Theorem istlm 24141

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 468	. 2 ⊢ (((𝑊 ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) ↔ (𝑊 ∈ (TopMnd ∩ LMod) ∧ (𝐹 ∈ TopRing ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))))
2		df-3an 1089	. . . 4 ⊢ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod) ∧ 𝐹 ∈ TopRing))
3		elin 3919	. . . . 5 ⊢ (𝑊 ∈ (TopMnd ∩ LMod) ↔ (𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod))
4	3	anbi1i 625	. . . 4 ⊢ ((𝑊 ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing) ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod) ∧ 𝐹 ∈ TopRing))
5	2, 4	bitr4i 278	. . 3 ⊢ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ↔ (𝑊 ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing))
6	5	anbi1i 625	. 2 ⊢ (((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) ↔ ((𝑊 ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
7		fveq2 6842	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
8		istlm.f	. . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊)
9	7, 8	eqtr4di 2790	. . . . 5 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
10	9	eleq1d 2822	. . . 4 ⊢ (𝑤 = 𝑊 → ((Scalar‘𝑤) ∈ TopRing ↔ 𝐹 ∈ TopRing))
11		fveq2 6842	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( ·sf ‘𝑤) = ( ·sf ‘𝑊))
12		istlm.s	. . . . . 6 ⊢ · = ( ·sf ‘𝑊)
13	11, 12	eqtr4di 2790	. . . . 5 ⊢ (𝑤 = 𝑊 → ( ·sf ‘𝑤) = · )
14	9	fveq2d 6846	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (TopOpen‘(Scalar‘𝑤)) = (TopOpen‘𝐹))
15		istlm.k	. . . . . . . 8 ⊢ 𝐾 = (TopOpen‘𝐹)
16	14, 15	eqtr4di 2790	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (TopOpen‘(Scalar‘𝑤)) = 𝐾)
17		fveq2 6842	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (TopOpen‘𝑤) = (TopOpen‘𝑊))
18		istlm.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝑊)
19	17, 18	eqtr4di 2790	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (TopOpen‘𝑤) = 𝐽)
20	16, 19	oveq12d 7386	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) = (𝐾 ×t 𝐽))
21	20, 19	oveq12d 7386	. . . . 5 ⊢ (𝑤 = 𝑊 → (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)) = ((𝐾 ×t 𝐽) Cn 𝐽))
22	13, 21	eleq12d 2831	. . . 4 ⊢ (𝑤 = 𝑊 → (( ·sf ‘𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)) ↔ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
23	10, 22	anbi12d 633	. . 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 24118	. . 3 ⊢ TopMod = {𝑤 ∈ (TopMnd ∩ LMod) ∣ ((Scalar‘𝑤) ∈ TopRing ∧ ( ·sf ‘𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)))}
25	23, 24	elrab2 3651	. 2 ⊢ (𝑊 ∈ TopMod ↔ (𝑊 ∈ (TopMnd ∩ LMod) ∧ (𝐹 ∈ TopRing ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))))
26	1, 6, 25	3bitr4ri 304	1 ⊢ (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ∩ cin 3902  ‘cfv 6500  (class class class)co 7368  Scalarcsca 17192  TopOpenctopn 17353  LModclmod 20823   ·_sf cscaf 20824   Cn ccn 23180   ×_t ctx 23516  TopMndctmd 24026  TopRingctrg 24112  TopModctlm 24114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-tlm 24118

This theorem is referenced by:  vscacn  24142  tlmtmd  24143  tlmlmod  24145  tlmtrg  24146  nlmtlm  24650