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Theorem subrgpropd 15595
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 15388 . . . . 5  |-  ( ph  ->  ( K  e.  Ring  <->  L  e.  Ring ) )
61ineq2d 3383 . . . . . . 7  |-  ( ph  ->  ( s  i^i  B
)  =  ( s  i^i  ( Base `  K
) ) )
7 vex 2804 . . . . . . . 8  |-  s  e. 
_V
8 eqid 2296 . . . . . . . . 9  |-  ( Ks  s )  =  ( Ks  s )
9 eqid 2296 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
108, 9ressbas 13214 . . . . . . . 8  |-  ( s  e.  _V  ->  (
s  i^i  ( Base `  K ) )  =  ( Base `  ( Ks  s ) ) )
117, 10ax-mp 8 . . . . . . 7  |-  ( s  i^i  ( Base `  K
) )  =  (
Base `  ( Ks  s
) )
126, 11syl6eq 2344 . . . . . 6  |-  ( ph  ->  ( s  i^i  B
)  =  ( Base `  ( Ks  s ) ) )
132ineq2d 3383 . . . . . . 7  |-  ( ph  ->  ( s  i^i  B
)  =  ( s  i^i  ( Base `  L
) ) )
14 eqid 2296 . . . . . . . . 9  |-  ( Ls  s )  =  ( Ls  s )
15 eqid 2296 . . . . . . . . 9  |-  ( Base `  L )  =  (
Base `  L )
1614, 15ressbas 13214 . . . . . . . 8  |-  ( s  e.  _V  ->  (
s  i^i  ( Base `  L ) )  =  ( Base `  ( Ls  s ) ) )
177, 16ax-mp 8 . . . . . . 7  |-  ( s  i^i  ( Base `  L
) )  =  (
Base `  ( Ls  s
) )
1813, 17syl6eq 2344 . . . . . 6  |-  ( ph  ->  ( s  i^i  B
)  =  ( Base `  ( Ls  s ) ) )
19 inss2 3403 . . . . . . . . 9  |-  ( s  i^i  B )  C_  B
2019sseli 3189 . . . . . . . 8  |-  ( x  e.  ( s  i^i 
B )  ->  x  e.  B )
2119sseli 3189 . . . . . . . 8  |-  ( y  e.  ( s  i^i 
B )  ->  y  e.  B )
2220, 21anim12i 549 . . . . . . 7  |-  ( ( x  e.  ( s  i^i  B )  /\  y  e.  ( s  i^i  B ) )  -> 
( x  e.  B  /\  y  e.  B
) )
23 eqid 2296 . . . . . . . . . . 11  |-  ( +g  `  K )  =  ( +g  `  K )
248, 23ressplusg 13266 . . . . . . . . . 10  |-  ( s  e.  _V  ->  ( +g  `  K )  =  ( +g  `  ( Ks  s ) ) )
257, 24ax-mp 8 . . . . . . . . 9  |-  ( +g  `  K )  =  ( +g  `  ( Ks  s ) )
2625oveqi 5887 . . . . . . . 8  |-  ( x ( +g  `  K
) y )  =  ( x ( +g  `  ( Ks  s ) ) y )
27 eqid 2296 . . . . . . . . . . 11  |-  ( +g  `  L )  =  ( +g  `  L )
2814, 27ressplusg 13266 . . . . . . . . . 10  |-  ( s  e.  _V  ->  ( +g  `  L )  =  ( +g  `  ( Ls  s ) ) )
297, 28ax-mp 8 . . . . . . . . 9  |-  ( +g  `  L )  =  ( +g  `  ( Ls  s ) )
3029oveqi 5887 . . . . . . . 8  |-  ( x ( +g  `  L
) y )  =  ( x ( +g  `  ( Ls  s ) ) y )
313, 26, 303eqtr3g 2351 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  ( Ks  s ) ) y )  =  ( x ( +g  `  ( Ls  s ) ) y ) )
3222, 31sylan2 460 . . . . . 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 2296 . . . . . . . . . . 11  |-  ( .r
`  K )  =  ( .r `  K
)
