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Theorem rspsn 14014
Description: Membership in principal ideals is closely related to divisibility. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
Hypotheses
Ref Expression
rspsn.b  |-  B  =  ( Base `  R
)
rspsn.k  |-  K  =  (RSpan `  R )
rspsn.d  |-  .||  =  (
||r `  R )
Assertion
Ref Expression
rspsn  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  ( K `  { G } )  =  {
x  |  G  .||  x } )
Distinct variable groups:    x, R    x, G    x, B    x, K    x,  .||

Proof of Theorem rspsn
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 eqcom 2195 . . . . 5  |-  ( x  =  ( a ( .r `  R ) G )  <->  ( a
( .r `  R
) G )  =  x )
21a1i 9 . . . 4  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  (
x  =  ( a ( .r `  R
) G )  <->  ( a
( .r `  R
) G )  =  x ) )
32rexbidv 2495 . . 3  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  ( E. a  e.  B  x  =  ( a
( .r `  R
) G )  <->  E. a  e.  B  ( a
( .r `  R
) G )  =  x ) )
4 rlmlmod 13944 . . . . 5  |-  ( R  e.  Ring  ->  (ringLMod `  R
)  e.  LMod )
5 simpr 110 . . . . . 6  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  G  e.  B )
6 rspsn.b . . . . . . . 8  |-  B  =  ( Base `  R
)
7 rlmbasg 13935 . . . . . . . 8  |-  ( R  e.  Ring  ->  ( Base `  R )  =  (
Base `  (ringLMod `  R
) ) )
86, 7eqtrid 2238 . . . . . . 7  |-  ( R  e.  Ring  ->  B  =  ( Base `  (ringLMod `  R ) ) )
98adantr 276 . . . . . 6  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  B  =  ( Base `  (ringLMod `  R ) ) )
105, 9eleqtrd 2272 . . . . 5  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  G  e.  ( Base `  (ringLMod `  R ) ) )
11 eqid 2193 . . . . . 6  |-  (Scalar `  (ringLMod `  R ) )  =  (Scalar `  (ringLMod `  R ) )
12 eqid 2193 . . . . . 6  |-  ( Base `  (Scalar `  (ringLMod `  R
) ) )  =  ( Base `  (Scalar `  (ringLMod `  R )
) )
13 eqid 2193 . . . . . 6  |-  ( Base `  (ringLMod `  R )
)  =  ( Base `  (ringLMod `  R )
)
14 eqid 2193 . . . . . 6  |-  ( .s
`  (ringLMod `  R )
)  =  ( .s
`  (ringLMod `  R )
)
15 eqid 2193 . . . . . 6  |-  ( LSpan `  (ringLMod `  R )
)  =  ( LSpan `  (ringLMod `  R )
)
1611, 12, 13, 14, 15ellspsn 13897 . . . . 5  |-  ( ( (ringLMod `  R )  e.  LMod  /\  G  e.  ( Base `  (ringLMod `  R
) ) )  -> 
( x  e.  ( ( LSpan `  (ringLMod `  R
) ) `  { G } )  <->  E. a  e.  ( Base `  (Scalar `  (ringLMod `  R )
) ) x  =  ( a ( .s
`  (ringLMod `  R )
) G ) ) )
174, 10, 16syl2an2r 595 . . . 4  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  (
x  e.  ( (
LSpan `  (ringLMod `  R
) ) `  { G } )  <->  E. a  e.  ( Base `  (Scalar `  (ringLMod `  R )
) ) x  =  ( a ( .s
`  (ringLMod `  R )
) G ) ) )
18 rspsn.k . . . . . . . 8  |-  K  =  (RSpan `  R )
19 rspvalg 13952 . . . . . . . 8  |-  ( R  e.  Ring  ->  (RSpan `  R )  =  (
LSpan `  (ringLMod `  R
) ) )
2018, 19eqtrid 2238 . . . . . . 7  |-  ( R  e.  Ring  ->  K  =  ( LSpan `  (ringLMod `  R
) ) )
2120adantr 276 . . . . . 6  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  K  =  ( LSpan `  (ringLMod `  R ) ) )
2221fveq1d 5548 . . . . 5  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  ( K `  { G } )  =  ( ( LSpan `  (ringLMod `  R
) ) `  { G } ) )
2322eleq2d 2263 . . . 4  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  (
x  e.  ( K `
 { G }
)  <->  x  e.  (
( LSpan `  (ringLMod `  R
) ) `  { G } ) ) )
24 rlmscabas 13940 . . . . . . 7  |-  ( R  e.  Ring  ->  ( Base `  R )  =  (
Base `  (Scalar `  (ringLMod `  R ) ) ) )
256, 24eqtrid 2238 . . . . . 6  |-  ( R  e.  Ring  ->  B  =  ( Base `  (Scalar `  (ringLMod `  R )
) ) )
