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Theorem lidlrsppropd 15982
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 15948 . . . . 5  |-  ( Base `  K )  =  (
Base `  (ringLMod `  K
) )
31, 2syl6eq 2331 . . . 4  |-  ( ph  ->  B  =  ( Base `  (ringLMod `  K )
) )
4 lidlpropd.2 . . . . 5  |-  ( ph  ->  B  =  ( Base `  L ) )
5 rlmbas 15948 . . . . 5  |-  ( Base `  L )  =  (
Base `  (ringLMod `  L
) )
64, 5syl6eq 2331 . . . 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 15949 . . . . . 6  |-  ( +g  `  K )  =  ( +g  `  (ringLMod `  K
) )
109oveqi 5871 . . . . 5  |-  ( x ( +g  `  K
) y )  =  ( x ( +g  `  (ringLMod `  K )
) y )
11 rlmplusg 15949 . . . . . 6  |-  ( +g  `  L )  =  ( +g  `  (ringLMod `  L
) )
1211oveqi 5871 . . . . 5  |-  ( x ( +g  `  L
) y )  =  ( x ( +g  `  (ringLMod `  L )
) y )
138, 10, 123eqtr3g 2338 . . . 4  |-  ( (
ph  /\  ( x  e.  W  /\  y  e.  W ) )  -> 
( x ( +g  `  (ringLMod `  K )
) y )  =  ( x ( +g  `  (ringLMod `  L )
) y ) )
14 rlmvsca 15954 . . . . . 6  |-  ( .r
`  K )  =  ( .s `  (ringLMod `  K ) )
1514oveqi 5871 . . . . 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 2368 . . . 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 15954 . . . . . 6  |-  ( .r
`  L )  =  ( .s `  (ringLMod `  L ) )
2019oveqi 5871 . . . . 5  |-  ( x ( .r `  L
) y )  =  ( x ( .s
`  (ringLMod `  L )
) y )
2118, 15, 203eqtr3g 2338 . . . 4  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .s
`  (ringLMod `  K )
) y )  =  ( x ( .s
`  (ringLMod `  L )
) y ) )
22 baseid 13190 . . . . . . 7  |-  Base  = Slot  ( Base `  ndx )
23 eqid 2283 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
2422, 23strfvi 13186 . . . . . 6  |-  ( Base `  K )  =  (
Base `  (  _I  `  K ) )
25 rlmsca2 15953 . . . . . . 7  |-  (  _I 
`  K )  =  (Scalar `  (ringLMod `  K
) )
2625fveq2i 5528 . . . . . 6  |-  ( Base `  (  _I  `  K
) )  =  (
Base `  (Scalar `  (ringLMod `  K ) ) )
2724, 26eqtri 2303 . . . . 5  |-  ( Base `  K )  =  (
Base `  (Scalar `  (ringLMod `  K ) ) )
281, 27syl6eq 2331 . . . 4  |-  ( ph  ->  B  =  ( Base `  (Scalar `  (ringLMod `  K
) ) ) )
29 eqid 2283 . . . . . . 7  |-  ( Base `  L )  =  (
Base `  L )
3022, 29strfvi 13186 . . . . . 6  |-  ( Base `  L )  =  (
Base `  (  _I  `  L ) )
31 rlmsca2 15953 . . . . . . 7  |-  (  _I 
`  L )  =  (Scalar `  (ringLMod `  L
) )
3231fveq2i 5528 . . . . . 6  |-  ( Base `  (  _I  `  L
) )  =  (
Base `  (Scalar `  (ringLMod `  L ) ) )
3330, 32eqtri 2303 . . . . 5  |-  ( Base `  L )  =  (
Base `  (Scalar `  (ringLMod `  L ) ) )
344, 33syl6eq 2331 . . . 4  |-  ( ph  ->  B  =  ( Base `  (Scalar `  (ringLMod `  L
) ) ) )
353, 6, 7, 13, 17, 21, 28, 34lsspropd 15774 . . 3  |-  ( ph  ->  ( LSubSp `  (ringLMod `  K
) )  =  (
LSubSp `  (ringLMod `  L
) ) )
36 lidlval 15946 . . 3  |-  (LIdeal `  K )  =  (
LSubSp `  (ringLMod `  K
) )
37 lidlval 15946 . . 3  |-  (LIdeal `  L )  =  (
LSubSp `  (ringLMod `  L
) )
3835, 36, 373eqtr4g 2340 . 2  |-  ( ph  ->  (LIdeal `  K )  =  (LIdeal `  L )
)
39 fvex 5539 . . . . 5  |-  (ringLMod `  K
)  e.  _V
4039a1i 10 . . . 4  |-  ( ph  ->  (ringLMod `  K )  e.  _V )
41 fvex 5539 . . . . 5  |-  (ringLMod `  L
)  e.  _V
4241a1i 10 . . . 4  |-  ( ph  ->  (ringLMod `  L )  e.  _V )
433, 6, 7, 13, 17, 21, 28, 34, 40, 42lsppropd 15775 . . 3  |-  ( ph  ->  ( LSpan `  (ringLMod `  K
) )  =  (
LSpan `  (ringLMod `  L
) ) )
44 rspval 15947 . . 3  |-  (RSpan `  K )  =  (
LSpan `  (ringLMod `  K
) )
45 rspval 15947 . . 3  |-  (RSpan `  L )  =  (
LSpan `  (ringLMod `  L
) )
4643, 44, 453eqtr4g 2340 . 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 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152    _I cid 4304   ` cfv 5255  (class class class)co 5858   ndxcnx 13145   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212   LSubSpclss 15689   LSpanclspn 15728  ringLModcrglmod 15922  LIdealclidl 15923  RSpancrsp 15924
This theorem is referenced by:  crngridl  15990
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-rep 4131  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-int 3863  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-4 9806  df-5 9807  df-6 9808  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-sca 13224  df-vsca 13225  df-lss 15690  df-lsp 15729  df-sra 15925  df-rgmod 15926  df-lidl 15927  df-rsp 15928
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