X e. B /\ X =/= .0. ) /\ ( Y e. B /\ Y =/= .0. ) ) -> ( F ` X ) e. RR )
11	10	recnd 10068	. . . 4 |- ( ( F e. A /\ ( X e. B /\ X =/= .0. ) /\ ( Y e. B /\ Y =/= .0. ) ) -> ( F ` X ) e. CC )
12	4, 5	abvcl 18824	. . . . . 6 |- ( ( F e. A /\ Y e. B ) -> ( F ` Y ) e. RR )
13	1, 3, 12	syl2anc 693	. . . . 5 |- ( ( F e. A /\ ( X e. B /\ X =/= .0. ) /\ ( Y e. B /\ Y =/= .0. ) ) -> ( F ` Y ) e. RR )
14	13	recnd 10068	. . . 4 |- ( ( F e. A /\ ( X e. B /\ X =/= .0. ) /\ ( Y e. B /\ Y =/= .0. ) ) -> ( F ` Y ) e. CC )
15		simp2r 1088	. . . . 5 |- ( ( F e. A /\ ( X e. B /\ X =/= .0. ) /\ ( Y e. B /\ Y =/= .0. ) ) -> X =/= .0. )
16		abvrec.z	. . . . . 6 |- .0. = ( 0g ` R )
17	4, 5, 16	abvne0 18827	. . . . 5 |- ( ( F e. A /\ X e. B /\ X =/= .0. ) -> ( F ` X ) =/= 0 )
18	1, 2, 15, 17	syl3anc 1326	. . . 4 |- ( ( F e. A /\ ( X e. B /\ X =/= .0. ) /\ ( Y e. B /\ Y =/= .0. ) ) -> ( F ` X ) =/= 0 )
19		simp3r 1090	. . . . 5 |- ( ( F e. A /\ ( X e. B /\ X =/= .0. ) /\ ( Y e. B /\ Y =/= .0. ) ) -> Y =/= .0. )
20	4, 5, 16	abvne0 18827	. . . . 5 |- ( ( F e. A /\ Y e. B /\ Y =/= .0. ) -> ( F ` Y ) =/= 0 )
21	1, 3, 19, 20	syl3anc 1326	. . . 4 |- ( ( F e. A /\ ( X e. B /\ X =/= .0. ) /\ ( Y e. B /\ Y =/= .0. ) ) -> ( F ` Y ) =/= 0 )
22	11, 14, 18, 21	mulne0d 10679	. . 3 |- ( ( F e. A /\ ( X e. B /\ X =/= .0. ) /\ ( Y e. B /\ Y =/= .0. ) ) -> ( ( F ` X ) x. ( F ` Y ) ) =/= 0 )
23	8, 22	eqnetrd 2861	. 2 |- ( ( F e. A /\ ( X e. B /\ X =/= .0. ) /\ ( Y e. B /\ Y =/= .0. ) ) -> ( F ` ( X .x. Y ) ) =/= 0 )
24	4, 16	abv0 18831	. . . . 5 |- ( F e. A -> ( F ` .0. ) = 0 )
25	1, 24	syl 17	. . . 4 |- ( ( F e. A /\ ( X e. B /\ X =/= .0. ) /\ ( Y e. B /\ Y =/= .0. ) ) -> ( F ` .0. ) = 0 )
26		fveq2 6191	. . . . 5 |- ( ( X .x. Y ) = .0. -> ( F ` ( X .x. Y ) ) = ( F ` .0. ) )
27	26	eqeq1d 2624	. . . 4 |- ( ( X .x. Y ) = .0. -> ( ( F ` ( X .x. Y ) ) = 0 <-> ( F ` .0. ) = 0 ) )
28	25, 27	syl5ibrcom 237	. . 3 |- ( ( F e. A /\ ( X e. B /\ X =/= .0. ) /\ ( Y e. B /\ Y =/= .0. ) ) -> ( ( X .x. Y ) = .0. -> ( F ` ( X .x. Y ) ) = 0 ) )
29	28	necon3d 2815	. 2 |- ( ( F e. A /\ ( X e. B /\ X =/= .0. ) /\ ( Y e. B /\ Y =/= .0. ) ) -> ( ( F ` ( X .x. Y ) ) =/= 0 -> ( X .x. Y ) =/= .0. ) )
30	23, 29	mpd 15	1 |- ( ( F e. A /\ ( X e. B /\ X =/= .0. ) /\ ( Y e. B /\ Y =/= .0. ) ) -> ( X .x. Y ) =/= .0. )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   x. cmul 9941   Basecbs 15857   .rcmulr 15942   0gc0g 16100  AbsValcabv 18816

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-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-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-map 7859  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-ico 12181  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-ring 18549  df-abv 18817

This theorem is referenced by:  abvn0b  19302