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Theorem lidlrsppropd 15998
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 15964 . . . . 5  |-  ( Base `  K )  =  (
Base `  (ringLMod `  K
) )
31, 2syl6eq 2344 . . . 4  |-  ( ph  ->  B  =  ( Base `  (ringLMod `  K )
) )
4 lidlpropd.2 . . . . 5  |-  ( ph  ->  B  =  ( Base `  L ) )
5 rlmbas 15964 . . . . 5  |-  ( Base `  L )  =  (
Base `  (ringLMod `  L
) )
64, 5syl6eq 2344 . . . 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 15965 . . . . . 6  |-  ( +g  `  K )  =  ( +g  `  (ringLMod `  K
) )
109oveqi 5887 . . . . 5  |-  ( x ( +g  `  K
) y )  =  ( x ( +g  `  (ringLMod `  K )
) y )
11 rlmplusg 15965 . . . . . 6  |-  ( +g  `  L )  =  ( +g  `  (ringLMod `  L
) )
1211oveqi 5887 . . . . 5  |-  ( x ( +g  `  L
) y )  =  ( x ( +g  `  (ringLMod `  L )
) y )
138, 10, 123eqtr3g 2351 . . . 4  |-  ( (
ph  /\  ( x  e.  W  /\  y  e.  W ) )  -> 
( x ( +g  `  (ringLMod `  K )
) y )  =  ( x ( +g  `  (ringLMod `  L )
) y ) )
14 rlmvsca 15970 . . . . . 6  |-  ( .r
`  K )  =  ( .s `  (ringLMod `  K ) )
1514oveqi 5887 . . . . 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 2381 . . . 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 15970 . . . . . 6  |-  ( .r
`  L )  =  ( .s `  (ringLMod `  L ) )
2019oveqi 5887 . . . . 5  |-  ( x ( .r `  L
) y )  =  ( x ( .s
`  (ringLMod `  L )
) y )
2118, 15, 203eqtr3g 2351 . . . 4  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .s
`  (ringLMod `  K )
) y )  =  ( x ( .s
`  (ringLMod `  L )
) y ) )
22 baseid 13206 . . . . . . 7  |-  Base  = Slot  ( Base `  ndx )
23 eqid 2296 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
2422, 23strfvi 13202 . . . . . 6  |-  ( Base `  K )  =  (
Base `  (  _I  `  K ) )
25 rlmsca2 15969 . . . . . . 7  |-  (  _I 
`  K )  =  (Scalar `  (ringLMod `  K
) )
2625fveq2i 5544 . . . . . 6  |-  ( Base `  (  _I  `  K
) )  =  (
Base `  (Scalar `  (ringLMod `  K ) ) )
2724, 26eqtri 2316 . . . . 5  |-  ( Base `  K )  =  (
Base `  (Scalar `  (ringLMod `  K ) ) )
281, 27syl6eq 2344 . . . 4  |-  ( ph  ->  B  =  ( Base `  (Scalar `  (ringLMod `  K
) ) ) )
29 eqid 2296 . . . . . . 7  |-  ( Base `  L )  =  (
Base `  L )
3022, 29strfvi 13202 . . . . . 6  |-  ( Base `  L )  =  (
Base `  (  _I  `  L ) )
31 rlmsca2 15969 . . . . . . 7  |-  (  _I 
`  L )  =  (Scalar `  (ringLMod `  L
) )
3231fveq2i 5544 . . . . . 6  |-  ( Base `  (  _I  `  L
) )  =  (
Base `  (Scalar `  (ringLMod `  L ) ) )
3330, 32eqtri 2316 . . . . 5  |-  ( Base `  L )  =  (
Base `  (Scalar `  (ringLMod `  L ) ) )
344, 33syl6eq 2344 . . . 4  |-  ( ph  ->  B  =  ( Base `  (Scalar `  (ringLMod `  L
) ) ) )
353, 6, 7, 13, 17, 21, 28, 34lsspropd 15790 . . 3  |-  ( ph  ->  ( LSubSp `  (ringLMod `  K
) )  =  (
LSubSp `  (ringLMod `  L
) ) )
36 lidlval 15962 . . 3  |-  (LIdeal `  K )  =  (
LSubSp `  (ringLMod `  K
) )
37 lidlval 15962 . . 3  |-  (LIdeal `  L )  =  (
LSubSp `  (ringLMod `  L
) )
3835, 36, 373eqtr4g 2353 . 2  |-  ( ph  ->  (LIdeal `  K )  =  (LIdeal `  L )
)
39 fvex 5555 . . . . 5  |-  (ringLMod `  K
)  e.  _V
4039a1i 10 . . . 4  |-  ( ph  ->  (ringLMod `  K )  e.  _V )
41 fvex 5555 . . . . 5  |-  (ringLMod `  L
)  e.  _V
4241a1i 10 . . . 4  |-  ( ph  ->  (ringLMod `  L )  e.  _V )
433, 6, 7, 13, 17, 21, 28, 34, 40, 42lsppropd 15791 . . 3  |-  ( ph  ->  ( LSpan `  (ringLMod `  K
) )  =  (
LSpan `  (ringLMod `  L
) ) )
44 rspval 15963 . . 3  |-  (RSpan `  K )  =  (
LSpan `  (ringLMod `  K
) )
45 rspval 15963 . . 3  |-  (RSpan `  L )  =  (
LSpan `  (ringLMod `  L
) )
4643, 44, 453eqtr4g 2353 . 2  |-  ( ph  ->  (RSpan `  K )  =  (RSpan `  L )
)
4738, 46jca 518 1  |-  ( ph  ->  ( (LIdeal `  K
)  =  (LIdeal `  L )  /\  (RSpan `  K )  =  (RSpan `  L ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165    _I cid 4320   ` cfv 5271  (class class class)co 5874   ndxcnx 13161   Basecbs 13164   +g cplusg 13224   .rcmulr 13225  Scalarcsca 13227   .scvsca 13228   LSubSpclss 15705   LSpanclspn 15744  ringLModcrglmod 15938  LIdealclidl 15939  RSpancrsp 15940
This theorem is referenced by:  crngridl  16006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-sca 13240  df-vsca 13241  df-lss 15706  df-lsp 15745  df-sra 15941  df-rgmod 15942  df-lidl 15943  df-rsp 15944
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