Theorem dvdsrmul1 13734

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 13679	. . . . 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 13726	. . 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 13645	. . . . . . . . 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 13728	. . . . . . 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 13648	. . . . . . . 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 4062	. . . . . 6 |- ( ( ( ( R e. Ring /\ Z e. B ) /\ X e. B ) /\ x e. B ) -> ( X .x. Z ) .|| ( ( x .x. X ) .x. Z ) )
24		oveq1 5932	. . . . . . 7 |- ( ( x .x. X ) = Y -> ( ( x .x. X ) .x. Z ) = ( Y .x. Z ) )
25	24	breq2d 4046	. . . . . 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 2614	. . . 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 2167   E.wrex 2476   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   Basecbs 12703   .rcmulr 12781  SRingcsrg 13595   Ringcrg 13628   ||_rcdsr 13718

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-3 9067  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-plusg 12793  df-mulr 12794  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-minusg 13206  df-cmn 13492  df-abl 13493  df-mgp 13553  df-ur 13592  df-srg 13596  df-ring 13630  df-dvdsr 13721

This theorem is referenced by:  unitmulcl  13745