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Theorem istlm 18206
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 3522 . . . . 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 5720 . . . . . 6  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
8 istlm.f . . . . . 6  |-  F  =  (Scalar `  W )
97, 8syl6eqr 2485 . . . . 5  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
109eleq1d 2501 . . . 4  |-  ( w  =  W  ->  (
(Scalar `  w )  e.  TopRing 
<->  F  e.  TopRing ) )
11 fveq2 5720 . . . . . 6  |-  ( w  =  W  ->  ( .s f `  w )  =  ( .s f `  W ) )
12 istlm.s . . . . . 6  |-  .x.  =  ( .s f `  W
)
1311, 12syl6eqr 2485 . . . . 5  |-  ( w  =  W  ->  ( .s f `  w )  =  .x.  )
149fveq2d 5724 . . . . . . . 8  |-  ( w  =  W  ->  ( TopOpen
`  (Scalar `  w )
)  =  ( TopOpen `  F ) )
15 istlm.k . . . . . . . 8  |-  K  =  ( TopOpen `  F )
1614, 15syl6eqr 2485 . . . . . . 7  |-  ( w  =  W  ->  ( TopOpen
`  (Scalar `  w )
)  =  K )
17 fveq2 5720 . . . . . . . 8  |-  ( w  =  W  ->  ( TopOpen
`  w )  =  ( TopOpen `  W )
)
18 istlm.j . . . . . . . 8  |-  J  =  ( TopOpen `  W )
1917, 18syl6eqr 2485 . . . . . . 7  |-  ( w  =  W  ->  ( TopOpen
`  w )  =  J )
2016, 19oveq12d 6091 . . . . . 6  |-  ( w  =  W  ->  (
( TopOpen `  (Scalar `  w
) )  tX  ( TopOpen
`  w ) )  =  ( K  tX  J ) )
2120, 19oveq12d 6091 . . . . 5  |-  ( w  =  W  ->  (
( ( TopOpen `  (Scalar `  w ) )  tX  ( TopOpen `  w )
)  Cn  ( TopOpen `  w ) )  =  ( ( K  tX  J )  Cn  J
) )
2213, 21eleq12d 2503 . . . 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 18183 . . 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 3086 . 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 1652    e. wcel 1725    i^i cin 3311   ` cfv 5446  (class class class)co 6073  Scalarcsca 13524   TopOpenctopn 13641   LModclmod 15942   .s fcscaf 15943    Cn ccn 17280    tX ctx 17584  TopMndctmd 18092   TopRingctrg 18177  TopModctlm 18179
This theorem is referenced by:  vscacn  18207  tlmtmd  18208  tlmlmod  18210  tlmtrg  18211  nlmtlm  18721
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076  df-tlm 18183
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