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Theorem lidlrsppropd 17430
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 17394 . . . . 5  |-  ( Base `  K )  =  (
Base `  (ringLMod `  K
) )
31, 2syl6eq 2509 . . . 4  |-  ( ph  ->  B  =  ( Base `  (ringLMod `  K )
) )
4 lidlpropd.2 . . . . 5  |-  ( ph  ->  B  =  ( Base `  L ) )
5 rlmbas 17394 . . . . 5  |-  ( Base `  L )  =  (
Base `  (ringLMod `  L
) )
64, 5syl6eq 2509 . . . 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 17395 . . . . . 6  |-  ( +g  `  K )  =  ( +g  `  (ringLMod `  K
) )
109oveqi 6208 . . . . 5  |-  ( x ( +g  `  K
) y )  =  ( x ( +g  `  (ringLMod `  K )
) y )
11 rlmplusg 17395 . . . . . 6  |-  ( +g  `  L )  =  ( +g  `  (ringLMod `  L
) )
1211oveqi 6208 . . . . 5  |-  ( x ( +g  `  L
) y )  =  ( x ( +g  `  (ringLMod `  L )
) y )
138, 10, 123eqtr3g 2516 . . . 4  |-  ( (
ph  /\  ( x  e.  W  /\  y  e.  W ) )  -> 
( x ( +g  `  (ringLMod `  K )
) y )  =  ( x ( +g  `  (ringLMod `  L )
) y ) )
14 rlmvsca 17401 . . . . . 6  |-  ( .r
`  K )  =  ( .s `  (ringLMod `  K ) )
1514oveqi 6208 . . . . 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 2545 . . . 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 17401 . . . . . 6  |-  ( .r
`  L )  =  ( .s `  (ringLMod `  L ) )
2019oveqi 6208 . . . . 5  |-  ( x ( .r `  L
) y )  =  ( x ( .s
`  (ringLMod `  L )
) y )
2118, 15, 203eqtr3g 2516 . . . 4  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .s
`  (ringLMod `  K )
) y )  =  ( x ( .s
`  (ringLMod `  L )
) y ) )
22 baseid 14333 . . . . . . 7  |-  Base  = Slot  ( Base `  ndx )
23 eqid 2452 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
2422, 23strfvi 14327 . . . . . 6  |-  ( Base `  K )  =  (
Base `  (  _I  `  K ) )
25 rlmsca2 17400 . . . . . . 7  |-  (  _I 
`  K )  =  (Scalar `  (ringLMod `  K
) )
2625fveq2i 5797 . . . . . 6  |-  ( Base `  (  _I  `  K
) )  =  (
Base `  (Scalar `  (ringLMod `  K ) ) )
2724, 26eqtri 2481 . . . . 5  |-  ( Base `  K )  =  (
Base `  (Scalar `  (ringLMod `  K ) ) )
281, 27syl6eq 2509 . . . 4  |-  ( ph  ->  B  =  ( Base `  (Scalar `  (ringLMod `  K
) ) ) )
29 eqid 2452 . . . . . . 7  |-  ( Base `  L )  =  (
Base `  L )
3022, 29strfvi 14327 . . . . . 6  |-  ( Base `  L )  =  (
Base `  (  _I  `  L ) )
31 rlmsca2 17400 . . . . . . 7  |-  (  _I 
`  L )  =  (Scalar `  (ringLMod `  L
) )
3231fveq2i 5797 . . . . . 6  |-  ( Base `  (  _I  `  L
) )  =  (
Base `  (Scalar `  (ringLMod `  L ) ) )
3330, 32eqtri 2481 . . . . 5  |-  ( Base `  L )  =  (
Base `  (Scalar `  (ringLMod `  L ) ) )
344, 33syl6eq 2509 . . . 4  |-  ( ph  ->  B  =  ( Base `  (Scalar `  (ringLMod `  L
) ) ) )
353, 6, 7, 13, 17, 21, 28, 34lsspropd 17216 . . 3  |-  ( ph  ->  ( LSubSp `  (ringLMod `  K
) )  =  (
LSubSp `  (ringLMod `  L
) ) )
36 lidlval 17391 . . 3  |-  (LIdeal `  K )  =  (
LSubSp `  (ringLMod `  K
) )
37 lidlval 17391 . . 3  |-  (LIdeal `  L )  =  (
LSubSp `  (ringLMod `  L
) )
3835, 36, 373eqtr4g 2518 . 2  |-  ( ph  ->  (LIdeal `  K )  =  (LIdeal `  L )
)
39 fvex 5804 . . . . 5  |-  (ringLMod `  K
)  e.  _V
4039a1i 11 . . . 4  |-  ( ph  ->  (ringLMod `  K )  e.  _V )
41 fvex 5804 . . . . 5  |-  (ringLMod `  L
)  e.  _V
4241a1i 11 . . . 4  |-  ( ph  ->  (ringLMod `  L )  e.  _V )
433, 6, 7, 13, 17, 21, 28, 34, 40, 42lsppropd 17217 . . 3  |-  ( ph  ->  ( LSpan `  (ringLMod `  K
) )  =  (
LSpan `  (ringLMod `  L
) ) )
44 rspval 17392 . . 3  |-  (RSpan `  K )  =  (
LSpan `  (ringLMod `  K
) )
45 rspval 17392 . . 3  |-  (RSpan `  L )  =  (
LSpan `  (ringLMod `  L
) )
4643, 44, 453eqtr4g 2518 . 2  |-  ( ph  ->  (RSpan `  K )  =  (RSpan `  L )
)
4738, 46jca 532 1  |-  ( ph  ->  ( (LIdeal `  K
)  =  (LIdeal `  L )  /\  (RSpan `  K )  =  (RSpan `  L ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3072    C_ wss 3431    _I cid 4734   ` cfv 5521  (class class class)co 6195   ndxcnx 14284   Basecbs 14287   +g cplusg 14352   .rcmulr 14353  Scalarcsca 14355   .scvsca 14356   LSubSpclss 17131   LSpanclspn 17170  ringLModcrglmod 17368  LIdealclidl 17369  RSpancrsp 17370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-recs 6937  df-rdg 6971  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-3 10487  df-4 10488  df-5 10489  df-6 10490  df-7 10491  df-8 10492  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ress 14294  df-plusg 14365  df-sca 14368  df-vsca 14369  df-ip 14370  df-lss 17132  df-lsp 17171  df-sra 17371  df-rgmod 17372  df-lidl 17373  df-rsp 17374
This theorem is referenced by:  crngridl  17438
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