MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subrgdvds Structured version   Unicode version

Theorem subrgdvds 15884
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 15751 . . 3  |-  Rel  E
32a1i 11 . 2  |-  ( A  e.  (SubRing `  R
)  ->  Rel  E )
4 subrgdvds.1 . . . . . . . 8  |-  S  =  ( Rs  A )
54subrgbas 15879 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
6 eqid 2438 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
76subrgss 15871 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
85, 7eqsstr3d 3385 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( Base `  S )  C_  ( Base `  R ) )
98sseld 3349 . . . . 5  |-  ( A  e.  (SubRing `  R
)  ->  ( x  e.  ( Base `  S
)  ->  x  e.  ( Base `  R )
) )
10 eqid 2438 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
114, 10ressmulr 13584 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
1211oveqd 6100 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  ( z
( .r `  R
) x )  =  ( z ( .r
`  S ) x ) )
1312eqeq1d 2446 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( (
z ( .r `  R ) x )  =  y  <->  ( z
( .r `  S
) x )  =  y ) )
1413rexbidv 2728 . . . . . 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 3410 . . . . . . 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 228 . . . . 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 548 . . . 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 2438 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
20 eqid 2438 . . . . 5  |-  ( .r
`  S )  =  ( .r `  S
)
2119, 1, 20dvdsr 15753 . . . 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 15753 . . . 4  |-  ( x 
.||  y  <->  ( x  e.  ( Base `  R
)  /\  E. z  e.  ( Base `  R
) ( z ( .r `  R ) x )  =  y ) )
2418, 21, 233imtr4g 263 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  ( x E y  ->  x  .||  y ) )
25 df-br 4215 . . 3  |-  ( x E y  <->  <. x ,  y >.  e.  E
)
26 df-br 4215 . . 3  |-  ( x 
.||  y  <->  <. x ,  y >.  e.  .||  )
2724, 25, 263imtr3g 262 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( <. x ,  y >.  e.  E  -> 
<. x ,  y >.  e.  .||  ) )
283, 27relssdv 4970 1  |-  ( A  e.  (SubRing `  R
)  ->  E  C_  .||  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708    C_ wss 3322   <.cop 3819   class class class wbr 4214   Rel wrel 4885   ` cfv 5456  (class class class)co 6083   Basecbs 13471   ↾s cress 13472   .rcmulr 13532   ||rcdsr 15745  SubRingcsubrg 15866
This theorem is referenced by:  subrguss  15885
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-mulr 13545  df-subg 14943  df-rng 15665  df-dvdsr 15748  df-subrg 15868
  Copyright terms: Public domain W3C validator