MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lidlrsppropd Structured version   Unicode version

Theorem lidlrsppropd 16293
Description: The left ideals and ring span of a ring depend only on the ring components. Here  W is expected to be either 
B (when closure is available) or  _V (when strong equality is available). (Contributed by Mario Carneiro, 14-Jun-2015.)
Hypotheses
Ref Expression
lidlpropd.1  |-  ( ph  ->  B  =  ( Base `  K ) )
lidlpropd.2  |-  ( ph  ->  B  =  ( Base `  L ) )
lidlpropd.3  |-  ( ph  ->  B  C_  W )
lidlpropd.4  |-  ( (
ph  /\  ( x  e.  W  /\  y  e.  W ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
lidlpropd.5  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .r
`  K ) y )  e.  W )
lidlpropd.6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .r
`  K ) y )  =  ( x ( .r `  L
) y ) )
Assertion
Ref Expression
lidlrsppropd  |-  ( ph  ->  ( (LIdeal `  K
)  =  (LIdeal `  L )  /\  (RSpan `  K )  =  (RSpan `  L ) ) )
Distinct variable groups:    x, y, B    x, K, y    x, L, y    ph, x, y   
x, W, y

Proof of Theorem lidlrsppropd
StepHypRef Expression
1 lidlpropd.1 . . . . 5  |-  ( ph  ->  B  =  ( Base `  K ) )
2 rlmbas 16259 . . . . 5  |-  ( Base `  K )  =  (
Base `  (ringLMod `  K
) )
31, 2syl6eq 2483 . . . 4  |-  ( ph  ->  B  =  ( Base `  (ringLMod `  K )
) )
4 lidlpropd.2 . . . . 5  |-  ( ph  ->  B  =  ( Base `  L ) )
5 rlmbas 16259 . . . . 5  |-  ( Base `  L )  =  (
Base `  (ringLMod `  L
) )
64, 5syl6eq 2483 . . . 4  |-  ( ph  ->  B  =  ( Base `  (ringLMod `  L )
) )
7 lidlpropd.3 . . . 4  |-  ( ph  ->  B  C_  W )
8 lidlpropd.4 . . . . 5  |-  ( (
ph  /\  ( x  e.  W  /\  y  e.  W ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
9 rlmplusg 16260 . . . . . 6  |-  ( +g  `  K )  =  ( +g  `  (ringLMod `  K
) )
109oveqi 6086 . . . . 5  |-  ( x ( +g  `  K
) y )  =  ( x ( +g  `  (ringLMod `  K )
) y )
11 rlmplusg 16260 . . . . . 6  |-  ( +g  `  L )  =  ( +g  `  (ringLMod `  L
) )
1211oveqi 6086 . . . . 5  |-  ( x ( +g  `  L
) y )  =  ( x ( +g  `  (ringLMod `  L )
) y )
138, 10, 123eqtr3g 2490 . . . 4  |-  ( (
ph  /\  ( x  e.  W  /\  y  e.  W ) )  -> 
( x ( +g  `  (ringLMod `  K )
) y )  =  ( x ( +g  `  (ringLMod `  L )
) y ) )
14 rlmvsca 16265 . . . . . 6  |-  ( .r
`  K )  =  ( .s `  (ringLMod `  K ) )
1514oveqi 6086 . . . . 5  |-  ( x ( .r `  K
) y )  =  ( x ( .s
`  (ringLMod `  K )
) y )
16 lidlpropd.5 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .r
`  K ) y )  e.  W )
1715, 16syl5eqelr 2520 . . . 4  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .s
`  (ringLMod `  K )
) y )  e.  W )
18 lidlpropd.6 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .r
`  K ) y )  =  ( x ( .r `  L
) y ) )
19 rlmvsca 16265 . . . . . 6  |-  ( .r
`  L )  =  ( .s `  (ringLMod `  L ) )
2019oveqi 6086 . . . . 5  |-  ( x ( .r `  L
) y )  =  ( x ( .s
`  (ringLMod `  L )
) y )
