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Theorem subrgpropd 13562
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 subrgrcl 13540 . . . 4  |-  ( s  e.  (SubRing `  K
)  ->  K  e.  Ring )
21a1i 9 . . 3  |-  ( ph  ->  ( s  e.  (SubRing `  K )  ->  K  e.  Ring ) )
3 subrgrcl 13540 . . . 4  |-  ( s  e.  (SubRing `  L
)  ->  L  e.  Ring )
4 subrgpropd.1 . . . . 5  |-  ( ph  ->  B  =  ( Base `  K ) )
5 subrgpropd.2 . . . . 5  |-  ( ph  ->  B  =  ( Base `  L ) )
6 subrgpropd.3 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
7 subrgpropd.4 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .r
`  K ) y )  =  ( x ( .r `  L
) y ) )
84, 5, 6, 7ringpropd 13359 . . . 4  |-  ( ph  ->  ( K  e.  Ring  <->  L  e.  Ring ) )
93, 8imbitrrid 156 . . 3  |-  ( ph  ->  ( s  e.  (SubRing `  L )  ->  K  e.  Ring ) )
108adantr 276 . . . . . . 7  |-  ( (
ph  /\  K  e.  Ring )  ->  ( K  e.  Ring  <->  L  e.  Ring ) )
114ineq2d 3351 . . . . . . . . . 10  |-  ( ph  ->  ( s  i^i  B
)  =  ( s  i^i  ( Base `  K
) ) )
1211adantr 276 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  Ring )  ->  ( s  i^i  B )  =  ( s  i^i  ( Base `  K ) ) )
13 eqidd 2190 . . . . . . . . . . 11  |-  ( ( ( ph  /\  K  e.  Ring )  /\  s  e.  _V )  ->  ( Ks  s )  =  ( Ks  s ) )
14 eqidd 2190 . . . . . . . . . . 11  |-  ( ( ( ph  /\  K  e.  Ring )  /\  s  e.  _V )  ->  ( Base `  K )  =  ( Base `  K
) )
15 simplr 528 . . . . . . . . . . 11  |-  ( ( ( ph  /\  K  e.  Ring )  /\  s  e.  _V )  ->  K  e.  Ring )
16 simpr 110 . . . . . . . . . . 11  |-  ( ( ( ph  /\  K  e.  Ring )  /\  s  e.  _V )  ->  s  e.  _V )
1713, 14, 15, 16ressbasd 12551 . . . . . . . . . 10  |-  ( ( ( ph  /\  K  e.  Ring )  /\  s  e.  _V )  ->  (
s  i^i  ( Base `  K ) )  =  ( Base `  ( Ks  s ) ) )
1817elvd 2757 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  Ring )  ->  ( s  i^i  ( Base `  K
) )  =  (
Base `  ( Ks  s
) ) )
1912, 18eqtrd 2222 . . . . . . . 8  |-  ( (
ph  /\  K  e.  Ring )  ->  ( s  i^i  B )  =  (
Base `  ( Ks  s
) ) )
205ineq2d 3351 . . . . . . . . . 10  |-  ( ph  ->  ( s  i^i  B
)  =  ( s  i^i  ( Base `  L
) ) )
2120adantr 276 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  Ring )  ->  ( s  i^i  B )  =  ( s  i^i  ( Base `  L ) ) )
22 eqidd 2190 . . . . . . . . . . 11  |-  ( ( ( ph  /\  K  e.  Ring )  /\  s  e.  _V )  ->  ( Ls  s )  =  ( Ls  s ) )
23 eqidd 2190 . . . . . . . . . . 11  |-  ( ( ( ph  /\  K  e.  Ring )  /\  s  e.  _V )  ->  ( Base `  L )  =  ( Base `  L
) )
248biimpa 296 . . . . . . . . . . . 12  |-  ( (
ph  /\  K  e.  Ring )  ->  L  e.  Ring )
2524adantr 276 . . . . . . . . . . 11  |-  ( ( ( ph  /\  K  e.  Ring )  /\  s  e.  _V )  ->  L  e.  Ring )
2622, 23, 25, 16ressbasd 12551 . . . . . . . . . 10  |-  ( ( ( ph  /\  K  e.  Ring )  /\  s  e.  _V )  ->  (
s  i^i  ( Base `  L ) )  =  ( Base `  ( Ls  s ) ) )
2726elvd 2757 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  Ring )  ->  ( s  i^i  ( Base `  L
) )  =  (
Base `  ( Ls  s
) ) )
2821, 27eqtrd 2222 . . . . . . . 8  |-  ( (
ph  /\  K  e.  Ring )  ->  ( s  i^i  B )  =  (
Base `  ( Ls  s
) ) )
29 elinel2 3337 . . . . . . . . . 10  |-  ( x  e.  ( s  i^i 
B )  ->  x  e.  B )
30 elinel2 3337 . . . . . . . . . 10  |-  ( y  e.  ( s  i^i 
B )  ->  y  e.  B )
3129, 30anim12i 338 . . . . . . . . 9  |-  ( ( x  e.  ( s  i^i  B )  /\  y  e.  ( s  i^i  B ) )  -> 
( x  e.  B  /\  y  e.  B
) )
326adantlr 477 . . . . . . . . . 10  |-  ( ( ( ph  /\  K  e.  Ring )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( +g  `  K ) y )  =  ( x ( +g  `  L
) y ) )
