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Theorem subrgpropd 15579
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 15372 . . . . 5  |-  ( ph  ->  ( K  e.  Ring  <->  L  e.  Ring ) )
61ineq2d 3370 . . . . . . 7  |-  ( ph  ->  ( s  i^i  B
)  =  ( s  i^i  ( Base `  K
) ) )
7 vex 2791 . . . . . . . 8  |-  s  e. 
_V
8 eqid 2283 . . . . . . . . 9  |-  ( Ks  s )  =  ( Ks  s )
9 eqid 2283 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
108, 9ressbas 13198 . . . . . . . 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 2331 . . . . . 6  |-  ( ph  ->  ( s  i^i  B
)  =  ( Base `  ( Ks  s ) ) )
132ineq2d 3370 . . . . . . 7  |-  ( ph  ->  ( s  i^i  B
)  =  ( s  i^i  ( Base `  L
) ) )
14 eqid 2283 . . . . . . . . 9  |-  ( Ls  s )  =  ( Ls  s )
15 eqid 2283 . . . . . . . . 9  |-  ( Base `  L )  =  (
Base `  L )
1614, 15ressbas 13198 . . . . . . . 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 2331 . . . . . 6  |-  ( ph  ->  ( s  i^i  B
)  =  ( Base `  ( Ls  s ) ) )
19 inss2 3390 . . . . . . . . 9  |-  ( s  i^i  B )  C_  B
2019sseli 3176 . . . . . . . 8  |-  ( x  e.  ( s  i^i 
B )  ->  x  e.  B )
2119sseli 3176 . . . . . . . 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 2283 . . . . . . . . . . 11  |-  ( +g  `  K )  =  ( +g  `  K )
248, 23ressplusg 13250 . . . . . . . . . 10  |-  ( s  e.  _V  ->  ( +g  `  K )  =  ( +g  `  ( Ks  s ) ) )
257, 24ax-mp 8 . . . . . . . . 9  |-  ( +g  `  K )  =  ( +g  `  ( Ks  s ) )
2625oveqi 5871 . . . . . . . 8  |-  ( x ( +g  `  K
) y )  =  ( x ( +g  `  ( Ks  s ) ) y )
27 eqid 2283 . . . . . . . . . . 11  |-  ( +g  `  L )  =  ( +g  `  L )
2814, 27ressplusg 13250 . . . . . . . . . 10  |-  ( s  e.  _V  ->  ( +g  `  L )  =  ( +g  `  ( Ls  s ) ) )
297, 28ax-mp 8 . . . . . . . . 9  |-  ( +g  `  L )  =  ( +g  `  ( Ls  s ) )
3029oveqi 5871 . . . . . . . 8  |-  ( x ( +g  `  L
) y )  =  ( x ( +g  `  ( Ls  s ) ) y )
313, 26, 303eqtr3g 2338 . . . . . . 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 2283 . . . . . . . . . . 11  |-  ( .r
`  K )  =  ( .r `  K
)
348, 33ressmulr 13261 . . . . . . . . . 10  |-  ( s  e.  _V  ->  ( .r `  K )  =  ( .r `  ( Ks  s ) ) )
357, 34ax-mp 8 . . . . . . . . 9  |-  ( .r
`  K )  =  ( .r `  ( Ks  s ) )
3635oveqi 5871 . . . . . . . 8  |-  ( x ( .r `  K
) y )  =  ( x ( .r
`  ( Ks  s ) ) y )
37 eqid 2283 . . . . . . . . . . 11  |-  ( .r
`  L )  =  ( .r `  L
)
3814, 37ressmulr 13261 . . . . . . . . . 10  |-  ( s  e.  _V  ->  ( .r `  L )  =  ( .r `  ( Ls  s ) ) )
397, 38ax-mp 8 . . . . . . . . 9  |-  ( .r
`  L )  =  ( .r `  ( Ls  s ) )
4039oveqi 5871 . . . . . . . 8  |-  ( x ( .r `  L
) y )  =  ( x ( .r
`  ( Ls  s ) ) y )
414, 36, 403eqtr3g 2338 . . . . . . 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 15372 . . . . 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 2317 . . . . . 6  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
4645sseq2d 3206 . . . . 5  |-  ( ph  ->  ( s  C_  ( Base `  K )  <->  s  C_  ( Base `  L )
) )
471, 2, 4rngidpropd 15477 . . . . . 6  |-  ( ph  ->  ( 1r `  K
)  =  ( 1r
`  L ) )
4847eleq1d 2349 . . . . 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 2283 . . . 4  |-  ( 1r
`  K )  =  ( 1r `  K
)
529, 51issubrg 15545 . . 3  |-  ( s  e.  (SubRing `  K
)  <->  ( ( K  e.  Ring  /\  ( Ks  s )  e.  Ring )  /\  ( s  C_  ( Base `  K )  /\  ( 1r `  K
)  e.  s ) ) )
53 eqid 2283 . . . 4  |-  ( 1r
`  L )  =  ( 1r `  L
)
5415, 53issubrg 15545 . . 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 2281 1  |-  ( ph  ->  (SubRing `  K )  =  (SubRing `  L )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149   +g cplusg 13208   .rcmulr 13209   Ringcrg 15337   1rcur 15339  SubRingcsubrg 15541
This theorem is referenced by:  ply1subrg  16276  subrgply1  16311
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mnd 14367  df-grp 14489  df-mgp 15326  df-rng 15340  df-ur 15342  df-subrg 15543
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