Theorem subrgdvds 13512

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 13501	. . . 4 |- ( A e. (SubRing ` R ) -> S e. Ring )
3		ringsrg 13354	. . . 4 |- ( S e. Ring -> S e. SRing )
4	2, 3	syl 14	. . 3 |- ( A e. (SubRing ` R ) -> S e. SRing )
5		reldvdsrsrg 13397	. . . 4 |- ( S e. SRing -> Rel ( ||r ` S ) )
6		subrgdvds.3	. . . . 5 |- E = ( ||r ` S )
7	6	releqi 4721	. . . 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 13507	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A = ( Base ` S ) )
11		eqid 2187	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
12	11	subrgss 13499	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
13	10, 12	eqsstrrd 3204	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( Base ` S ) C_ ( Base ` R ) )
14	13	sseld 3166	. . . . 5 |- ( A e. (SubRing ` R ) -> ( x e. ( Base ` S ) -> x e. ( Base ` R ) ) )
15		subrgrcl 13503	. . . . . . . . . 10 |- ( A e. (SubRing ` R ) -> R e. Ring )
16		eqid 2187	. . . . . . . . . . 11 |- ( .r ` R ) = ( .r ` R )
17	1, 16	ressmulrg 12618	. . . . . . . . . 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 5905	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( z ( .r ` R ) x ) = ( z ( .r ` S ) x ) )
20	19	eqeq1d 2196	. . . . . . 7 |- ( A e. (SubRing ` R ) -> ( ( z ( .r ` R ) x ) = y <-> ( z ( .r ` S ) x ) = y ) )
21	20	rexbidv 2488	. . . . . 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 3232	. . . . . . 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 2188	. . . . 5 |- ( A e. (SubRing ` R ) -> ( Base ` S ) = ( Base ` S ) )
27	6	a1i 9	. . . . 5 |- ( A e. (SubRing ` R ) -> E = ( ||r ` S ) )
28		eqidd 2188	. . . . 5 |- ( A e. (SubRing ` R ) -> ( .r ` S ) = ( .r ` S ) )
29	26, 27, 4, 28	dvdsrd 13399	. . . 4 |- ( A e. (SubRing ` R ) -> ( x E y <-> ( x e. ( Base ` S ) /\ E. z e. ( Base ` S ) ( z ( .r ` S ) x ) = y ) ) )
30		eqidd 2188	. . . . 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 13354	. . . . . 6 |- ( R e. Ring -> R e. SRing )
34	15, 33	syl 14	. . . . 5 |- ( A e. (SubRing ` R ) -> R e. SRing )
35		eqidd 2188	. . . . 5 |- ( A e. (SubRing ` R ) -> ( .r ` R ) = ( .r ` R ) )
36	30, 32, 34, 35	dvdsrd 13399	. . . 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 4016	. . 3 |- ( x E y <-> <. x , y >. e. E )
39		df-br 4016	. . 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 4730	1 |- ( A e. (SubRing ` R ) -> E C_ .|| )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1363   e. wcel 2158   E.wrex 2466   C_ wss 3141   <.cop 3607   class class class wbr 4015   Rel wrel 4643   ` cfv 5228  (class class class)co 5888   Basecbs 12476   ↾_s cress 12477   .rcmulr 12552  SRingcsrg 13272   Ringcrg 13305   ||_rcdsr 13391  SubRingcsubrg 13494

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-addcom 7925  ax-addass 7927  ax-i2m1 7930  ax-0lt1 7931  ax-0id 7933  ax-rnegex 7934  ax-pre-ltirr 7937  ax-pre-lttrn 7939  ax-pre-ltadd 7941

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8008  df-mnf 8009  df-ltxr 8011  df-inn 8934  df-2 8992  df-3 8993  df-ndx 12479  df-slot 12480  df-base 12482  df-sets 12483  df-iress 12484  df-plusg 12564  df-mulr 12565  df-0g 12725  df-mgm 12794  df-sgrp 12827  df-mnd 12840  df-grp 12909  df-minusg 12910  df-subg 13070  df-cmn 13180  df-abl 13181  df-mgp 13230  df-ur 13269  df-srg 13273  df-ring 13307  df-dvdsr 13394  df-subrg 13496

This theorem is referenced by:  subrguss  13513