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Theorem subrgpropd 15902
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 15695 . . . . 5  |-  ( ph  ->  ( K  e.  Ring  <->  L  e.  Ring ) )
61ineq2d 3542 . . . . . . 7  |-  ( ph  ->  ( s  i^i  B
)  =  ( s  i^i  ( Base `  K
) ) )
7 vex 2959 . . . . . . . 8  |-  s  e. 
_V
8 eqid 2436 . . . . . . . . 9  |-  ( Ks  s )  =  ( Ks  s )
9 eqid 2436 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
108, 9ressbas 13519 . . . . . . . 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 2484 . . . . . 6  |-  ( ph  ->  ( s  i^i  B
)  =  ( Base `  ( Ks  s ) ) )
132ineq2d 3542 . . . . . . 7  |-  ( ph  ->  ( s  i^i  B
)  =  ( s  i^i  ( Base `  L
) ) )
14 eqid 2436 . . . . . . . . 9  |-  ( Ls  s )  =  ( Ls  s )
15 eqid 2436 . . . . . . . . 9  |-  ( Base `  L )  =  (
Base `  L )
1614, 15ressbas 13519 . . . . . . . 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 2484 . . . . . 6  |-  ( ph  ->  ( s  i^i  B
)  =  ( Base `  ( Ls  s ) ) )
19 inss2 3562 . . . . . . . . 9  |-  ( s  i^i  B )  C_  B
2019sseli 3344 . . . . . . . 8  |-  ( x  e.  ( s  i^i 
B )  ->  x  e.  B )
2119sseli 3344 . . . . . . . 8  |-  ( y  e.  ( s  i^i 
B )  ->  y  e.  B )
2220, 21anim12i 550 . . . . . . 7  |-  ( ( x  e.  ( s  i^i  B )  /\  y  e.  ( s  i^i  B ) )  -> 
( x  e.  B  /\  y  e.  B
) )
23 eqid 2436 . . . . . . . . . . 11  |-  ( +g  `  K )  =  ( +g  `  K )
248, 23ressplusg 13571 . . . . . . . . . 10  |-  ( s  e.  _V  ->  ( +g  `  K )  =  ( +g  `  ( Ks  s ) ) )
257, 24ax-mp 8 . . . . . . . . 9  |-  ( +g  `  K )  =  ( +g  `  ( Ks  s ) )
2625oveqi 6094 . . . . . . . 8  |-  ( x ( +g  `  K
) y )  =  ( x ( +g  `  ( Ks  s ) ) y )
27 eqid 2436 . . . . . . . . . . 11  |-  ( +g  `  L )  =  ( +g  `  L )
2814, 27ressplusg 13571 . . . . . . . . . 10  |-  ( s  e.  _V  ->  ( +g  `  L )  =  ( +g  `  ( Ls  s ) ) )
297, 28ax-mp 8 . . . . . . . . 9  |-  ( +g  `  L )  =  ( +g  `  ( Ls  s ) )
3029oveqi 6094 . . . . . . . 8  |-  ( x ( +g  `  L
) y )  =  ( x ( +g  `  ( Ls  s ) ) y )
313, 26, 303eqtr3g 2491 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  ( Ks  s ) ) y )  =  ( x ( +g  `  ( Ls  s ) ) y ) )
3222, 31sylan2 461 . . . . . 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 2436 . . . . . . . . . . 11  |-  ( .r
`  K )  =  ( .r `  K
)
348, 33ressmulr 13582 . . . . . . . . . 10  |-  ( s  e.  _V  ->  ( .r `  K )  =  ( .r `  ( Ks  s ) ) )
357, 34ax-mp 8 . . . . . . . . 9  |-  ( .r
`  K )  =  ( .r `  ( Ks  s ) )
3635oveqi 6094 . . . . . . . 8  |-  ( x ( .r `  K
) y )  =  ( x ( .r
`  ( Ks  s ) ) y )
37 eqid 2436 . . . . . . . . . . 11  |-  ( .r
`  L )  =  ( .r `  L
)
3814, 37ressmulr 13582 . . . . . . . . . 10  |-  ( s  e.  _V  ->  ( .r `  L )  =  ( .r `  ( Ls  s ) ) )
397, 38ax-mp 8 . . . . . . . . 9  |-  ( .r
`  L )  =  ( .r `  ( Ls  s ) )
4039oveqi 6094 . . . . . . . 8  |-  ( x ( .r `  L
) y )  =  ( x ( .r
`  ( Ls  s ) ) y )
414, 36, 403eqtr3g 2491 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .r
`  ( Ks  s ) ) y )  =  ( x ( .r
`  ( Ls  s ) ) y ) )
4222, 41sylan2 461 . . . . . 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 15695 . . . . 5  |-  ( ph  ->  ( ( Ks  s )  e.  Ring  <->  ( Ls  s )  e.  Ring ) )
445, 43anbi12d 692 . . . 4  |-  ( ph  ->  ( ( K  e. 
Ring  /\  ( Ks  s )  e.  Ring )  <->  ( L  e.  Ring  /\  ( Ls  s
)  e.  Ring )
) )
451, 2eqtr3d 2470 . . . . . 6  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
4645sseq2d 3376 . . . . 5  |-  ( ph  ->  ( s  C_  ( Base `  K )  <->  s  C_  ( Base `  L )
) )
471, 2, 4rngidpropd 15800 . . . . . 6  |-  ( ph  ->  ( 1r `  K
)  =  ( 1r
`  L ) )
4847eleq1d 2502 . . . . 5  |-  ( ph  ->  ( ( 1r `  K )  e.  s  <-> 
( 1r `  L
)  e.  s ) )
4946, 48anbi12d 692 . . . 4  |-  ( ph  ->  ( ( s  C_  ( Base `  K )  /\  ( 1r `  K
)  e.  s )  <-> 
( s  C_  ( Base `  L )  /\  ( 1r `  L )  e.  s ) ) )
5044, 49anbi12d 692 . . 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 2436 . . . 4  |-  ( 1r
`  K )  =  ( 1r `  K
)
529, 51issubrg 15868 . . 3  |-  ( s  e.  (SubRing `  K
)  <->  ( ( K  e.  Ring  /\  ( Ks  s )  e.  Ring )  /\  ( s  C_  ( Base `  K )  /\  ( 1r `  K
)  e.  s ) ) )
53 eqid 2436 . . . 4  |-  ( 1r
`  L )  =  ( 1r `  L
)
5415, 53issubrg 15868 . . 3  |-  ( s  e.  (SubRing `  L
)  <->  ( ( L  e.  Ring  /\  ( Ls  s )  e.  Ring )  /\  ( s  C_  ( Base `  L )  /\  ( 1r `  L
)  e.  s ) ) )
5550, 52, 543bitr4g 280 . 2  |-  ( ph  ->  ( s  e.  (SubRing `  K )  <->  s  e.  (SubRing `  L ) ) )
5655eqrdv 2434 1  |-  ( ph  ->  (SubRing `  K )  =  (SubRing `  L )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    i^i cin 3319    C_ wss 3320   ` cfv 5454  (class class class)co 6081   Basecbs 13469   ↾s cress 13470   +g cplusg 13529   .rcmulr 13530   Ringcrg 15660   1rcur 15662  SubRingcsubrg 15864
This theorem is referenced by:  ply1subrg  16595  subrgply1  16627
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-0g 13727  df-mnd 14690  df-grp 14812  df-mgp 15649  df-rng 15663  df-ur 15665  df-subrg 15866
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