Theorem subrgdvds 14545

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 14534	. . . 4 |- ( A e. (SubRing ` R ) -> S e. Ring )
3		ringsrg 14354	. . . 4 |- ( S e. Ring -> S e. SRing )
4	2, 3	syl 14	. . 3 |- ( A e. (SubRing ` R ) -> S e. SRing )
5		reldvdsrsrg 14401	. . . 4 |- ( S e. SRing -> Rel ( ||r ` S ) )
6		subrgdvds.3	. . . . 5 |- E = ( ||r ` S )
7	6	releqi 4858	. . . 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 14540	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A = ( Base ` S ) )
11		eqid 2238	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
12	11	subrgss 14532	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
13	10, 12	eqsstrrd 3285	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( Base ` S ) C_ ( Base ` R ) )
14	13	sseld 3247	. . . . 5 |- ( A e. (SubRing ` R ) -> ( x e. ( Base ` S ) -> x e. ( Base ` R ) ) )
15		subrgrcl 14536	. . . . . . . . . 10 |- ( A e. (SubRing ` R ) -> R e. Ring )
16		eqid 2238	. . . . . . . . . . 11 |- ( .r ` R ) = ( .r ` R )
17	1, 16	ressmulrg 13501	. . . . . . . . . 10 |- ( ( A e. (SubRing ` R ) /\ R e. Ring ) -> ( .r ` R ) = ( .r ` S ) )
18	15, 17	mpdan 425	. . . . . . . . 9 |- ( A e. (SubRing ` R ) -> ( .r ` R ) = ( .r ` S ) )
19	18	oveqd 6102	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( z ( .r ` R ) x ) = ( z ( .r ` S ) x ) )
20	19	eqeq1d 2247	. . . . . . 7 |- ( A e. (SubRing ` R ) -> ( ( z ( .r ` R ) x ) = y <-> ( z ( .r ` S ) x ) = y ) )
21	20	rexbidv 2551	. . . . . 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 3313	. . . . . . 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 2239	. . . . 5 |- ( A e. (SubRing ` R ) -> ( Base ` S ) = ( Base ` S ) )
27	6	a1i 9	. . . . 5 |- ( A e. (SubRing ` R ) -> E = ( ||r ` S ) )
28		eqidd 2239	. . . . 5 |- ( A e. (SubRing ` R ) -> ( .r ` S ) = ( .r ` S ) )
29	26, 27, 4, 28	dvdsrd 14403	. . . 4 |- ( A e. (SubRing ` R ) -> ( x E y <-> ( x e. ( Base ` S ) /\ E. z e. ( Base ` S ) ( z ( .r ` S ) x ) = y ) ) )
30		eqidd 2239	. . . . 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 14354	. . . . . 6 |- ( R e. Ring -> R e. SRing )
34	15, 33	syl 14	. . . . 5 |- ( A e. (SubRing ` R ) -> R e. SRing )
35		eqidd 2239	. . . . 5 |- ( A e. (SubRing ` R ) -> ( .r ` R ) = ( .r ` R ) )
36	30, 32, 34, 35	dvdsrd 14403	. . . 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 4131	. . 3 |- ( x E y <-> <. x , y >. e. E )
39		df-br 4131	. . 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 4867	1 |- ( A e. (SubRing ` R ) -> E C_ .|| )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   E.wrex 2529   C_ wss 3220   <.cop 3712   class class class wbr 4130   Rel wrel 4779   ` cfv 5377  (class class class)co 6085   Basecbs 13354   ↾_s cress 13355   .rcmulr 13434  SRingcsrg 14269   Ringcrg 14302   ||_rcdsr 14394  SubRingcsubrg 14527

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9306  df-2 9364  df-3 9365  df-ndx 13357  df-slot 13358  df-base 13360  df-sets 13361  df-iress 13362  df-plusg 13446  df-mulr 13447  df-0g 13614  df-mgm 13678  df-sgrp 13719  df-mnd 13732  df-grp 13810  df-minusg 13811  df-subg 13975  df-cmn 14091  df-abl 14092  df-mgp 14220  df-ur 14265  df-srg 14270  df-ring 14304  df-dvdsr 14397  df-subrg 14529

This theorem is used by:  subrguss  14546