Theorem istlm 21988

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.

Step	Hyp	Ref	Expression
1		anass 681	. 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 1039	. . . 4 |- ( ( W e. TopMnd /\ W e. LMod /\ F e. TopRing ) <-> ( ( W e. TopMnd /\ W e. LMod ) /\ F e. TopRing ) )
3		elin 3796	. . . . 5 |- ( W e. (TopMnd i^i LMod ) <-> ( W e. TopMnd /\ W e. LMod ) )
4	3	anbi1i 731	. . . 4 |- ( ( W e. (TopMnd i^i LMod ) /\ F e. TopRing ) <-> ( ( W e. TopMnd /\ W e. LMod ) /\ F e. TopRing ) )
5	2, 4	bitr4i 267	. . 3 |- ( ( W e. TopMnd /\ W e. LMod /\ F e. TopRing ) <-> ( W e. (TopMnd i^i LMod ) /\ F e. TopRing ) )
6	5	anbi1i 731	. 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 6191	. . . . . 6 |- ( w = W -> (Scalar ` w ) = (Scalar ` W ) )
8		istlm.f	. . . . . 6 |- F = (Scalar ` W )
9	7, 8	syl6eqr 2674	. . . . 5 |- ( w = W -> (Scalar ` w ) = F )
10	9	eleq1d 2686	. . . 4 |- ( w = W -> ( (Scalar ` w ) e. TopRing <-> F e. TopRing ) )
11		fveq2 6191	. . . . . 6 |- ( w = W -> ( .sf ` w ) = ( .sf ` W ) )
12		istlm.s	. . . . . 6 |- .x. = ( .sf ` W )
13	11, 12	syl6eqr 2674	. . . . 5 |- ( w = W -> ( .sf ` w ) = .x. )
14	9	fveq2d 6195	. . . . . . . 8 |- ( w = W -> ( TopOpen ` (Scalar ` w ) ) = ( TopOpen ` F ) )
15		istlm.k	. . . . . . . 8 |- K = ( TopOpen ` F )
16	14, 15	syl6eqr 2674	. . . . . . 7 |- ( w = W -> ( TopOpen ` (Scalar ` w ) ) = K )
17		fveq2 6191	. . . . . . . 8 |- ( w = W -> ( TopOpen ` w ) = ( TopOpen ` W ) )
18		istlm.j	. . . . . . . 8 |- J = ( TopOpen ` W )
19	17, 18	syl6eqr 2674	. . . . . . 7 |- ( w = W -> ( TopOpen ` w ) = J )
20	16, 19	oveq12d 6668	. . . . . 6 |- ( w = W -> ( ( TopOpen ` (Scalar ` w ) ) tX ( TopOpen ` w ) ) = ( K tX J ) )
21	20, 19	oveq12d 6668	. . . . 5 |- ( w = W -> ( ( ( TopOpen ` (Scalar ` w ) ) tX ( TopOpen ` w ) ) Cn ( TopOpen ` w ) ) = ( ( K tX J ) Cn J ) )
22	13, 21	eleq12d 2695	. . . 4 |- ( w = W -> ( ( .sf ` w ) e. ( ( ( TopOpen ` (Scalar ` w ) ) tX ( TopOpen ` w ) ) Cn ( TopOpen ` w ) ) <-> .x. e. ( ( K tX J ) Cn J ) ) )
23	10, 22	anbi12d 747	. . 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 21965	. . 3 |- TopMod = { w e. (TopMnd i^i LMod ) | ( (Scalar ` w ) e. TopRing /\ ( .sf ` w ) e. ( ( ( TopOpen ` (Scalar ` w ) ) tX ( TopOpen ` w ) ) Cn ( TopOpen ` w ) ) ) }
25	23, 24	elrab2 3366	. 2 |- ( W e. TopMod <-> ( W e. (TopMnd i^i LMod ) /\ ( F e. TopRing /\ .x. e. ( ( K tX J ) Cn J ) ) ) )
26	1, 6, 25	3bitr4ri 293	1 |- ( W e. TopMod <-> ( ( W e. TopMnd /\ W e. LMod /\ F e. TopRing ) /\ .x. e. ( ( K tX J ) Cn J ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   ` cfv 5888  (class class class)co 6650  Scalarcsca 15944   TopOpenctopn 16082   LModclmod 18863   .sfcscaf 18864   Cn ccn 21028   tX ctx 21363  TopMndctmd 21874   TopRingctrg 21959  TopModctlm 21961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-tlm 21965

This theorem is referenced by:  vscacn  21989  tlmtmd  21990  tlmlmod  21992  tlmtrg  21993  nlmtlm  22498