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Theorem istlm 17883
Description: The predicate " W is a topological left module". (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypotheses
Ref Expression
istlm.s  |-  .x.  =  ( .s f `  W
)
istlm.j  |-  J  =  ( TopOpen `  W )
istlm.f  |-  F  =  (Scalar `  W )
istlm.k  |-  K  =  ( TopOpen `  F )
Assertion
Ref Expression
istlm  |-  ( W  e. TopMod 
<->  ( ( W  e. TopMnd  /\  W  e.  LMod  /\  F  e.  TopRing )  /\  .x. 
e.  ( ( K 
tX  J )  Cn  J ) ) )

Proof of Theorem istlm
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 anass 630 . 2  |-  ( ( ( W  e.  (TopMnd 
i^i  LMod )  /\  F  e.  TopRing )  /\  .x.  e.  ( ( K  tX  J )  Cn  J
) )  <->  ( W  e.  (TopMnd  i^i  LMod )  /\  ( F  e.  TopRing  /\  .x.  e.  ( ( K  tX  J )  Cn  J
) ) ) )
2 df-3an 936 . . . 4  |-  ( ( W  e. TopMnd  /\  W  e. 
LMod  /\  F  e.  TopRing )  <-> 
( ( W  e. TopMnd  /\  W  e.  LMod )  /\  F  e.  TopRing ) )
3 elin 3371 . . . . 5  |-  ( W  e.  (TopMnd  i^i  LMod ) 
<->  ( W  e. TopMnd  /\  W  e.  LMod ) )
43anbi1i 676 . . . 4  |-  ( ( W  e.  (TopMnd  i^i  LMod )  /\  F  e.  TopRing )  <->  ( ( W  e. TopMnd  /\  W  e.  LMod )  /\  F  e.  TopRing ) )
52, 4bitr4i 243 . . 3  |-  ( ( W  e. TopMnd  /\  W  e. 
LMod  /\  F  e.  TopRing )  <-> 
( W  e.  (TopMnd 
i^i  LMod )  /\  F  e.  TopRing ) )
65anbi1i 676 . 2  |-  ( ( ( W  e. TopMnd  /\  W  e.  LMod  /\  F  e.  TopRing )  /\  .x.  e.  (
( K  tX  J
)  Cn  J ) )  <->  ( ( W  e.  (TopMnd  i^i  LMod )  /\  F  e.  TopRing )  /\  .x.  e.  (
( K  tX  J
)  Cn  J ) ) )
7 fveq2 5541 . . . . . 6  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
8 istlm.f . . . . . 6  |-  F  =  (Scalar `  W )
97, 8syl6eqr 2346 . . . . 5  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
109eleq1d 2362 . . . 4  |-  ( w  =  W  ->  (
(Scalar `  w )  e.  TopRing 
<->  F  e.  TopRing ) )
11 fveq2 5541 . . . . . 6  |-  ( w  =  W  ->  ( .s f `  w )  =  ( .s f `  W ) )
12 istlm.s . . . . . 6  |-  .x.  =  ( .s f `  W
)
1311, 12syl6eqr 2346 . . . . 5  |-  ( w  =  W  ->  ( .s f `  w )  =  .x.  )
149fveq2d 5545 . . . . . . . 8  |-  ( w  =  W  ->  ( TopOpen
`  (Scalar `  w )
)  =  ( TopOpen `  F ) )
15 istlm.k . . . . . . . 8  |-  K  =  ( TopOpen `  F )
1614, 15syl6eqr 2346 . . . . . . 7  |-  ( w  =  W  ->  ( TopOpen
`  (Scalar `  w )
)  =  K )
17 fveq2 5541 . . . . . . . 8  |-  ( w  =  W  ->  ( TopOpen
`  w )  =  ( TopOpen `  W )
)
18 istlm.j . . . . . . . 8  |-  J  =  ( TopOpen `  W )
1917, 18syl6eqr 2346 . . . . . . 7  |-  ( w  =  W  ->  ( TopOpen
`  w )  =  J )
2016, 19oveq12d 5892 . . . . . 6  |-  ( w  =  W  ->  (
( TopOpen `  (Scalar `  w
) )  tX  ( TopOpen
`  w ) )  =  ( K  tX  J ) )
2120, 19oveq12d 5892 . . . . 5  |-  ( w  =  W  ->  (
( ( TopOpen `  (Scalar `  w ) )  tX  ( TopOpen `  w )
)  Cn  ( TopOpen `  w ) )  =  ( ( K  tX  J )  Cn  J
) )
2213, 21eleq12d 2364 . . . 4  |-  ( w  =  W  ->  (
( .s f `  w )  e.  ( ( ( TopOpen `  (Scalar `  w ) )  tX  ( TopOpen `  w )
)  Cn  ( TopOpen `  w ) )  <->  .x.  e.  ( ( K  tX  J
)  Cn  J ) ) )
2310, 22anbi12d 691 . . 3  |-  ( w  =  W  ->  (
( (Scalar `  w
)  e.  TopRing  /\  ( .s f `  w )  e.  ( ( (
TopOpen `  (Scalar `  w
) )  tX  ( TopOpen
`  w ) )  Cn  ( TopOpen `  w
) ) )  <->  ( F  e.  TopRing  /\  .x.  e.  ( ( K  tX  J
)  Cn  J ) ) ) )
24 df-tlm 17860 . . 3  |- TopMod  =  {
w  e.  (TopMnd  i^i  LMod )  |  ( (Scalar `  w )  e.  TopRing  /\  ( .s f `  w
)  e.  ( ( ( TopOpen `  (Scalar `  w
) )  tX  ( TopOpen
`  w ) )  Cn  ( TopOpen `  w
) ) ) }
2523, 24elrab2 2938 . 2  |-  ( W  e. TopMod 
<->  ( W  e.  (TopMnd 
i^i  LMod )  /\  ( F  e.  TopRing  /\  .x.  e.  ( ( K  tX  J )  Cn  J
) ) ) )
261, 6, 253bitr4ri 269 1  |-  ( W  e. TopMod 
<->  ( ( W  e. TopMnd  /\  W  e.  LMod  /\  F  e.  TopRing )  /\  .x. 
e.  ( ( K 
tX  J )  Cn  J ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    i^i cin 3164   ` cfv 5271  (class class class)co 5874  Scalarcsca 13227   TopOpenctopn 13342   LModclmod 15643   .s fcscaf 15644    Cn ccn 16970    tX ctx 17271  TopMndctmd 17769   TopRingctrg 17854  TopModctlm 17856
This theorem is referenced by:  vscacn  17884  tlmtmd  17885  tlmlmod  17887  tlmtrg  17888  nlmtlm  18220
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-tlm 17860
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