348, 33ressmulr 13277 . . . . . . . . . 10  |-  ( s  e.  _V  ->  ( .r `  K )  =  ( .r `  ( Ks  s ) ) )
357, 34ax-mp 8 . . . . . . . . 9  |-  ( .r
`  K )  =  ( .r `  ( Ks  s ) )
3635oveqi 5887 . . . . . . . 8  |-  ( x ( .r `  K
) y )  =  ( x ( .r
`  ( Ks  s ) ) y )
37 eqid 2296 . . . . . . . . . . 11  |-  ( .r
`  L )  =  ( .r `  L
)
3814, 37ressmulr 13277 . . . . . . . . . 10  |-  ( s  e.  _V  ->  ( .r `  L )  =  ( .r `  ( Ls  s ) ) )
397, 38ax-mp 8 . . . . . . . . 9  |-  ( .r
`  L )  =  ( .r `  ( Ls  s ) )
4039oveqi 5887 . . . . . . . 8  |-  ( x ( .r `  L
) y )  =  ( x ( .r
`  ( Ls  s ) ) y )
414, 36, 403eqtr3g 2351 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .r
`  ( Ks  s ) ) y )  =  ( x ( .r
`  ( Ls  s ) ) y ) )
4222, 41sylan2 460 . . . . . 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 15388 . . . . 5  |-  ( ph  ->  ( ( Ks  s )  e.  Ring  <->  ( Ls  s )  e.  Ring ) )
445, 43anbi12d 691 . . . 4  |-  ( ph  ->  ( ( K  e. 
Ring  /\  ( Ks  s )  e.  Ring )  <->  ( L  e.  Ring  /\  ( Ls  s
)  e.  Ring )
) )
451, 2eqtr3d 2330 . . . . . 6  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
4645sseq2d 3219 . . . . 5  |-  ( ph  ->  ( s  C_  ( Base `  K )  <->  s  C_  ( Base `  L )
) )
471, 2, 4rngidpropd 15493 . . . . . 6  |-  ( ph  ->  ( 1r `  K
)  =  ( 1r
`  L ) )
4847eleq1d 2362 . . . . 5  |-  ( ph  ->  ( ( 1r `  K )  e.  s  <-> 
( 1r `  L
)  e.  s ) )
4946, 48anbi12d 691 . . . 4  |-  ( ph  ->  ( ( s  C_  ( Base `  K )  /\  ( 1r `  K
)  e.  s )  <-> 
( s  C_  ( Base `  L )  /\  ( 1r `  L )  e.  s ) ) )
5044, 49anbi12d 691 . . 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 2296 . . . 4  |-  ( 1r
`  K )  =  ( 1r `  K
)
529, 51issubrg 15561 . . 3  |-  ( s  e.  (SubRing `  K
)  <->  ( ( K  e.  Ring  /\  ( Ks  s )  e.  Ring )  /\  ( s  C_  ( Base `  K )  /\  ( 1r `  K
)  e.  s ) ) )
53 eqid 2296 . . . 4  |-  ( 1r
`  L )  =  ( 1r `  L
)
5415, 53issubrg 15561 . . 3  |-  ( s  e.  (SubRing `  L
)  <->  ( ( L  e.  Ring  /\  ( Ls  s )  e.  Ring )  /\  ( s  C_  ( Base `  L )  /\  ( 1r `  L
)  e.  s ) ) )
5550, 52, 543bitr4g 279 . 2  |-  ( ph  ->  ( s  e.  (SubRing `  K )  <->  s  e.  (SubRing `  L ) ) )
5655eqrdv 2294 1  |-  ( ph  ->  (SubRing `  K )  =  (SubRing `  L )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165   +g cplusg 13224   .rcmulr 13225   Ringcrg 15353   1rcur 15355  SubRingcsubrg 15557
This theorem is referenced by:  ply1subrg  16292  subrgply1  16327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mnd 14383  df-grp 14505  df-mgp 15342  df-rng 15356  df-ur 15358  df-subrg 15559
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