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Theorem subrgdvds 16987
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.3 . . . 4  |-  E  =  ( ||r `
 S )
21reldvdsr 16844 . . 3  |-  Rel  E
32a1i 11 . 2  |-  ( A  e.  (SubRing `  R
)  ->  Rel  E )
4 subrgdvds.1 . . . . . . . 8  |-  S  =  ( Rs  A )
54subrgbas 16982 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
6 eqid 2451 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
76subrgss 16974 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
85, 7eqsstr3d 3491 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( Base `  S )  C_  ( Base `  R ) )
98sseld 3455 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  ( x  e.  ( Base `  S
)  ->  x  e.  ( Base `  R )
) )
10 eqid 2451 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
114, 10ressmulr 14395 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
1211oveqd 6209 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  ( z
( .r `  R
) x )  =  ( z ( .r
`  S ) x ) )
1312eqeq1d 2453 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( (
z ( .r `  R ) x )  =  y  <->  ( z
( .r `  S
) x )  =  y ) )
1413rexbidv 2848 . . . . . 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 ) )
15 ssrexv 3517 . . . . . . 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 ) )
168, 15syl 16 . . . . . 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 ) )
1714, 16sylbird 235 . . . . 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 ) )
189, 17anim12d 563 . . . 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 ) ) )
19 eqid 2451 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
20 eqid 2451 . . . . 5  |-  ( .r
`  S )  =  ( .r `  S
)
2119, 1, 20dvdsr 16846 . . . 4  |-  ( x E y  <->  ( x  e.  ( Base `  S
)  /\  E. z  e.  ( Base `  S
) ( z ( .r `  S ) x )  =  y ) )
22 subrgdvds.2 . . . . 5  |-  .||  =  (
||r `  R )
236, 22, 10dvdsr 16846 . . . 4  |-  ( x 
.||  y  <->  ( x  e.  ( Base `  R
)  /\  E. z  e.  ( Base `  R
) ( z ( .r `  R ) x )  =  y ) )
2418, 21, 233imtr4g 270 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  ( x E y  ->  x  .||  y ) )
25 df-br 4393 . . 3  |-  ( x E y  <->  <. x ,  y >.  e.  E
)
26 df-br 4393 . . 3  |-  ( x 
.||  y  <->  <. x ,  y >.  e.  .||  )
2724, 25, 263imtr3g 269 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( <. x ,  y >.  e.  E  -> 
<. x ,  y >.  e.  .||  ) )
283, 27relssdv 5032 1  |-  ( A  e.  (SubRing `  R
)  ->  E  C_  .||  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2796    C_ wss 3428   <.cop 3983   class class class wbr 4392   Rel wrel 4945   ` cfv 5518  (class class class)co 6192   Basecbs 14278   ↾s cress 14279   .rcmulr 14343   ||rcdsr 16838  SubRingcsubrg 16969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-recs 6934  df-rdg 6968  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-ress 14285  df-mulr 14356  df-subg 15782  df-rng 16755  df-dvdsr 16841  df-subrg 16971
This theorem is referenced by:  subrguss  16988
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