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Theorem subrgpropd 13307
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 13285 . . . 4  |-  ( s  e.  (SubRing `  K
)  ->  K  e.  Ring )
21a1i 9 . . 3  |-  ( ph  ->  ( s  e.  (SubRing `  K )  ->  K  e.  Ring ) )
3 subrgrcl 13285 . . . 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 13148 . . . 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 3336 . . . . . . . . . 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 2178 . . . . . . . . . . 11  |-  ( ( ( ph  /\  K  e.  Ring )  /\  s  e.  _V )  ->  ( Ks  s )  =  ( Ks  s ) )
14 eqidd 2178 . . . . . . . . . . 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 12519 . . . . . . . . . 10  |-  ( ( ( ph  /\  K  e.  Ring )  /\  s  e.  _V )  ->  (
s  i^i  ( Base `  K ) )  =  ( Base `  ( Ks  s ) ) )
1817elvd 2742 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  Ring )  ->  ( s  i^i  ( Base `  K
) )  =  (
Base `  ( Ks  s
) ) )
1912, 18eqtrd 2210 . . . . . . . 8  |-  ( (
ph  /\  K  e.  Ring )  ->  ( s  i^i  B )  =  (
Base `  ( Ks  s
) ) )
205ineq2d 3336 . . . . . . . . . 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 2178 . . . . . . . . . . 11  |-  ( ( ( ph  /\  K  e.  Ring )  /\  s  e.  _V )  ->  ( Ls  s )  =  ( Ls  s ) )
23 eqidd 2178 . . . . . . . . . . 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 12519 . . . . . . . . . 10  |-  ( ( ( ph  /\  K  e.  Ring )  /\  s  e.  _V )  ->  (
s  i^i  ( Base `  L ) )  =  ( Base `  ( Ls  s ) ) )
2726elvd 2742 . . . . . . . . 9  |-  ( (
ph  /\  K  e.  Ring )  ->  ( s  i^i  ( Base `  L
) )  =  (
Base `  ( Ls  s
) ) )
2821, 27eqtrd 2210 . . . . . . . 8  |-  ( (
ph  /\  K  e.  Ring )  ->  ( s  i^i  B )  =  (
Base `  ( Ls  s
) ) )
29 elinel2 3322 . . . . . . . . . 10  |-  ( x  e.  ( s  i^i 
B )  ->  x  e.  B )
30 elinel2 3322 . . . . . . . . . 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 2178 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  K  e.  Ring )  /\  s  e.  _V )  ->  ( +g  `  K )  =  ( +g  `  K
) )
3413, 33, 16, 15ressplusgd 12579 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  K  e.  Ring )  /\  s  e.  _V )  ->  ( +g  `  K )  =  ( +g  `  ( Ks  s ) ) )
3534elvd 2742 . . . . . . . . . . 11  |-  ( (
ph  /\  K  e.  Ring )  ->  ( +g  `  K )  =  ( +g  `  ( Ks  s ) ) )
3635oveqdr 5900 . . . . . . . . . 10  |-  ( ( ( ph  /\  K  e.  Ring )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( +g  `  K ) y )  =  ( x ( +g  `  ( Ks  s ) ) y ) )
37 eqidd 2178 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  K  e.  Ring )  /\  s  e.  _V )  ->  ( +g  `  L )  =  ( +g  `  L
) )
3822, 37, 16, 25ressplusgd 12579 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  K  e.  Ring )  /\  s  e.  _V )  ->  ( +g  `  L )  =  ( +g  `  ( Ls  s ) ) )
3938elvd 2742 . . . . . . . . . . 11  |-  ( (
ph  /\  K  e.  Ring )  ->  ( +g  `  L )  =  ( +g  `  ( Ls  s ) ) )
4039oveqdr 5900 . . . . . . . . . 10  |-  ( ( ( ph  /\  K  e.  Ring )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( +g  `  L ) y )  =  ( x ( +g  `  ( Ls  s ) ) y ) )
4132, 36, 403eqtr3d 2218 . . . . . . . . 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 2740 . . . . . . . . . . . . 13  |-  s  e. 
_V
45 eqid 2177 . . . . . . . . . . . . . 14  |-  ( Ks  s )  =  ( Ks  s )
46 eqid 2177 . . . . . . . . . . . . . 14  |-  ( .r
`  K )  =  ( .r `  K
)
4745, 46ressmulrg 12595 . . . . . . . . . . . . 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 5900 . . . . . . . . . 10  |-  ( ( ( ph  /\  K  e.  Ring )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( .r `  K
) y )  =  ( x ( .r
`  ( Ks  s ) ) y ) )
51 eqid 2177 . . . . . . . . . . . . 13  |-  ( Ls  s )  =  ( Ls  s )
52 eqid 2177 . . . . . . . . . . . . 13  |-  ( .r
`  L )  =  ( .r `  L
)
5351, 52ressmulrg 12595 . . . . . . . . . . . 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 5900 . . . . . . . . . 10  |-  ( ( ( ph  /\  K  e.  Ring )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( .r `  L
) y )  =  ( x ( .r
`  ( Ls  s ) ) y ) )
5643, 50, 553eqtr3d 2218 . . . . . . . . 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 13148 . . . . . . 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 2212 . . . . . . . . 9  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
6160sseq2d 3185 . . . . . . . 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 13246 . . . . . . . 8  |-  ( (
ph  /\  K  e.  Ring )  ->  ( 1r `  K )  =  ( 1r `  L ) )
6766eleq1d 2246 . . . . . . 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 2177 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
71 eqid 2177 . . . . . 6  |-  ( 1r
`  K )  =  ( 1r `  K
)
7270, 71issubrg 13280 . . . . 5  |-  ( s  e.  (SubRing `  K
)  <->  ( ( K  e.  Ring  /\  ( Ks  s )  e.  Ring )  /\  ( s  C_  ( Base `  K )  /\  ( 1r `  K
)  e.  s ) ) )
73 eqid 2177 . . . . . 6  |-  ( Base `  L )  =  (
Base `  L )
74 eqid 2177 . . . . . 6  |-  ( 1r
`  L )  =  ( 1r `  L
)
7573, 74issubrg 13280 . . . . 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 705 . 2  |-  ( ph  ->  ( s  e.  (SubRing `  K )  <->  s  e.  (SubRing `  L ) ) )
7978eqrdv 2175 1  |-  ( ph  ->  (SubRing `  K )  =  (SubRing `  L )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   _Vcvv 2737    i^i cin 3128    C_ wss 3129   ` cfv 5215  (class class class)co 5872   Basecbs 12454   ↾s cress 12455   +g cplusg 12528   .rcmulr 12529   1rcur 13073   Ringcrg 13110  SubRingcsubrg 13276
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-pre-ltirr 7920  ax-pre-lttrn 7922  ax-pre-ltadd 7924
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-inn 8916  df-2 8974  df-3 8975  df-ndx 12457  df-slot 12458  df-base 12460  df-sets 12461  df-iress 12462  df-plusg 12541  df-mulr 12542  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-mgp 13062  df-ur 13074  df-ring 13112  df-subrg 13278
This theorem is referenced by: (None)
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