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Theorem istlm 19759
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 967 . . . 4  |-  ( ( W  e. TopMnd  /\  W  e. 
LMod  /\  F  e.  TopRing )  <-> 
( ( W  e. TopMnd  /\  W  e.  LMod )  /\  F  e.  TopRing ) )
3 elin 3539 . . . . 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 5691 . . . . . 6  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
8 istlm.f . . . . . 6  |-  F  =  (Scalar `  W )
97, 8syl6eqr 2493 . . . . 5  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
109eleq1d 2509 . . . 4  |-  ( w  =  W  ->  (
(Scalar `  w )  e.  TopRing 
<->  F  e.  TopRing ) )
11 fveq2 5691 . . . . . 6  |-  ( w  =  W  ->  ( .sf `  w )  =  ( .sf `  W ) )
12 istlm.s . . . . . 6  |-  .x.  =  ( .sf `  W
)
1311, 12syl6eqr 2493 . . . . 5  |-  ( w  =  W  ->  ( .sf `  w )  =  .x.  )
149fveq2d 5695 . . . . . . . 8  |-  ( w  =  W  ->  ( TopOpen
`  (Scalar `  w )
)  =  ( TopOpen `  F ) )
15 istlm.k . . . . . . . 8  |-  K  =  ( TopOpen `  F )
1614, 15syl6eqr 2493 . . . . . . 7  |-  ( w  =  W  ->  ( TopOpen
`  (Scalar `  w )
)  =  K )
17 fveq2 5691 . . . . . . . 8  |-  ( w  =  W  ->  ( TopOpen
`  w )  =  ( TopOpen `  W )
)
18 istlm.j . . . . . . . 8  |-  J  =  ( TopOpen `  W )
1917, 18syl6eqr 2493 . . . . . . 7  |-  ( w  =  W  ->  ( TopOpen
`  w )  =  J )
2016, 19oveq12d 6109 . . . . . 6  |-  ( w  =  W  ->  (
( TopOpen `  (Scalar `  w
) )  tX  ( TopOpen
`  w ) )  =  ( K  tX  J ) )
2120, 19oveq12d 6109 . . . . 5  |-  ( w  =  W  ->  (
( ( TopOpen `  (Scalar `  w ) )  tX  ( TopOpen `  w )
)  Cn  ( TopOpen `  w ) )  =  ( ( K  tX  J )  Cn  J
) )
2213, 21eleq12d 2511 . . . 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 19736 . . 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 3119 . 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 965    = wceq 1369    e. wcel 1756    i^i cin 3327   ` cfv 5418  (class class class)co 6091  Scalarcsca 14241   TopOpenctopn 14360   LModclmod 16948   .sfcscaf 16949    Cn ccn 18828    tX ctx 19133  TopMndctmd 19641   TopRingctrg 19730  TopModctlm 19732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-iota 5381  df-fv 5426  df-ov 6094  df-tlm 19736
This theorem is referenced by:  vscacn  19760  tlmtmd  19761  tlmlmod  19763  tlmtrg  19764  nlmtlm  20274
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