Theorem subrgdvds 14207

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 14196	. . . 4 |- ( A e. (SubRing ` R ) -> S e. Ring )
3		ringsrg 14018	. . . 4 |- ( S e. Ring -> S e. SRing )
4	2, 3	syl 14	. . 3 |- ( A e. (SubRing ` R ) -> S e. SRing )
5		reldvdsrsrg 14064	. . . 4 |- ( S e. SRing -> Rel ( ||r ` S ) )
6		subrgdvds.3	. . . . 5 |- E = ( ||r ` S )
7	6	releqi 4802	. . . 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 14202	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A = ( Base ` S ) )
11		eqid 2229	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
12	11	subrgss 14194	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
13	10, 12	eqsstrrd 3261	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( Base ` S ) C_ ( Base ` R ) )
14	13	sseld 3223	. . . . 5 |- ( A e. (SubRing ` R ) -> ( x e. ( Base ` S ) -> x e. ( Base ` R ) ) )
15		subrgrcl 14198	. . . . . . . . . 10 |- ( A e. (SubRing ` R ) -> R e. Ring )
16		eqid 2229	. . . . . . . . . . 11 |- ( .r ` R ) = ( .r ` R )
17	1, 16	ressmulrg 13186	. . . . . . . . . 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 6024	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( z ( .r ` R ) x ) = ( z ( .r ` S ) x ) )
20	19	eqeq1d 2238	. . . . . . 7 |- ( A e. (SubRing ` R ) -> ( ( z ( .r ` R ) x ) = y <-> ( z ( .r ` S ) x ) = y ) )
21	20	rexbidv 2531	. . . . . 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 3289	. . . . . . 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 2230	. . . . 5 |- ( A e. (SubRing ` R ) -> ( Base ` S ) = ( Base ` S ) )
27	6	a1i 9	. . . . 5 |- ( A e. (SubRing ` R ) -> E = ( ||r ` S ) )
28		eqidd 2230	. . . . 5 |- ( A e. (SubRing ` R ) -> ( .r ` S ) = ( .r ` S ) )
29	26, 27, 4, 28	dvdsrd 14066	. . . 4 |- ( A e. (SubRing ` R ) -> ( x E y <-> ( x e. ( Base ` S ) /\ E. z e. ( Base ` S ) ( z ( .r ` S ) x ) = y ) ) )
30		eqidd 2230	. . . . 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 14018	. . . . . 6 |- ( R e. Ring -> R e. SRing )
34	15, 33	syl 14	. . . . 5 |- ( A e. (SubRing ` R ) -> R e. SRing )
35		eqidd 2230	. . . . 5 |- ( A e. (SubRing ` R ) -> ( .r ` R ) = ( .r ` R ) )
36	30, 32, 34, 35	dvdsrd 14066	. . . 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 4084	. . 3 |- ( x E y <-> <. x , y >. e. E )
39		df-br 4084	. . 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 4811	1 |- ( A e. (SubRing ` R ) -> E C_ .|| )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   E.wrex 2509   C_ wss 3197   <.cop 3669   class class class wbr 4083   Rel wrel 4724   ` cfv 5318  (class class class)co 6007   Basecbs 13040   ↾_s cress 13041   .rcmulr 13119  SRingcsrg 13934   Ringcrg 13967   ||_rcdsr 14057  SubRingcsubrg 14189

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-addcom 8107  ax-addass 8109  ax-i2m1 8112  ax-0lt1 8113  ax-0id 8115  ax-rnegex 8116  ax-pre-ltirr 8119  ax-pre-lttrn 8121  ax-pre-ltadd 8123

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8191  df-mnf 8192  df-ltxr 8194  df-inn 9119  df-2 9177  df-3 9178  df-ndx 13043  df-slot 13044  df-base 13046  df-sets 13047  df-iress 13048  df-plusg 13131  df-mulr 13132  df-0g 13299  df-mgm 13397  df-sgrp 13443  df-mnd 13458  df-grp 13544  df-minusg 13545  df-subg 13715  df-cmn 13831  df-abl 13832  df-mgp 13892  df-ur 13931  df-srg 13935  df-ring 13969  df-dvdsr 14060  df-subrg 14191

This theorem is referenced by:  subrguss  14208