33 eqidd 2190 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  K  e.  Ring )  /\  s  e.  _V )  ->  ( +g  `  K )  =  ( +g  `  K
) )
3413, 33, 16, 15ressplusgd 12612 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  K  e.  Ring )  /\  s  e.  _V )  ->  ( +g  `  K )  =  ( +g  `  ( Ks  s ) ) )
3534elvd 2757 . . . . . . . . . . 11  |-  ( (
ph  /\  K  e.  Ring )  ->  ( +g  `  K )  =  ( +g  `  ( Ks  s ) ) )
3635oveqdr 5919 . . . . . . . . . 10  |-  ( ( ( ph  /\  K  e.  Ring )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( +g  `  K ) y )  =  ( x ( +g  `  ( Ks  s ) ) y ) )
37 eqidd 2190 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  K  e.  Ring )  /\  s  e.  _V )  ->  ( +g  `  L )  =  ( +g  `  L
) )
3822, 37, 16, 25ressplusgd 12612 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  K  e.  Ring )  /\  s  e.  _V )  ->  ( +g  `  L )  =  ( +g  `  ( Ls  s ) ) )
3938elvd 2757 . . . . . . . . . . 11  |-  ( (
ph  /\  K  e.  Ring )  ->  ( +g  `  L )  =  ( +g  `  ( Ls  s ) ) )
4039oveqdr 5919 . . . . . . . . . 10  |-  ( ( ( ph  /\  K  e.  Ring )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( +g  `  L ) y )  =  ( x ( +g  `  ( Ls  s ) ) y ) )
4132, 36, 403eqtr3d 2230 . . . . . . . . 9  |-  ( ( ( ph  /\  K  e.  Ring )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( +g  `  ( Ks  s ) ) y )  =  ( x ( +g  `  ( Ls  s ) ) y ) )
4231, 41sylan2 286 . . . . . . . 8  |-  ( ( ( ph  /\  K  e.  Ring )  /\  (
x  e.  ( s  i^i  B )  /\  y  e.  ( s  i^i  B ) ) )  ->  ( x ( +g  `  ( Ks  s ) ) y )  =  ( x ( +g  `  ( Ls  s ) ) y ) )
437adantlr 477 . . . . . . . . . 10  |-  ( ( ( ph  /\  K  e.  Ring )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( .r `  K
) y )  =  ( x ( .r
`  L ) y ) )
44 vex 2755 . . . . . . . . . . . . 13  |-  s  e. 
_V
45 eqid 2189 . . . . . . . . . . . . . 14  |-  ( Ks  s )  =  ( Ks  s )
46 eqid 2189 . . . . . . . . . . . . . 14  |-  ( .r
`  K )  =  ( .r `  K
)
4745, 46ressmulrg 12628 . . . . . . . . . . . . 13  |-  ( ( s  e.  _V  /\  K  e.  Ring )  -> 
( .r `  K
)  =  ( .r
`  ( Ks  s ) ) )
4844, 47mpan 424 . . . . . . . . . . . 12  |-  ( K  e.  Ring  ->  ( .r
`  K )  =  ( .r `  ( Ks  s ) ) )
4948adantl 277 . . . . . . . . . . 11  |-  ( (
ph  /\  K  e.  Ring )  ->  ( .r `  K )  =  ( .r `  ( Ks  s ) ) )
5049oveqdr 5919 . . . . . . . . . 10  |-  ( ( ( ph  /\  K  e.  Ring )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( .r `  K
) y )  =  ( x ( .r
`  ( Ks  s ) ) y ) )
51 eqid 2189 . . . . . . . . . . . . 13  |-  ( Ls  s )  =  ( Ls  s )
52 eqid 2189 . . . . . . . . . . . . 13  |-  ( .r
`  L )  =  ( .r `  L
)
5351, 52ressmulrg 12628 . . . . . . . . . . . 12  |-  ( ( s  e.  _V  /\  L  e.  Ring )  -> 
( .r `  L
)  =  ( .r
`  ( Ls  s ) ) )
5444, 24, 53sylancr 414 . . . . . . . . . . 11  |-  ( (
ph  /\  K  e.  Ring )  ->  ( .r `  L )  =  ( .r `  ( Ls  s ) ) )
5554oveqdr 5919 . . . . . . . . . 10  |-  ( ( ( ph  /\  K  e.  Ring )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( .r `  L
) y )  =  ( x ( .r
`  ( Ls  s ) ) y ) )
5643, 50, 553eqtr3d 2230 . . . . . . . . 9  |-  ( ( ( ph  /\  K  e.  Ring )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( .r `  ( Ks  s ) ) y )  =  ( x ( .r `  ( Ls  s ) ) y ) )
5731, 56sylan2 286 . . . . . . . 8  |-  ( ( ( ph  /\  K  e.  Ring )  /\  (
x  e.  ( s  i^i  B )  /\  y  e.  ( s  i^i  B ) ) )  ->  ( x ( .r `  ( Ks  s ) ) y )  =  ( x ( .r `  ( Ls  s ) ) y ) )
5819, 28, 42, 57ringpropd 13359 . . . . . . 7  |-  ( (
ph  /\  K  e.  Ring )  ->  ( ( Ks  s )  e.  Ring  <->  ( Ls  s )  e.  Ring ) )
5910, 58anbi12d 473 . . . . . 6  |-  ( (
ph  /\  K  e.  Ring )  ->  ( ( K  e.  Ring  /\  ( Ks  s )  e.  Ring ) 
<->  ( L  e.  Ring  /\  ( Ls  s )  e. 
