Theorem subrgdvds 13731

Description: If an element divides another in a subring, then it also divides the other in the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.)

Hypotheses

Ref	Expression
subrgdvds.1	|- S = ( R ↾s A )
subrgdvds.2	|- .|| = ( ||r ` R )
subrgdvds.3	|- E = ( ||r ` S )

Assertion

Ref	Expression
subrgdvds	|- ( A e. (SubRing ` R ) -> E C_ .|| )

Proof of Theorem subrgdvds

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		subrgdvds.1	. . . . 5 |- S = ( R ↾s A )
2	1	subrgring 13720	. . . 4 |- ( A e. (SubRing ` R ) -> S e. Ring )
3		ringsrg 13543	. . . 4 |- ( S e. Ring -> S e. SRing )
4	2, 3	syl 14	. . 3 |- ( A e. (SubRing ` R ) -> S e. SRing )
5		reldvdsrsrg 13588	. . . 4 |- ( S e. SRing -> Rel ( ||r ` S ) )
6		subrgdvds.3	. . . . 5 |- E = ( ||r ` S )
7	6	releqi 4742	. . . 4 |- ( Rel E <-> Rel ( ||r ` S ) )
8	5, 7	sylibr 134	. . 3 |- ( S e. SRing -> Rel E )
9	4, 8	syl 14	. 2 |- ( A e. (SubRing ` R ) -> Rel E )
10	1	subrgbas 13726	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A = ( Base ` S ) )
11		eqid 2193	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
12	11	subrgss 13718	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
13	10, 12	eqsstrrd 3216	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( Base ` S ) C_ ( Base ` R ) )
14	13	sseld 3178	. . . . 5 |- ( A e. (SubRing ` R ) -> ( x e. ( Base ` S ) -> x e. ( Base ` R ) ) )
15		subrgrcl 13722	. . . . . . . . . 10 |- ( A e. (SubRing ` R ) -> R e. Ring )
16		eqid 2193	. . . . . . . . . . 11 |- ( .r ` R ) = ( .r ` R )
17	1, 16	ressmulrg 12762	. . . . . . . . . 10 |- ( ( A e. (SubRing ` R ) /\ R e. Ring ) -> ( .r ` R ) = ( .r ` S ) )
18	15, 17	mpdan 421	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> ( .r ` R ) = ( .r ` S ) )
19	18	oveqd 5935	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( z ( .r ` R ) x ) = ( z ( .r ` S ) x ) )
20	19	eqeq1d 2202	. . . . . . 7 |- ( A e. (SubRing ` R ) -> ( ( z ( .r ` R ) x ) = y <-> ( z ( .r ` S ) x ) = y ) )
21	20	rexbidv 2495	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( E. z e. ( Base ` S ) ( z ( .r ` R ) x ) = y <-> E. z e. ( Base ` S ) ( z ( .r ` S ) x ) = y ) )
22		ssrexv 3244	. . . . . . 7 |- ( ( Base ` S ) C_ ( Base ` R ) -> ( E. z e. ( Base ` S ) ( z ( .r ` R ) x ) = y -> E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) )
23	13, 22	syl 14	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( E. z e. ( Base ` S ) ( z ( .r ` R ) x ) = y -> E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) )
24	21, 23	sylbird 170	. . . . 5 |- ( A e. (SubRing ` R ) -> ( E. z e. ( Base ` S ) ( z ( .r ` S ) x ) = y -> E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) )
25	14, 24	anim12d 335	. . . 4 |- ( A e. (SubRing ` R ) -> ( ( x e. ( Base ` S ) /\ E. z e. ( Base ` S ) ( z ( .r ` S ) x ) = y ) -> ( x e. ( Base ` R ) /\ E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) ) )
26		eqidd 2194	. . . . 5 |- ( A e. (SubRing ` R ) -> ( Base ` S ) = ( Base ` S ) )
27	6	a1i 9	. . . . 5 |- ( A e. (SubRing ` R ) -> E = ( ||r ` S ) )
28		eqidd 2194	. . . . 5 |- ( A e. (SubRing ` R ) -> ( .r ` S ) = ( .r ` S ) )
29	26, 27, 4, 28	dvdsrd 13590	. . . 4 |- ( A e. (SubRing ` R ) -> ( x E y <-> ( x e. ( Base ` S ) /\ E. z e. ( Base ` S ) ( z ( .r ` S ) x ) = y ) ) )
30		eqidd 2194	. . . . 5 |- ( A e. (SubRing ` R ) -> ( Base ` R ) = ( Base ` R ) )
31		subrgdvds.2	. . . . . 6 |- .|| = ( ||r ` R )
32	31	a1i 9	. . . . 5 |- ( A e. (SubRing ` R ) -> .|| = ( ||r ` R ) )
33		ringsrg 13543	. . . . . 6 |- ( R e. Ring -> R e. SRing )
34	15, 33	syl 14	. . . . 5 |- ( A e. (SubRing ` R ) -> R e. SRing )
35		eqidd 2194	. . . . 5 |- ( A e. (SubRing ` R ) -> ( .r ` R ) = ( .r ` R ) )
36	30, 32, 34, 35	dvdsrd 13590	. . . 4 |- ( A e. (SubRing ` R ) -> ( x .|| y <-> ( x e. ( Base ` R ) /\ E. z e. ( Base ` R ) ( z ( .r ` R ) x ) = y ) ) )
37	25, 29, 36	3imtr4d 203	. . 3 |- ( A e. (SubRing ` R ) -> ( x E y -> x .|| y ) )
38		df-br 4030	. . 3 |- ( x E y <-> <. x , y >. e. E )
39		df-br 4030	. . 3 |- ( x .|| y <-> <. x , y >. e. .|| )
40	37, 38, 39	3imtr3g 204	. 2 |- ( A e. (SubRing ` R ) -> ( <. x , y >. e. E -> <. x , y >. e. .|| ) )
41	9, 40	relssdv 4751	1 |- ( A e. (SubRing ` R ) -> E C_ .|| )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   E.wrex 2473   C_ wss 3153   <.cop 3621   class class class wbr 4029   Rel wrel 4664   ` cfv 5254  (class class class)co 5918   Basecbs 12618   ↾_s cress 12619   .rcmulr 12696  SRingcsrg 13459   Ringcrg 13492   ||_rcdsr 13582  SubRingcsubrg 13713

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 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988

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 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-mulr 12709  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-subg 13240  df-cmn 13356  df-abl 13357  df-mgp 13417  df-ur 13456  df-srg 13460  df-ring 13494  df-dvdsr 13585  df-subrg 13715

This theorem is referenced by:  subrguss  13732