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Theorem subrgdvds 13294
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 13283 . . . 4  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
3 ringsrg 13155 . . . 4  |-  ( S  e.  Ring  ->  S  e. SRing
)
42, 3syl 14 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  S  e. SRing )
5 reldvdsrsrg 13192 . . . 4  |-  ( S  e. SRing  ->  Rel  ( ||r `  S
) )
6 subrgdvds.3 . . . . 5  |-  E  =  ( ||r `
 S )
76releqi 4708 . . . 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 13289 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
11 eqid 2177 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
1211subrgss 13281 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
1310, 12eqsstrrd 3192 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( Base `  S )  C_  ( Base `  R ) )
1413sseld 3154 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  ( x  e.  ( Base `  S
)  ->  x  e.  ( Base `  R )
) )
15 subrgrcl 13285 . . . . . . . . . 10  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
16 eqid 2177 . . . . . . . . . . 11  |-  ( .r
`  R )  =  ( .r `  R
)
171, 16ressmulrg 12595 . . . . . . . . . 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 5889 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  ( z
( .r `  R
) x )  =  ( z ( .r
`  S ) x ) )
2019eqeq1d 2186 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( (
z ( .r `  R ) x )  =  y  <->  ( z
( .r `  S
) x )  =  y ) )
2120rexbidv 2478 . . . . . 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 3220 . . . . . . 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 2178 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  ( Base `  S )  =  (
Base `  S )
)
276a1i 9 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  E  =  ( ||r `
 S ) )
28 eqidd 2178 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  S )  =  ( .r `  S ) )
2926, 27, 4, 28dvdsrd 13194 . . . 4  |-  ( A  e.  (SubRing `  R
)  ->  ( x E y  <->  ( x  e.  ( Base `  S
)  /\  E. z  e.  ( Base `  S
) ( z ( .r `  S ) x )  =  y ) ) )
30 eqidd 2178 . . . . 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 13155 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. SRing
)
3415, 33syl 14 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  R  e. SRing )
35 eqidd 2178 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  R ) )
3630, 32, 34, 35dvdsrd 13194 . . . 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 4003 . . 3  |-  ( x E y  <->  <. x ,  y >.  e.  E
)
39 df-br 4003 . . 3  |-  ( x 
.||  y  <->  <. x ,  y >.  e.  .||  )
4037, 38, 393imtr3g 204 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( <. x ,  y >.  e.  E  -> 
<. x ,  y >.  e.  .||  ) )
419, 40relssdv 4717 1  |-  ( A  e.  (SubRing `  R
)  ->  E  C_  .||  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   E.wrex 2456    C_ wss 3129   <.cop 3595   class class class wbr 4002   Rel wrel 4630   ` cfv 5215  (class class class)co 5872   Basecbs 12454   ↾s cress 12455   .rcmulr 12529  SRingcsrg 13077   Ringcrg 13110   ||rcdsr 13186  SubRingcsubrg 13276
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-pre-ltirr 7920  ax-pre-lttrn 7922  ax-pre-ltadd 7924
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-inn 8916  df-2 8974  df-3 8975  df-ndx 12457  df-slot 12458  df-base 12460  df-sets 12461  df-iress 12462  df-plusg 12541  df-mulr 12542  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-minusg 12813  df-subg 12961  df-cmn 13021  df-abl 13022  df-mgp 13062  df-ur 13074  df-srg 13078  df-ring 13112  df-dvdsr 13189  df-subrg 13278
This theorem is referenced by:  subrguss  13295
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