2625adantr 276 . . . . 5  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  B  =  ( Base `  (Scalar `  (ringLMod `  R )
) ) )
27 rlmvscag 13941 . . . . . . . 8  |-  ( R  e.  Ring  ->  ( .r
`  R )  =  ( .s `  (ringLMod `  R ) ) )
2827adantr 276 . . . . . . 7  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  ( .r `  R )  =  ( .s `  (ringLMod `  R ) ) )
2928oveqd 5927 . . . . . 6  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  (
a ( .r `  R ) G )  =  ( a ( .s `  (ringLMod `  R
) ) G ) )
3029eqeq2d 2205 . . . . 5  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  (
x  =  ( a ( .r `  R
) G )  <->  x  =  ( a ( .s
`  (ringLMod `  R )
) G ) ) )
3126, 30rexeqbidv 2707 . . . 4  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  ( E. a  e.  B  x  =  ( a
( .r `  R
) G )  <->  E. a  e.  ( Base `  (Scalar `  (ringLMod `  R )
) ) x  =  ( a ( .s
`  (ringLMod `  R )
) G ) ) )
3217, 23, 313bitr4d 220 . . 3  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  (
x  e.  ( K `
 { G }
)  <->  E. a  e.  B  x  =  ( a
( .r `  R
) G ) ) )
336a1i 9 . . . 4  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  B  =  ( Base `  R
) )
34 rspsn.d . . . . 5  |-  .||  =  (
||r `  R )
3534a1i 9 . . . 4  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  .||  =  (
||r `  R ) )
36 ringsrg 13527 . . . . 5  |-  ( R  e.  Ring  ->  R  e. SRing
)
3736adantr 276 . . . 4  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  R  e. SRing )
38 eqid 2193 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
3938a1i 9 . . . 4  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  ( .r `  R )  =  ( .r `  R
) )
4033, 35, 37, 39, 5dvdsr2d 13575 . . 3  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  ( G  .||  x  <->  E. a  e.  B  ( a
( .r `  R
) G )  =  x ) )
413, 32, 403bitr4d 220 . 2  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  (
x  e.  ( K `
 { G }
)  <->  G  .||  x ) )
4241eqabdv 2322 1  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  ( K `  { G } )  =  {
x  |  G  .||  x } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2164   {cab 2179   E.wrex 2473   {csn 3618   class class class wbr 4029   ` cfv 5246  (class class class)co 5910   Basecbs 12608   .rcmulr 12686  Scalarcsca 12688   .scvsca 12689  SRingcsrg 13443   Ringcrg 13476   ||rcdsr 13566   LModclmod 13767   LSpanclspn 13866  ringLModcrglmod 13914  RSpancrsp 13948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4462  ax-setind 4565  ax-cnex 7953  ax-resscn 7954  ax-1cn 7955  ax-1re 7956  ax-icn 7957  ax-addcl 7958  ax-addrcl 7959  ax-mulcl 7960  ax-addcom 7962  ax-addass 7964  ax-i2m1 7967  ax-0lt1 7968  ax-0id 7970  ax-rnegex 7971  ax-pre-ltirr 7974  ax-pre-lttrn 7976  ax-pre-ltadd 7978
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-iota 5207  df-fun 5248  df-fn 5249  df-f 5250  df-f1 5251  df-fo 5252  df-f1o 5253  df-fv 5254  df-riota 5865  df-ov 5913  df-oprab 5914  df-mpo 5915  df-1st 6184  df-2nd 6185  df-pnf 8046  df-mnf 8047  df-ltxr 8049  df-inn 8973  df-2 9031  df-3 9032  df-4 9033  df-5 9034  df-6 9035  df-7 9036  df-8 9037  df-ndx 12611  df-slot 12612  df-base 12614  df-sets 12615  df-iress 12616  df-plusg 12698  df-mulr 12699  df-sca 12701  df-vsca 12702  df-ip 12703  df-0g 12859  df-mgm 12929  df-sgrp 12975  df-mnd 12988  df-grp 13065  df-minusg 13066  df-sbg 13067  df-subg 13229  df-cmn 13345  df-abl 13346  df-mgp 13401  df-ur 13440  df-srg 13444  df-ring 13478  df-dvdsr 13569  df-subrg 13699  df-lmod 13769  df-lssm 13833  df-lsp 13867  df-sra 13915  df-rgmod 13916  df-rsp 13950
This theorem is referenced by:  zndvds  14114
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