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

Theorem subrgpropd 17032
Description: If two structures have the same group components (properties), they have the same set of subrings. (Contributed by Mario Carneiro, 9-Feb-2015.)
Hypotheses
Ref Expression
subrgpropd.1  |-  ( ph  ->  B  =  ( Base `  K ) )
subrgpropd.2  |-  ( ph  ->  B  =  ( Base `  L ) )
subrgpropd.3  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
subrgpropd.4  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .r
`  K ) y )  =  ( x ( .r `  L
) y ) )
Assertion
Ref Expression
subrgpropd  |-  ( ph  ->  (SubRing `  K )  =  (SubRing `  L )
)
Distinct variable groups:    x, y, B    x, K, y    ph, x, y    x, L, y

Proof of Theorem subrgpropd
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 subrgpropd.1 . . . . . 6  |-  ( ph  ->  B  =  ( Base `  K ) )
2 subrgpropd.2 . . . . . 6  |-  ( ph  ->  B  =  ( Base `  L ) )
3 subrgpropd.3 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
4 subrgpropd.4 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .r
`  K ) y )  =  ( x ( .r `  L
) y ) )
51, 2, 3, 4rngpropd 16809 . . . . 5  |-  ( ph  ->  ( K  e.  Ring  <->  L  e.  Ring ) )
61ineq2d 3663 . . . . . . 7  |-  ( ph  ->  ( s  i^i  B
)  =  ( s  i^i  ( Base `  K
) ) )
7 vex 3081 . . . . . . . 8  |-  s  e. 
_V
8 eqid 2454 . . . . . . . . 9  |-  ( Ks  s )  =  ( Ks  s )
9 eqid 2454 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
108, 9ressbas 14351 . . . . . . . 8  |-  ( s  e.  _V  ->  (
s  i^i  ( Base `  K ) )  =  ( Base `  ( Ks  s ) ) )
117, 10ax-mp 5 . . . . . . 7  |-  ( s  i^i  ( Base `  K
) )  =  (
Base `  ( Ks  s
) )
126, 11syl6eq 2511 . . . . . 6  |-  ( ph  ->  ( s  i^i  B
)  =  ( Base `  ( Ks  s ) ) )
132ineq2d 3663 . . . . . . 7  |-  ( ph  ->  ( s  i^i  B
)  =  ( s  i^i  ( Base `  L
) ) )
14 eqid 2454 . . . . . . . . 9  |-  ( Ls  s )  =  ( Ls  s )
15 eqid 2454 . . . . . . . . 9  |-  ( Base `  L )  =  (
Base `  L )
1614, 15ressbas 14351 . . . . . . . 8  |-  ( s  e.  _V  ->  (
s  i^i  ( Base `  L ) )  =  ( Base `  ( Ls  s ) ) )
177, 16ax-mp 5 . . . . . . 7  |-  ( s  i^i  ( Base `  L
) )  =  (
Base `  ( Ls  s
) )
1813, 17syl6eq 2511 . . . . . 6  |-  ( ph  ->  ( s  i^i  B
)  =  ( Base `  ( Ls  s ) ) )
19 inss2 3682 . . . . . . . . 9  |-  ( s  i^i  B )  C_  B
2019sseli 3463 . . . . . . . 8  |-  ( x  e.  ( s  i^i 
B )  ->  x  e.  B )
2119sseli 3463 . . . . . . . 8  |-  ( y  e.  ( s  i^i 
B )  ->  y  e.  B )
2220, 21anim12i 566 . . . . . . 7  |-  ( ( x  e.  ( s  i^i  B )  /\  y  e.  ( s  i^i  B ) )  -> 
( x  e.  B  /\  y  e.  B
) )
23 eqid 2454 . . . . . . . . . . 11  |-  ( +g  `  K )  =  ( +g  `  K )
248, 23ressplusg 14403 . . . . . . . . . 10  |-  ( s  e.  _V  ->  ( +g  `  K )  =  ( +g  `  ( Ks  s ) ) )
257, 24ax-mp 5 . . . . . . . . 9  |-  ( +g  `  K )  =  ( +g  `  ( Ks  s ) )
2625oveqi 6216 . . . . . . . 8  |-  ( x ( +g  `  K
) y )  =  ( x ( +g  `  ( Ks  s ) ) y )
27 eqid 2454 . . . . . . . . . . 11  |-  ( +g  `  L )  =  ( +g  `  L )
2814, 27ressplusg 14403 . . . . . . . . . 10  |-  ( s  e.  _V  ->  ( +g  `  L )  =  ( +g  `  ( Ls  s ) ) )
297, 28ax-mp 5 . . . . . . . . 9  |-  ( +g  `  L )  =  ( +g  `  ( Ls  s ) )
3029oveqi 6216 . . . . . . . 8  |-  ( x ( +g  `  L
) y )  =  ( x ( +g  `  ( Ls  s ) ) y )
313, 26, 303eqtr3g 2518 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  ( Ks  s ) ) y )  =  ( x ( +g  `  ( Ls  s ) ) y ) )
3222, 31sylan2 474 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( s  i^i  B
