Theorem dvdsrmul1 13276

Description: The divisibility relation is preserved under right-multiplication. (Contributed by Mario Carneiro, 1-Dec-2014.)

Hypotheses

Ref	Expression
dvdsr.1	|- B = ( Base ` R )
dvdsr.2	|- .|| = ( ||r ` R )
dvdsrmul1.3	|- .x. = ( .r ` R )

Assertion

Ref	Expression
dvdsrmul1	|- ( ( R e. Ring /\ Z e. B /\ X .|| Y ) -> ( X .x. Z ) .|| ( Y .x. Z ) )

Proof of Theorem dvdsrmul1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvdsr.1	. . . . 5 |- B = ( Base ` R )
2	1	a1i 9	. . . 4 |- ( ( R e. Ring /\ Z e. B ) -> B = ( Base ` R ) )
3		dvdsr.2	. . . . 5 |- .|| = ( ||r ` R )
4	3	a1i 9	. . . 4 |- ( ( R e. Ring /\ Z e. B ) -> .|| = ( ||r ` R ) )
5		ringsrg 13229	. . . . 5 |- ( R e. Ring -> R e. SRing )
6	5	adantr 276	. . . 4 |- ( ( R e. Ring /\ Z e. B ) -> R e. SRing )
7		dvdsrmul1.3	. . . . 5 |- .x. = ( .r ` R )
8	7	a1i 9	. . . 4 |- ( ( R e. Ring /\ Z e. B ) -> .x. = ( .r ` R ) )
9	2, 4, 6, 8	dvdsrd 13268	. . 3 |- ( ( R e. Ring /\ Z e. B ) -> ( X .|| Y <-> ( X e. B /\ E. x e. B ( x .x. X ) = Y ) ) )
10	1	a1i 9	. . . . . . . 8 |- ( ( ( ( R e. Ring /\ Z e. B ) /\ X e. B ) /\ x e. B ) -> B = ( Base ` R ) )
11	3	a1i 9	. . . . . . . 8 |- ( ( ( ( R e. Ring /\ Z e. B ) /\ X e. B ) /\ x e. B ) -> .|| = ( ||r ` R ) )
12		simplll 533	. . . . . . . . 9 |- ( ( ( ( R e. Ring /\ Z e. B ) /\ X e. B ) /\ x e. B ) -> R e. Ring )
13	12, 5	syl 14	. . . . . . . 8 |- ( ( ( ( R e. Ring /\ Z e. B ) /\ X e. B ) /\ x e. B ) -> R e. SRing )
14	7	a1i 9	. . . . . . . 8 |- ( ( ( ( R e. Ring /\ Z e. B ) /\ X e. B ) /\ x e. B ) -> .x. = ( .r ` R ) )
15		simplr 528	. . . . . . . . 9 |- ( ( ( ( R e. Ring /\ Z e. B ) /\ X e. B ) /\ x e. B ) -> X e. B )
16		simpllr 534	. . . . . . . . 9 |- ( ( ( ( R e. Ring /\ Z e. B ) /\ X e. B ) /\ x e. B ) -> Z e. B )
17	1, 7	ringcl 13201	. . . . . . . . 9 |- ( ( R e. Ring /\ X e. B /\ Z e. B ) -> ( X .x. Z ) e. B )
18	12, 15, 16, 17	syl3anc 1238	. . . . . . . 8 |- ( ( ( ( R e. Ring /\ Z e. B ) /\ X e. B ) /\ x e. B ) -> ( X .x. Z ) e. B )
19		simpr 110	. . . . . . . 8 |- ( ( ( ( R e. Ring /\ Z e. B ) /\ X e. B ) /\ x e. B ) -> x e. B )
20	10, 11, 13, 14, 18, 19	dvdsrmuld 13270	. . . . . . 7 |- ( ( ( ( R e. Ring /\ Z e. B ) /\ X e. B ) /\ x e. B ) -> ( X .x. Z ) .|| ( x .x. ( X .x. Z ) ) )
21	1, 7	ringass 13204	. . . . . . . 8 |- ( ( R e. Ring /\ ( x e. B /\ X e. B /\ Z e. B ) ) -> ( ( x .x. X ) .x. Z ) = ( x .x. ( X .x. Z ) ) )
22	12, 19, 15, 16, 21	syl13anc 1240	. . . . . . 7 |- ( ( ( ( R e. Ring /\ Z e. B ) /\ X e. B ) /\ x e. B ) -> ( ( x .x. X ) .x. Z ) = ( x .x. ( X .x. Z ) ) )
23	20, 22	breqtrrd 4033	. . . . . 6 |- ( ( ( ( R e. Ring /\ Z e. B ) /\ X e. B ) /\ x e. B ) -> ( X .x. Z ) .|| ( ( x .x. X ) .x. Z ) )
24		oveq1 5884	. . . . . . 7 |- ( ( x .x. X ) = Y -> ( ( x .x. X ) .x. Z ) = ( Y .x. Z ) )
25	24	breq2d 4017	. . . . . 6 |- ( ( x .x. X ) = Y -> ( ( X .x. Z ) .|| ( ( x .x. X ) .x. Z ) <-> ( X .x. Z ) .|| ( Y .x. Z ) ) )
26	23, 25	syl5ibcom 155	. . . . 5 |- ( ( ( ( R e. Ring /\ Z e. B ) /\ X e. B ) /\ x e. B ) -> ( ( x .x. X ) = Y -> ( X .x. Z ) .|| ( Y .x. Z ) ) )
27	26	rexlimdva 2594	. . . 4 |- ( ( ( R e. Ring /\ Z e. B ) /\ X e. B ) -> ( E. x e. B ( x .x. X ) = Y -> ( X .x. Z ) .|| ( Y .x. Z ) ) )
28	27	expimpd 363	. . 3 |- ( ( R e. Ring /\ Z e. B ) -> ( ( X e. B /\ E. x e. B ( x .x. X ) = Y ) -> ( X .x. Z ) .|| ( Y .x. Z ) ) )
29	9, 28	sylbid 150	. 2 |- ( ( R e. Ring /\ Z e. B ) -> ( X .|| Y -> ( X .x. Z ) .|| ( Y .x. Z ) ) )
30	29	3impia 1200	1 |- ( ( R e. Ring /\ Z e. B /\ X .|| Y ) -> ( X .x. Z ) .|| ( Y .x. Z ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   E.wrex 2456   class class class wbr 4005   ` cfv 5218  (class class class)co 5877   Basecbs 12464   .rcmulr 12539  SRingcsrg 13151   Ringcrg 13184   ||_rcdsr 13260

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-3 8981  df-ndx 12467  df-slot 12468  df-base 12470  df-sets 12471  df-plusg 12551  df-mulr 12552  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885  df-minusg 12886  df-cmn 13095  df-abl 13096  df-mgp 13136  df-ur 13148  df-srg 13152  df-ring 13186  df-dvdsr 13263

This theorem is referenced by:  unitmulcl  13287