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Theorem istlm 20813
Description: The predicate " W is a topological left module". (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypotheses
Ref Expression
istlm.s  |-  .x.  =  ( .sf `  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 649 . 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 975 . . . 4  |-  ( ( W  e. TopMnd  /\  W  e. 
LMod  /\  F  e.  TopRing )  <-> 
( ( W  e. TopMnd  /\  W  e.  LMod )  /\  F  e.  TopRing ) )
3 elin 3683 . . . . 5  |-  ( W  e.  (TopMnd  i^i  LMod ) 
<->  ( W  e. TopMnd  /\  W  e.  LMod ) )
43anbi1i 695 . . . 4  |-  ( ( W  e.  (TopMnd  i^i  LMod )  /\  F  e.  TopRing )  <->  ( ( W  e. TopMnd  /\  W  e.  LMod )  /\  F  e.  TopRing ) )
52, 4bitr4i 252 . . 3  |-  ( ( W  e. TopMnd  /\  W  e. 
LMod  /\  F  e.  TopRing )  <-> 
( W  e.  (TopMnd 
i^i  LMod )  /\  F  e.  TopRing ) )
65anbi1i 695 . 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 5872 . . . . . 6  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
8 istlm.f . . . . . 6  |-  F  =  (Scalar `  W )
97, 8syl6eqr 2516 . . . . 5  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
109eleq1d 2526 . . . 4  |-  ( w  =  W  ->  (
(Scalar `  w )  e.  TopRing 
<->  F  e.  TopRing ) )
11 fveq2 5872 . . . . . 6  |-  ( w  =  W  ->  ( .sf `  w )  =  ( .sf `  W ) )
12 istlm.s . . . . . 6  |-  .x.  =  ( .sf `  W
)
1311, 12syl6eqr 2516 . . . . 5  |-  ( w  =  W  ->  ( .sf `  w )  =  .x.  )
149fveq2d 5876 . . . . . . . 8  |-  ( w  =  W  ->  ( TopOpen
`  (Scalar `  w )
)  =  ( TopOpen `  F ) )
15 istlm.k . . . . . . . 8  |-  K  =  ( TopOpen `  F )
1614, 15syl6eqr 2516 . . . . . . 7  |-  ( w  =  W  ->  ( TopOpen
`  (Scalar `  w )
)  =  K )
17 fveq2 5872 . . . . . . . 8  |-  ( w  =  W  ->  ( TopOpen
`  w )  =  ( TopOpen `  W )
)
18 istlm.j . . . . . . . 8  |-  J  =  ( TopOpen `  W )
1917, 18syl6eqr 2516 . . . . . . 7  |-  ( w  =  W  ->  ( TopOpen
`  w )  =  J )
2016, 19oveq12d 6314 . . . . . 6  |-  ( w  =  W  ->  (
( TopOpen `  (Scalar `  w
) )  tX  ( TopOpen
`  w ) )  =  ( K  tX  J ) )
2120, 19oveq12d 6314 . . . . 5  |-  ( w  =  W  ->  (
( ( TopOpen `  (Scalar `  w ) )  tX  ( TopOpen `  w )
)  Cn  ( TopOpen `  w ) )  =  ( ( K  tX  J )  Cn  J
) )
2213, 21eleq12d 2539 . . . 4  |-  ( w  =  W  ->  (
( .sf `  w )  e.  ( ( ( TopOpen `  (Scalar `  w ) )  tX  ( TopOpen `  w )
)  Cn  ( TopOpen `  w ) )  <->  .x.  e.  ( ( K  tX  J
)  Cn  J ) ) )
2310, 22anbi12d 710 . . 3  |-  ( w  =  W  ->  (
( (Scalar `  w
)  e.  TopRing  /\  ( .sf `  w )  e.  ( ( (
TopOpen `  (Scalar `  w
) )  tX  ( TopOpen
`  w ) )  Cn  ( TopOpen `  w
) ) )  <->  ( F  e.  TopRing  /\  .x.  e.  ( ( K  tX  J
)  Cn  J ) ) ) )
24 df-tlm 20790 . . 3  |- TopMod  =  {
w  e.  (TopMnd  i^i  LMod )  |  ( (Scalar `  w )  e.  TopRing  /\  ( .sf `  w
)  e.  ( ( ( TopOpen `  (Scalar `  w
) )  tX  ( TopOpen
`  w ) )  Cn  ( TopOpen `  w
) ) ) }
2523, 24elrab2 3259 . 2  |-  ( W  e. TopMod 
<->  ( W  e.  (TopMnd 
i^i  LMod )  /\  ( F  e.  TopRing  /\  .x.  e.  ( ( K  tX  J )  Cn  J
) ) ) )
261, 6, 253bitr4ri 278 1  |-  ( W  e. TopMod 
<->  ( ( W  e. TopMnd  /\  W  e.  LMod  /\  F  e.  TopRing )  /\  .x. 
e.  ( ( K 
tX  J )  Cn  J ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    i^i cin 3470   ` cfv 5594  (class class class)co 6296  Scalarcsca 14715   TopOpenctopn 14839   LModclmod 17639   .sfcscaf 17640    Cn ccn 19852    tX ctx 20187  TopMndctmd 20695   TopRingctrg 20784  TopModctlm 20786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-ov 6299  df-tlm 20790
This theorem is referenced by:  vscacn  20814  tlmtmd  20815  tlmlmod  20817  tlmtrg  20818  nlmtlm  21328
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