Theorem dvdsrmul1 13864

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 13809	. . . . 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 13856	. . 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 13775	. . . . . . . . 9 |- ( ( R e. Ring /\ X e. B /\ Z e. B ) -> ( X .x. Z ) e. B )
18	12, 15, 16, 17	syl3anc 1250	. . . . . . . 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 13858	. . . . . . 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 13778	. . . . . . . 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 1252	. . . . . . 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 4072	. . . . . 6 |- ( ( ( ( R e. Ring /\ Z e. B ) /\ X e. B ) /\ x e. B ) -> ( X .x. Z ) .|| ( ( x .x. X ) .x. Z ) )
24		oveq1 5951	. . . . . . 7 |- ( ( x .x. X ) = Y -> ( ( x .x. X ) .x. Z ) = ( Y .x. Z ) )
25	24	breq2d 4056	. . . . . 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 2623	. . . 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 1203	1 |- ( ( R e. Ring /\ Z e. B /\ X .|| Y ) -> ( X .x. Z ) .|| ( Y .x. Z ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2176   E.wrex 2485   class class class wbr 4044   ` cfv 5271  (class class class)co 5944   Basecbs 12832   .rcmulr 12910  SRingcsrg 13725   Ringcrg 13758   ||_rcdsr 13848

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-pre-ltirr 8037  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-inn 9037  df-2 9095  df-3 9096  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-plusg 12922  df-mulr 12923  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-grp 13335  df-minusg 13336  df-cmn 13622  df-abl 13623  df-mgp 13683  df-ur 13722  df-srg 13726  df-ring 13760  df-dvdsr 13851

This theorem is referenced by:  unitmulcl  13875