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Theorem lidlrsppropd 18196
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 18159 . . . . 5  |-  ( Base `  K )  =  (
Base `  (ringLMod `  K
) )
31, 2syl6eq 2459 . . . 4  |-  ( ph  ->  B  =  ( Base `  (ringLMod `  K )
) )
4 lidlpropd.2 . . . . 5  |-  ( ph  ->  B  =  ( Base `  L ) )
5 rlmbas 18159 . . . . 5  |-  ( Base `  L )  =  (
Base `  (ringLMod `  L
) )
64, 5syl6eq 2459 . . . 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 18160 . . . . . 6  |-  ( +g  `  K )  =  ( +g  `  (ringLMod `  K
) )
109oveqi 6290 . . . . 5  |-  ( x ( +g  `  K
) y )  =  ( x ( +g  `  (ringLMod `  K )
) y )
11 rlmplusg 18160 . . . . . 6  |-  ( +g  `  L )  =  ( +g  `  (ringLMod `  L
) )
1211oveqi 6290 . . . . 5  |-  ( x ( +g  `  L
) y )  =  ( x ( +g  `  (ringLMod `  L )
) y )
138, 10, 123eqtr3g 2466 . . . 4  |-  ( (
ph  /\  ( x  e.  W  /\  y  e.  W ) )  -> 
( x ( +g  `  (ringLMod `  K )
) y )  =  ( x ( +g  `  (ringLMod `  L )
) y ) )
14 rlmvsca 18166 . . . . . 6  |-  ( .r
`  K )  =  ( .s `  (ringLMod `  K ) )
1514oveqi 6290 . . . . 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 2495 . . . 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 18166 . . . . . 6  |-  ( .r
`  L )  =  ( .s `  (ringLMod `  L ) )
2019oveqi 6290 . . . . 5  |-  ( x ( .r `  L
) y )  =  ( x ( .s
`  (ringLMod `  L )
) y )
2118, 15, 203eqtr3g 2466 . . . 4  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .s
`  (ringLMod `  K )
) y )  =  ( x ( .s
`  (ringLMod `  L )
) y ) )
22 baseid 14887 . . . . . . 7  |-  Base  = Slot  ( Base `  ndx )
23 eqid 2402 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
2422, 23strfvi 14881 . . . . . 6  |-  ( Base `  K )  =  (
Base `  (  _I  `  K ) )
25 rlmsca2 18165 . . . . . . 7  |-  (  _I 
`  K )  =  (Scalar `  (ringLMod `  K
) )
2625fveq2i 5851 . . . . . 6  |-  ( Base `  (  _I  `  K
) )  =  (
Base `  (Scalar `  (ringLMod `  K ) ) )
2724, 26eqtri 2431 . . . . 5  |-  ( Base `  K )  =  (
Base `  (Scalar `  (ringLMod `  K ) ) )
281, 27syl6eq 2459 . . . 4  |-  ( ph  ->  B  =  ( Base `  (Scalar `  (ringLMod `  K
) ) ) )
29 eqid 2402 . . . . . . 7  |-  ( Base `  L )  =  (
Base `  L )
3022, 29strfvi 14881 . . . . . 6  |-  ( Base `  L )  =  (
Base `  (  _I  `  L ) )
31 rlmsca2 18165 . . . . . . 7  |-  (  _I 
`  L )  =  (Scalar `  (ringLMod `  L
) )
3231fveq2i 5851 . . . . . 6  |-  ( Base `  (  _I  `  L
) )  =  (
Base `  (Scalar `  (ringLMod `  L ) ) )
3330, 32eqtri 2431 . . . . 5  |-  ( Base `  L )  =  (
Base `  (Scalar `  (ringLMod `  L ) ) )
344, 33syl6eq 2459 . . . 4  |-  ( ph  ->  B  =  ( Base `  (Scalar `  (ringLMod `  L
) ) ) )
353, 6, 7, 13, 17, 21, 28, 34lsspropd 17981 . . 3  |-  ( ph  ->  ( LSubSp `  (ringLMod `  K
) )  =  (
LSubSp `  (ringLMod `  L
) ) )
36 lidlval 18156 . . 3  |-  (LIdeal `  K )  =  (
LSubSp `  (ringLMod `  K
) )
37 lidlval 18156 . . 3  |-  (LIdeal `  L )  =  (
LSubSp `  (ringLMod `  L
) )
3835, 36, 373eqtr4g 2468 . 2  |-  ( ph  ->  (LIdeal `  K )  =  (LIdeal `  L )
)
39 fvex 5858 . . . . 5  |-  (ringLMod `  K
)  e.  _V
4039a1i 11 . . . 4  |-  ( ph  ->  (ringLMod `  K )  e.  _V )
41 fvex 5858 . . . . 5  |-  (ringLMod `  L
)  e.  _V
4241a1i 11 . . . 4  |-  ( ph  ->  (ringLMod `  L )  e.  _V )
433, 6, 7, 13, 17, 21, 28, 34, 40, 42lsppropd 17982 . . 3  |-  ( ph  ->  ( LSpan `  (ringLMod `  K
) )  =  (
LSpan `  (ringLMod `  L
) ) )
44 rspval 18157 . . 3  |-  (RSpan `  K )  =  (
LSpan `  (ringLMod `  K
) )
45 rspval 18157 . . 3  |-  (RSpan `  L )  =  (
LSpan `  (ringLMod `  L
) )
4643, 44, 453eqtr4g 2468 . 2  |-  ( ph  ->  (RSpan `  K )  =  (RSpan `  L )
)
4738, 46jca 530 1  |-  ( ph  ->  ( (LIdeal `  K
)  =  (LIdeal `  L )  /\  (RSpan `  K )  =  (RSpan `  L ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3058    C_ wss 3413    _I cid 4732   ` cfv 5568  (class class class)co 6277   ndxcnx 14836   Basecbs 14839   +g cplusg 14907   .rcmulr 14908  Scalarcsca 14910   .scvsca 14911   LSubSpclss 17896   LSpanclspn 17935  ringLModcrglmod 18133  LIdealclidl 18134  RSpancrsp 18135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-7 10639  df-8 10640  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-sca 14923  df-vsca 14924  df-ip 14925  df-lss 17897  df-lsp 17936  df-sra 18136  df-rgmod 18137  df-lidl 18138  df-rsp 18139
This theorem is referenced by:  crngridl  18204
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