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Theorem istlm 18135
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 631 . 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 938 . . . 4  |-  ( ( W  e. TopMnd  /\  W  e. 
LMod  /\  F  e.  TopRing )  <-> 
( ( W  e. TopMnd  /\  W  e.  LMod )  /\  F  e.  TopRing ) )
3 elin 3473 . . . . 5  |-  ( W  e.  (TopMnd  i^i  LMod ) 
<->  ( W  e. TopMnd  /\  W  e.  LMod ) )
43anbi1i 677 . . . 4  |-  ( ( W  e.  (TopMnd  i^i  LMod )  /\  F  e.  TopRing )  <->  ( ( W  e. TopMnd  /\  W  e.  LMod )  /\  F  e.  TopRing ) )
52, 4bitr4i 244 . . 3  |-  ( ( W  e. TopMnd  /\  W  e. 
LMod  /\  F  e.  TopRing )  <-> 
( W  e.  (TopMnd 
i^i  LMod )  /\  F  e.  TopRing ) )
65anbi1i 677 . 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 5668 . . . . . 6  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
8 istlm.f . . . . . 6  |-  F  =  (Scalar `  W )
97, 8syl6eqr 2437 . . . . 5  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
109eleq1d 2453 . . . 4  |-  ( w  =  W  ->  (
(Scalar `  w )  e.  TopRing 
<->  F  e.  TopRing ) )
11 fveq2 5668 . . . . . 6  |-  ( w  =  W  ->  ( .s f `  w )  =  ( .s f `  W ) )
12 istlm.s . . . . . 6  |-  .x.  =  ( .s f `  W
)
1311, 12syl6eqr 2437 . . . . 5  |-  ( w  =  W  ->  ( .s f `  w )  =  .x.  )
149fveq2d 5672 . . . . . . . 8  |-  ( w  =  W  ->  ( TopOpen
`  (Scalar `  w )
)  =  ( TopOpen `  F ) )
15 istlm.k . . . . . . . 8  |-  K  =  ( TopOpen `  F )
1614, 15syl6eqr 2437 . . . . . . 7  |-  ( w  =  W  ->  ( TopOpen
`  (Scalar `  w )
)  =  K )
17 fveq2 5668 . . . . . . . 8  |-  ( w  =  W  ->  ( TopOpen
`  w )  =  ( TopOpen `  W )
)
18 istlm.j . . . . . . . 8  |-  J  =  ( TopOpen `  W )
1917, 18syl6eqr 2437 . . . . . . 7  |-  ( w  =  W  ->  ( TopOpen
`  w )  =  J )
2016, 19oveq12d 6038 . . . . . 6  |-  ( w  =  W  ->  (
( TopOpen `  (Scalar `  w
) )  tX  ( TopOpen
`  w ) )  =  ( K  tX  J ) )
2120, 19oveq12d 6038 . . . . 5  |-  ( w  =  W  ->  (
( ( TopOpen `  (Scalar `  w ) )  tX  ( TopOpen `  w )
)  Cn  ( TopOpen `  w ) )  =  ( ( K  tX  J )  Cn  J
) )
2213, 21eleq12d 2455 . . . 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 692 . . 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 18112 . . 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 3037 . 2  |-  ( W  e. TopMod 
<->  ( W  e.  (TopMnd 
i^i  LMod )  /\  ( F  e.  TopRing  /\  .x.  e.  ( ( K  tX  J )  Cn  J
) ) ) )
261, 6, 253bitr4ri 270 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    i^i cin 3262   ` cfv 5394  (class class class)co 6020  Scalarcsca 13459   TopOpenctopn 13576   LModclmod 15877   .s fcscaf 15878    Cn ccn 17210    tX ctx 17513  TopMndctmd 18021   TopRingctrg 18106  TopModctlm 18108
This theorem is referenced by:  vscacn  18136  tlmtmd  18137  tlmlmod  18139  tlmtrg  18140  nlmtlm  18600
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-iota 5358  df-fv 5402  df-ov 6023  df-tlm 18112
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