Theorem subrgdvds 14466

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 14455	. . . 4 |- ( A e. (SubRing ` R ) -> S e. Ring )
3		ringsrg 14275	. . . 4 |- ( S e. Ring -> S e. SRing )
4	2, 3	syl 14	. . 3 |- ( A e. (SubRing ` R ) -> S e. SRing )
5		reldvdsrsrg 14322	. . . 4 |- ( S e. SRing -> Rel ( ||r ` S ) )
6		subrgdvds.3	. . . . 5 |- E = ( ||r ` S )
7	6	releqi 4838	. . . 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 14461	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A = ( Base ` S ) )
11		eqid 2234	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
12	11	subrgss 14453	. . . . . . 7 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
13	10, 12	eqsstrrd 3279	. . . . . 6 |- ( A e. (SubRing ` R ) -> ( Base ` S ) C_ ( Base ` R ) )
14	13	sseld 3241	. . . . 5 |- ( A e. (SubRing ` R ) -> ( x e. ( Base ` S ) -> x e. ( Base ` R ) ) )
15		subrgrcl 14457	. . . . . . . . . 10 |- ( A e. (SubRing ` R ) -> R e. Ring )
16		eqid 2234	. . . . . . . . . . 11 |- ( .r ` R ) = ( .r ` R )
17	1, 16	ressmulrg 13442	. . . . . . . . . 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 6075	. . . . . . . 8 |- ( A e. (SubRing ` R ) -> ( z ( .r ` R ) x ) = ( z ( .r ` S ) x ) )
20	19	eqeq1d 2243	. . . . . . 7 |- ( A e. (SubRing ` R ) -> ( ( z ( .r ` R ) x ) = y <-> ( z ( .r ` S ) x ) = y ) )
21	20	rexbidv 2545	. . . . . 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 3307	. . . . . . 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 2235	. . . . 5 |- ( A e. (SubRing ` R ) -> ( Base ` S ) = ( Base ` S ) )
27	6	a1i 9	. . . . 5 |- ( A e. (SubRing ` R ) -> E = ( ||r ` S ) )
28		eqidd 2235	. . . . 5 |- ( A e. (SubRing ` R ) -> ( .r ` S ) = ( .r ` S ) )
29	26, 27, 4, 28	dvdsrd 14324	. . . 4 |- ( A e. (SubRing ` R ) -> ( x E y <-> ( x e. ( Base ` S ) /\ E. z e. ( Base ` S ) ( z ( .r ` S ) x ) = y ) ) )
30		eqidd 2235	. . . . 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 14275	. . . . . 6 |- ( R e. Ring -> R e. SRing )
34	15, 33	syl 14	. . . . 5 |- ( A e. (SubRing ` R ) -> R e. SRing )
35		eqidd 2235	. . . . 5 |- ( A e. (SubRing ` R ) -> ( .r ` R ) = ( .r ` R ) )
36	30, 32, 34, 35	dvdsrd 14324	. . . 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 4115	. . 3 |- ( x E y <-> <. x , y >. e. E )
39		df-br 4115	. . 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 4847	1 |- ( A e. (SubRing ` R ) -> E C_ .|| )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   E.wrex 2523   C_ wss 3214   <.cop 3697   class class class wbr 4114   Rel wrel 4759   ` cfv 5357  (class class class)co 6058   Basecbs 13296   ↾_s cress 13297   .rcmulr 13375  SRingcsrg 14191   Ringcrg 14224   ||_rcdsr 14315  SubRingcsubrg 14448

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgm 13653  df-sgrp 13699  df-mnd 13714  df-grp 13800  df-minusg 13801  df-subg 13971  df-cmn 14087  df-abl 14088  df-mgp 14149  df-ur 14188  df-srg 14192  df-ring 14226  df-dvdsr 14318  df-subrg 14450

This theorem is referenced by:  subrguss  14467