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Theorem subrgdvds 13549
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  =  ( Rs  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.
StepHypRef Expression
1 subrgdvds.1 . . . . 5  |-  S  =  ( Rs  A )
21subrgring 13538 . . . 4  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
3 ringsrg 13366 . . . 4  |-  ( S  e.  Ring  ->  S  e. SRing
)
42, 3syl 14 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  S  e. SRing )
5 reldvdsrsrg 13409 . . . 4  |-  ( S  e. SRing  ->  Rel  ( ||r `  S
) )
6 subrgdvds.3 . . . . 5  |-  E  =  ( ||r `
 S )
76releqi 4724 . . . 4  |-  ( Rel 
E  <->  Rel  ( ||r `
 S ) )
85, 7sylibr 134 . . 3  |-  ( S  e. SRing  ->  Rel  E )
94, 8syl 14 . 2  |-  ( A  e.  (SubRing `  R
)  ->  Rel  E )
101subrgbas 13544 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
11 eqid 2189 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
1211subrgss 13536 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
1310, 12eqsstrrd 3207 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( Base `  S )  C_  ( Base `  R ) )
1413sseld 3169 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  ( x  e.  ( Base `  S
)  ->  x  e.  ( Base `  R )
) )
15 subrgrcl 13540 . . . . . . . . . 10  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
16 eqid 2189 . . . . . . . . . . 11  |-  ( .r
`  R )  =  ( .r `  R
)
171, 16ressmulrg 12628 . . . . . . . . . 10  |-  ( ( A  e.  (SubRing `  R
)  /\  R  e.  Ring )  ->  ( .r `  R )  =  ( .r `  S ) )
1815, 17mpdan 421 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
1918oveqd 5908 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  ( z
( .r `  R
) x )  =  ( z ( .r
`  S ) x ) )
2019eqeq1d 2198 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( (
z ( .r `  R ) x )  =  y  <->  ( z
( .r `  S
) x )  =  y ) )
2120rexbidv 2491 . . . . . 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 3235 . . . . . . 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 ) )
2313, 22syl 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 ) )
2421, 23sylbird 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 ) )
2514, 24anim12d 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 2190 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  ( Base `  S )  =  (
Base `  S )
)
276a1i 9 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  E  =  ( ||r `
 S ) )
28 eqidd 2190 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  S )  =  ( .r `  S ) )
2926, 27, 4, 28dvdsrd 13411 . . . 4  |-  ( A  e.  (SubRing `  R
)  ->  ( x E y  <->  ( x  e.  ( Base `  S
)  /\  E. z  e.  ( Base `  S
) ( z ( .r `  S ) x )  =  y ) ) )
30 eqidd 2190 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  ( Base `  R )  =  (
Base `  R )
)
31 subrgdvds.2 . . . . . 6  |-  .||  =  (
||r `  R )
3231a1i 9 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  .||  =  (
||r `  R ) )
33 ringsrg 13366 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. SRing
)
3415, 33syl 14 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  R  e. SRing )
35 eqidd 2190 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  R ) )
3630, 32, 34, 35dvdsrd 13411 . . . 4  |-  ( A  e.  (SubRing `  R
)  ->  ( x  .||  y  <->  ( x  e.  ( Base `  R
)  /\  E. z  e.  ( Base `  R
) ( z ( .r `  R ) x )  =  y ) ) )
3725, 29, 363imtr4d 203 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  ( x E y  ->  x  .||  y ) )
38 df-br 4019 . . 3  |-  ( x E y  <->  <. x ,  y >.  e.  E
)
39 df-br 4019 . . 3  |-  ( x 
.||  y  <->  <. x ,  y >.  e.  .||  )
4037, 38, 393imtr3g 204 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( <. x ,  y >.  e.  E  -> 
<. x ,  y >.  e.  .||  ) )
419, 40relssdv 4733 1  |-  ( A  e.  (SubRing `  R
)  ->  E  C_  .||  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   E.wrex 2469    C_ wss 3144   <.cop 3610   class class class wbr 4018   Rel wrel 4646   ` cfv 5231  (class class class)co 5891   Basecbs 12486   ↾s cress 12487   .rcmulr 12562  SRingcsrg 13284   Ringcrg 13317   ||rcdsr 13403  SubRingcsubrg 13531
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 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-addcom 7930  ax-addass 7932  ax-i2m1 7935  ax-0lt1 7936  ax-0id 7938  ax-rnegex 7939  ax-pre-ltirr 7942  ax-pre-lttrn 7944  ax-pre-ltadd 7946
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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-pnf 8013  df-mnf 8014  df-ltxr 8016  df-inn 8939  df-2 8997  df-3 8998  df-ndx 12489  df-slot 12490  df-base 12492  df-sets 12493  df-iress 12494  df-plusg 12574  df-mulr 12575  df-0g 12735  df-mgm 12804  df-sgrp 12837  df-mnd 12850  df-grp 12920  df-minusg 12921  df-subg 13081  df-cmn 13192  df-abl 13193  df-mgp 13242  df-ur 13281  df-srg 13285  df-ring 13319  df-dvdsr 13406  df-subrg 13533
This theorem is referenced by:  subrguss  13550
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