Theorem istlm 24393

Description: The predicate "𝑊 is a topological left module". (Contributed by Mario Carneiro, 5-Oct-2015.)

Hypotheses

Ref	Expression
istlm.s	⊢ · = ( ·sf ‘𝑊)
istlm.j	⊢ 𝐽 = (TopOpen‘𝑊)
istlm.f	⊢ 𝐹 = (Scalar‘𝑊)
istlm.k	⊢ 𝐾 = (TopOpen‘𝐹)

Assertion

Ref	Expression
istlm	⊢ (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))

Proof of Theorem istlm

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		anass 474	. 2 ⊢ (((𝑊 ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) ↔ (𝑊 ∈ (TopMnd ∩ LMod) ∧ (𝐹 ∈ TopRing ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))))
2		df-3an 1105	. . . 4 ⊢ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod) ∧ 𝐹 ∈ TopRing))
3		elin 3922	. . . . 5 ⊢ (𝑊 ∈ (TopMnd ∩ LMod) ↔ (𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod))
4	3	anbi1i 636	. . . 4 ⊢ ((𝑊 ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing) ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod) ∧ 𝐹 ∈ TopRing))
5	2, 4	bitr4i 281	. . 3 ⊢ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ↔ (𝑊 ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing))
6	5	anbi1i 636	. 2 ⊢ (((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) ↔ ((𝑊 ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
7		fveq2 6885	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
8		istlm.f	. . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊)
9	7, 8	eqtr4di 2818	. . . . 5 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
10	9	eleq1d 2850	. . . 4 ⊢ (𝑤 = 𝑊 → ((Scalar‘𝑤) ∈ TopRing ↔ 𝐹 ∈ TopRing))
11		fveq2 6885	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( ·sf ‘𝑤) = ( ·sf ‘𝑊))
12		istlm.s	. . . . . 6 ⊢ · = ( ·sf ‘𝑊)
13	11, 12	eqtr4di 2818	. . . . 5 ⊢ (𝑤 = 𝑊 → ( ·sf ‘𝑤) = · )
14	9	fveq2d 6889	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (TopOpen‘(Scalar‘𝑤)) = (TopOpen‘𝐹))
15		istlm.k	. . . . . . . 8 ⊢ 𝐾 = (TopOpen‘𝐹)
16	14, 15	eqtr4di 2818	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (TopOpen‘(Scalar‘𝑤)) = 𝐾)
17		fveq2 6885	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (TopOpen‘𝑤) = (TopOpen‘𝑊))
18		istlm.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝑊)
19	17, 18	eqtr4di 2818	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (TopOpen‘𝑤) = 𝐽)
20	16, 19	oveq12d 7437	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) = (𝐾 ×t 𝐽))
21	20, 19	oveq12d 7437	. . . . 5 ⊢ (𝑤 = 𝑊 → (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)) = ((𝐾 ×t 𝐽) Cn 𝐽))
22	13, 21	eleq12d 2859	. . . 4 ⊢ (𝑤 = 𝑊 → (( ·sf ‘𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)) ↔ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
23	10, 22	anbi12d 644	. . 3 ⊢ (𝑤 = 𝑊 → (((Scalar‘𝑤) ∈ TopRing ∧ ( ·sf ‘𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤))) ↔ (𝐹 ∈ TopRing ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))))
24		df-tlm 24370	. . 3 ⊢ TopMod = {𝑤 ∈ (TopMnd ∩ LMod) ∣ ((Scalar‘𝑤) ∈ TopRing ∧ ( ·sf ‘𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)))}
25	23, 24	elrab2 3656	. 2 ⊢ (𝑊 ∈ TopMod ↔ (𝑊 ∈ (TopMnd ∩ LMod) ∧ (𝐹 ∈ TopRing ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))))
26	1, 6, 25	3bitr4ri 307	1 ⊢ (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ∩ cin 3905  ‘cfv 6540  (class class class)co 7419  Scalarcsca 17335  TopOpenctopn 17496  LModclmod 21031   ·_sf cscaf 21032   Cn ccn 23431   ×_t ctx 23768  TopMndctmd 24278  TopRingctrg 24364  TopModctlm 24366

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-ov 7422  df-tlm 24370

This theorem is used by:  vscacn  24394  tlmtmd  24395  tlmlmod  24397  tlmtrg  24398  nlmtlm  24902