)  /\  y  e.  ( s  i^i  B
) ) )  -> 
( x ( +g  `  ( Ks  s ) ) y )  =  ( x ( +g  `  ( Ls  s ) ) y ) )
33 eqid 2454 . . . . . . . . . . 11  |-  ( .r
`  K )  =  ( .r `  K
)
348, 33ressmulr 14414 . . . . . . . . . 10  |-  ( s  e.  _V  ->  ( .r `  K )  =  ( .r `  ( Ks  s ) ) )
357, 34ax-mp 5 . . . . . . . . 9  |-  ( .r
`  K )  =  ( .r `  ( Ks  s ) )
3635oveqi 6216 . . . . . . . 8  |-  ( x ( .r `  K
) y )  =  ( x ( .r
`  ( Ks  s ) ) y )
37 eqid 2454 . . . . . . . . . . 11  |-  ( .r
`  L )  =  ( .r `  L
)
3814, 37ressmulr 14414 . . . . . . . . . 10  |-  ( s  e.  _V  ->  ( .r `  L )  =  ( .r `  ( Ls  s ) ) )
397, 38ax-mp 5 . . . . . . . . 9  |-  ( .r
`  L )  =  ( .r `  ( Ls  s ) )
4039oveqi 6216 . . . . . . . 8  |-  ( x ( .r `  L
) y )  =  ( x ( .r
`  ( Ls  s ) ) y )
414, 36, 403eqtr3g 2518 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .r
`  ( Ks  s ) ) y )  =  ( x ( .r
`  ( Ls  s ) ) y ) )
4222, 41sylan2 474 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( s  i^i  B
)  /\  y  e.  ( s  i^i  B
) ) )  -> 
( x ( .r
`  ( Ks  s ) ) y )  =  ( x ( .r
`  ( Ls  s ) ) y ) )
4312, 18, 32, 42rngpropd 16809 . . . . 5  |-  ( ph  ->  ( ( Ks  s )  e.  Ring  <->  ( Ls  s )  e.  Ring ) )
445, 43anbi12d 710 . . . 4  |-  ( ph  ->  ( ( K  e. 
Ring  /\  ( Ks  s )  e.  Ring )  <->  ( L  e.  Ring  /\  ( Ls  s
)  e.  Ring )
) )
451, 2eqtr3d 2497 . . . . . 6  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
4645sseq2d 3495 . . . . 5  |-  ( ph  ->  ( s  C_  ( Base `  K )  <->  s  C_  ( Base `  L )
) )
471, 2, 4rngidpropd 16920 . . . . . 6  |-  ( ph  ->  ( 1r `  K
)  =  ( 1r
`  L ) )
4847eleq1d 2523 . . . . 5  |-  ( ph  ->  ( ( 1r `  K )  e.  s  <-> 
( 1r `  L
)  e.  s ) )
4946, 48anbi12d 710 . . . 4  |-  ( ph  ->  ( ( s  C_  ( Base `  K )  /\  ( 1r `  K
)  e.  s )  <-> 
( s  C_  ( Base `  L )  /\  ( 1r `  L )  e.  s ) ) )
5044, 49anbi12d 710 . . 3  |-  ( ph  ->  ( ( ( K  e.  Ring  /\  ( Ks  s )  e.  Ring )  /\  ( s  C_  ( Base `  K )  /\  ( 1r `  K
)  e.  s ) )  <->  ( ( L  e.  Ring  /\  ( Ls  s )  e.  Ring )  /\  ( s  C_  ( Base `  L )  /\  ( 1r `  L
)  e.  s ) ) ) )
51 eqid 2454 . . . 4  |-  ( 1r
`  K )  =  ( 1r `  K
)
529, 51issubrg 16998 . . 3  |-  ( s  e.  (SubRing `  K
)  <->  ( ( K  e.  Ring  /\  ( Ks  s )  e.  Ring )  /\  ( s  C_  ( Base `  K )  /\  ( 1r `  K
)  e.  s ) ) )
53 eqid 2454 . . . 4  |-  ( 1r
`  L )  =  ( 1r `  L
)
5415, 53issubrg 16998 . . 3  |-  ( s  e.  (SubRing `  L
)  <->  ( ( L  e.  Ring  /\  ( Ls  s )  e.  Ring )  /\  ( s  C_  ( Base `  L )  /\  ( 1r `  L
)  e.  s ) ) )
5550, 52, 543bitr4g 288 . 2  |-  ( ph  ->  ( s  e.  (SubRing `  K )  <->  s  e.  (SubRing `  L ) ) )
5655eqrdv 2451 1  |-  ( ph  ->  (SubRing `  K )  =  (SubRing `  L )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078    i^i cin 3438    C_ wss 3439   ` cfv 5529  (class class class)co 6203   Basecbs 14296   ↾s cress 14297   +g cplusg 14361   .rcmulr 14362   1rcur 16735   Ringcrg 16778  SubRingcsubrg 16994
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 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-3 10496  df-ndx 14299  df-slot 14300  df-base 14301  df-sets 14302  df-ress 14303  df-plusg 14374  df-mulr 14375  df-0g 14503  df-mnd 15538  df-grp 15668  df-mgp 16724  df-ur 16736  df-rng 16780  df-subrg 16996
This theorem is referenced by:  ply1subrg  17787  subrgply1  17821
  Copyright terms: Public domain W3C validator