Ring ) ) )
604, 5eqtr3d 2224 . . . . . . . . 9  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
6160sseq2d 3200 . . . . . . . 8  |-  ( ph  ->  ( s  C_  ( Base `  K )  <->  s  C_  ( Base `  L )
) )
6261adantr 276 . . . . . . 7  |-  ( (
ph  /\  K  e.  Ring )  ->  ( s  C_  ( Base `  K
)  <->  s  C_  ( Base `  L ) ) )
634adantr 276 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  Ring )  ->  B  =  ( Base `  K )
)
645adantr 276 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  Ring )  ->  B  =  ( Base `  L )
)
65 simpr 110 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  Ring )  ->  K  e.  Ring )
6663, 64, 43, 65, 24rngidpropdg 13463 . . . . . . . 8  |-  ( (
ph  /\  K  e.  Ring )  ->  ( 1r `  K )  =  ( 1r `  L ) )
6766eleq1d 2258 . . . . . . 7  |-  ( (
ph  /\  K  e.  Ring )  ->  ( ( 1r `  K )  e.  s  <->  ( 1r `  L )  e.  s ) )
6862, 67anbi12d 473 . . . . . 6  |-  ( (
ph  /\  K  e.  Ring )  ->  ( (
s  C_  ( Base `  K )  /\  ( 1r `  K )  e.  s )  <->  ( s  C_  ( Base `  L
)  /\  ( 1r `  L )  e.  s ) ) )
6959, 68anbi12d 473 . . . . 5  |-  ( (
ph  /\  K  e.  Ring )  ->  ( (
( 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 ) ) ) )
70 eqid 2189 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
71 eqid 2189 . . . . . 6  |-  ( 1r
`  K )  =  ( 1r `  K
)
7270, 71issubrg 13535 . . . . 5  |-  ( s  e.  (SubRing `  K
)  <->  ( ( K  e.  Ring  /\  ( Ks  s )  e.  Ring )  /\  ( s  C_  ( Base `  K )  /\  ( 1r `  K
)  e.  s ) ) )
73 eqid 2189 . . . . . 6  |-  ( Base `  L )  =  (
Base `  L )
74 eqid 2189 . . . . . 6  |-  ( 1r
`  L )  =  ( 1r `  L
)
7573, 74issubrg 13535 . . . . 5  |-  ( s  e.  (SubRing `  L
)  <->  ( ( L  e.  Ring  /\  ( Ls  s )  e.  Ring )  /\  ( s  C_  ( Base `  L )  /\  ( 1r `  L
)  e.  s ) ) )
7669, 72, 753bitr4g 223 . . . 4  |-  ( (
ph  /\  K  e.  Ring )  ->  ( s  e.  (SubRing `  K )  <->  s  e.  (SubRing `  L
) ) )
7776ex 115 . . 3  |-  ( ph  ->  ( K  e.  Ring  -> 
( s  e.  (SubRing `  K )  <->  s  e.  (SubRing `  L ) ) ) )
782, 9, 77pm5.21ndd 706 . 2  |-  ( ph  ->  ( s  e.  (SubRing `  K )  <->  s  e.  (SubRing `  L ) ) )
7978eqrdv 2187 1  |-  ( ph  ->  (SubRing `  K )  =  (SubRing `  L )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160   _Vcvv 2752    i^i cin 3143    C_ wss 3144   ` cfv 5231  (class class class)co 5891   Basecbs 12486   ↾s cress 12487   +g cplusg 12561   .rcmulr 12562   1rcur 13280   Ringcrg 13317  SubRingcsubrg 13531
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-addcom 7930  ax-addass 7932  ax-i2m1 7935  ax-0lt1 7936  ax-0id 7938  ax-rnegex 7939  ax-pre-ltirr 7942  ax-pre-lttrn 7944  ax-pre-ltadd 7946
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-pnf 8013  df-mnf 8014  df-ltxr 8016  df-inn 8939  df-2 8997  df-3 8998  df-ndx 12489  df-slot 12490  df-base 12492  df-sets 12493  df-iress 12494  df-plusg 12574  df-mulr 12575  df-0g 12735  df-mgm 12804  df-sgrp 12837  df-mnd 12850  df-grp 12920  df-mgp 13242  df-ur 13281  df-ring 13319  df-subrg 13533
This theorem is referenced by: (None)
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