2118, 15, 203eqtr3g 2490 . . . 4  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .s
`  (ringLMod `  K )
) y )  =  ( x ( .s
`  (ringLMod `  L )
) y ) )
22 baseid 13503 . . . . . . 7  |-  Base  = Slot  ( Base `  ndx )
23 eqid 2435 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
2422, 23strfvi 13499 . . . . . 6  |-  ( Base `  K )  =  (
Base `  (  _I  `  K ) )
25 rlmsca2 16264 . . . . . . 7  |-  (  _I 
`  K )  =  (Scalar `  (ringLMod `  K
) )
2625fveq2i 5723 . . . . . 6  |-  ( Base `  (  _I  `  K
) )  =  (
Base `  (Scalar `  (ringLMod `  K ) ) )
2724, 26eqtri 2455 . . . . 5  |-  ( Base `  K )  =  (
Base `  (Scalar `  (ringLMod `  K ) ) )
281, 27syl6eq 2483 . . . 4  |-  ( ph  ->  B  =  ( Base `  (Scalar `  (ringLMod `  K
) ) ) )
29 eqid 2435 . . . . . . 7  |-  ( Base `  L )  =  (
Base `  L )
3022, 29strfvi 13499 . . . . . 6  |-  ( Base `  L )  =  (
Base `  (  _I  `  L ) )
31 rlmsca2 16264 . . . . . . 7  |-  (  _I 
`  L )  =  (Scalar `  (ringLMod `  L
) )
3231fveq2i 5723 . . . . . 6  |-  ( Base `  (  _I  `  L
) )  =  (
Base `  (Scalar `  (ringLMod `  L ) ) )
3330, 32eqtri 2455 . . . . 5  |-  ( Base `  L )  =  (
Base `  (Scalar `  (ringLMod `  L ) ) )
344, 33syl6eq 2483 . . . 4  |-  ( ph  ->  B  =  ( Base `  (Scalar `  (ringLMod `  L
) ) ) )
353, 6, 7, 13, 17, 21, 28, 34lsspropd 16085 . . 3  |-  ( ph  ->  ( LSubSp `  (ringLMod `  K
) )  =  (
LSubSp `  (ringLMod `  L
) ) )
36 lidlval 16257 . . 3  |-  (LIdeal `  K )  =  (
LSubSp `  (ringLMod `  K
) )
37 lidlval 16257 . . 3  |-  (LIdeal `  L )  =  (
LSubSp `  (ringLMod `  L
) )
3835, 36, 373eqtr4g 2492 . 2  |-  ( ph  ->  (LIdeal `  K )  =  (LIdeal `  L )
)
39 fvex 5734 . . . . 5  |-  (ringLMod `  K
)  e.  _V
4039a1i 11 . . . 4  |-  ( ph  ->  (ringLMod `  K )  e.  _V )
41 fvex 5734 . . . . 5  |-  (ringLMod `  L
)  e.  _V
4241a1i 11 . . . 4  |-  ( ph  ->  (ringLMod `  L )  e.  _V )
433, 6, 7, 13, 17, 21, 28, 34, 40, 42lsppropd 16086 . . 3  |-  ( ph  ->  ( LSpan `  (ringLMod `  K
) )  =  (
LSpan `  (ringLMod `  L
) ) )
44 rspval 16258 . . 3  |-  (RSpan `  K )  =  (
LSpan `  (ringLMod `  K
) )
45 rspval 16258 . . 3  |-  (RSpan `  L )  =  (
LSpan `  (ringLMod `  L
) )
4643, 44, 453eqtr4g 2492 . 2  |-  ( ph  ->  (RSpan `  K )  =  (RSpan `  L )
)
4738, 46jca 519 1  |-  ( ph  ->  ( (LIdeal `  K
)  =  (LIdeal `  L )  /\  (RSpan `  K )  =  (RSpan `  L ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948    C_ wss 3312    _I cid 4485   ` cfv 5446  (class class class)co 6073   ndxcnx 13458   Basecbs 13461   +g cplusg 13521   .rcmulr 13522  Scalarcsca 13524   .scvsca 13525   LSubSpclss 16000   LSpanclspn 16039  ringLModcrglmod 16233  LIdealclidl 16234  RSpancrsp 16235
This theorem is referenced by:  crngridl  16301
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-sca 13537  df-vsca 13538  df-lss 16001  df-lsp 16040  df-sra 16236  df-rgmod 16237  df-lidl 16238  df-rsp 16239
  Copyright terms: Public domain W3C validator