Theorem dvdsrmul1 13601

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 13546	. . . . 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 13593	. . 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 13512	. . . . . . . . 9 |- ( ( R e. Ring /\ X e. B /\ Z e. B ) -> ( X .x. Z ) e. B )
18	12, 15, 16, 17	syl3anc 1249	. . . . . . . 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 13595	. . . . . . 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 13515	. . . . . . . 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 1251	. . . . . . 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 4058	. . . . . 6 |- ( ( ( ( R e. Ring /\ Z e. B ) /\ X e. B ) /\ x e. B ) -> ( X .x. Z ) .|| ( ( x .x. X ) .x. Z ) )
24		oveq1 5926	. . . . . . 7 |- ( ( x .x. X ) = Y -> ( ( x .x. X ) .x. Z ) = ( Y .x. Z ) )
25	24	breq2d 4042	. . . . . 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 2611	. . . 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 1202	1 |- ( ( R e. Ring /\ Z e. B /\ X .|| Y ) -> ( X .x. Z ) .|| ( Y .x. Z ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2164   E.wrex 2473   class class class wbr 4030   ` cfv 5255  (class class class)co 5919   Basecbs 12621   .rcmulr 12699  SRingcsrg 13462   Ringcrg 13495   ||_rcdsr 13585

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-3 9044  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-plusg 12711  df-mulr 12712  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078  df-minusg 13079  df-cmn 13359  df-abl 13360  df-mgp 13420  df-ur 13459  df-srg 13463  df-ring 13497  df-dvdsr 13588

This theorem is referenced by:  unitmulcl  13612