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Theorem subrgdvds 13450
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 13439 . . . 4  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
3 ringsrg 13292 . . . 4  |-  ( S  e.  Ring  ->  S  e. SRing
)
42, 3syl 14 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  S  e. SRing )
5 reldvdsrsrg 13335 . . . 4  |-  ( S  e. SRing  ->  Rel  ( ||r `  S
) )
6 subrgdvds.3 . . . . 5  |-  E  =  ( ||r `
 S )
76releqi 4721 . . . 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 13445 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
11 eqid 2187 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
1211subrgss 13437 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
1310, 12eqsstrrd 3204 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( Base `  S )  C_  ( Base `  R ) )
1413sseld 3166 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  ( x  e.  ( Base `  S
)  ->  x  e.  ( Base `  R )
) )
15 subrgrcl 13441 . . . . . . . . . 10  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
16 eqid 2187 . . . . . . . . . . 11  |-  ( .r
`  R )  =  ( .r `  R
)
171, 16ressmulrg 12617 . . . . . . . . . 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 5905 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  ( z
( .r `  R
) x )  =  ( z ( .r
`  S ) x ) )
2019eqeq1d 2196 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( (
z ( .r `  R ) x )  =  y  <->  ( z
( .r `  S
) x )  =  y ) )
2120rexbidv 2488 . . . . . 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 3232 . . . . . . 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 2188 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  ( Base `  S )  =  (
Base `  S )
)
276a1i 9 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  E  =  ( ||r `
 S ) )
28 eqidd 2188 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  S )  =  ( .r `  S ) )
2926, 27, 4, 28dvdsrd 13337 . . . 4  |-  ( A  e.  (SubRing `  R
)  ->  ( x E y  <->  ( x  e.  ( Base `  S
)  /\  E. z  e.  ( Base `  S
) ( z ( .r `  S ) x )  =  y ) ) )
30 eqidd 2188 . . . . 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 13292 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. SRing
)
3415, 33syl 14 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  R  e. SRing )
35 eqidd 2188 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  R ) )
3630, 32, 34, 35dvdsrd 13337 . . . 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 4016 . . 3  |-  ( x E y  <->  <. x ,  y >.  e.  E
)
39 df-br 4016 . . 3  |-  ( x 
.||  y  <->  <. x ,  y >.  e.  .||  )
4037, 38, 393imtr3g 204 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( <. x ,  y >.  e.  E  -> 
<. x ,  y >.  e.  .||  ) )
419, 40relssdv 4730 1  |-  ( A  e.  (SubRing `  R
)  ->  E  C_  .||  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1363    e. wcel 2158   E.wrex 2466    C_ wss 3141   <.cop 3607   class class class wbr 4015   Rel wrel 4643   ` cfv 5228  (class class class)co 5888   Basecbs 12475   ↾s cress 12476   .rcmulr 12551  SRingcsrg 13210   Ringcrg 13243   ||rcdsr 13329  SubRingcsubrg 13432
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-i2m1 7929  ax-0lt1 7930  ax-0id 7932  ax-rnegex 7933  ax-pre-ltirr 7936  ax-pre-lttrn 7938  ax-pre-ltadd 7940
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8007  df-mnf 8008  df-ltxr 8010  df-inn 8933  df-2 8991  df-3 8992  df-ndx 12478  df-slot 12479  df-base 12481  df-sets 12482  df-iress 12483  df-plusg 12563  df-mulr 12564  df-0g 12724  df-mgm 12793  df-sgrp 12826  df-mnd 12839  df-grp 12901  df-minusg 12902  df-subg 13061  df-cmn 13122  df-abl 13123  df-mgp 13171  df-ur 13207  df-srg 13211  df-ring 13245  df-dvdsr 13332  df-subrg 13434
This theorem is referenced by:  subrguss  13451
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