Theorem dvdsrmul1 14247

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 14191	. . . . 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 14239	. . 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 535	. . . . . . . . 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 529	. . . . . . . . 9 |- ( ( ( ( R e. Ring /\ Z e. B ) /\ X e. B ) /\ x e. B ) -> X e. B )
16		simpllr 536	. . . . . . . . 9 |- ( ( ( ( R e. Ring /\ Z e. B ) /\ X e. B ) /\ x e. B ) -> Z e. B )
17	1, 7	ringcl 14157	. . . . . . . . 9 |- ( ( R e. Ring /\ X e. B /\ Z e. B ) -> ( X .x. Z ) e. B )
18	12, 15, 16, 17	syl3anc 1274	. . . . . . . 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 14241	. . . . . . 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 14160	. . . . . . . 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 1276	. . . . . . 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 4137	. . . . . 6 |- ( ( ( ( R e. Ring /\ Z e. B ) /\ X e. B ) /\ x e. B ) -> ( X .x. Z ) .|| ( ( x .x. X ) .x. Z ) )
24		oveq1 6057	. . . . . . 7 |- ( ( x .x. X ) = Y -> ( ( x .x. X ) .x. Z ) = ( Y .x. Z ) )
25	24	breq2d 4121	. . . . . 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 2660	. . . 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 1227	1 |- ( ( R e. Ring /\ Z e. B /\ X .|| Y ) -> ( X .x. Z ) .|| ( Y .x. Z ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   E.wrex 2521   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   Basecbs 13212   .rcmulr 13291  SRingcsrg 14107   Ringcrg 14140   ||_rcdsr 14230

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-ltadd 8243

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-plusg 13303  df-mulr 13304  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-grp 13716  df-minusg 13717  df-cmn 14003  df-abl 14004  df-mgp 14065  df-ur 14104  df-srg 14108  df-ring 14142  df-dvdsr 14233

This theorem is referenced by:  unitmulcl  14258