Theorem istlm 24193

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 3967	. . . . 5 ⊢ (𝑊 ∈ (TopMnd ∩ LMod) ↔ (𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod))
4	3	anbi1i 624	. . . 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 624	. 2 ⊢ (((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) ↔ ((𝑊 ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
7		fveq2 6906	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
8		istlm.f	. . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊)
9	7, 8	eqtr4di 2795	. . . . 5 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
10	9	eleq1d 2826	. . . 4 ⊢ (𝑤 = 𝑊 → ((Scalar‘𝑤) ∈ TopRing ↔ 𝐹 ∈ TopRing))
11		fveq2 6906	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( ·sf ‘𝑤) = ( ·sf ‘𝑊))
12		istlm.s	. . . . . 6 ⊢ · = ( ·sf ‘𝑊)
13	11, 12	eqtr4di 2795	. . . . 5 ⊢ (𝑤 = 𝑊 → ( ·sf ‘𝑤) = · )
14	9	fveq2d 6910	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (TopOpen‘(Scalar‘𝑤)) = (TopOpen‘𝐹))
15		istlm.k	. . . . . . . 8 ⊢ 𝐾 = (TopOpen‘𝐹)
16	14, 15	eqtr4di 2795	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (TopOpen‘(Scalar‘𝑤)) = 𝐾)
17		fveq2 6906	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (TopOpen‘𝑤) = (TopOpen‘𝑊))
18		istlm.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝑊)
19	17, 18	eqtr4di 2795	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (TopOpen‘𝑤) = 𝐽)
20	16, 19	oveq12d 7449	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) = (𝐾 ×t 𝐽))
21	20, 19	oveq12d 7449	. . . . 5 ⊢ (𝑤 = 𝑊 → (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)) = ((𝐾 ×t 𝐽) Cn 𝐽))
22	13, 21	eleq12d 2835	. . . 4 ⊢ (𝑤 = 𝑊 → (( ·sf ‘𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)) ↔ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
23	10, 22	anbi12d 632	. . 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 24170	. . 3 ⊢ TopMod = {𝑤 ∈ (TopMnd ∩ LMod) ∣ ((Scalar‘𝑤) ∈ TopRing ∧ ( ·sf ‘𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)))}
25	23, 24	elrab2 3695	. 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 1540   ∈ wcel 2108   ∩ cin 3950  ‘cfv 6561  (class class class)co 7431  Scalarcsca 17300  TopOpenctopn 17466  LModclmod 20858   ·_sf cscaf 20859   Cn ccn 23232   ×_t ctx 23568  TopMndctmd 24078  TopRingctrg 24164  TopModctlm 24166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-tlm 24170

This theorem is referenced by:  vscacn  24194  tlmtmd  24195  tlmlmod  24197  tlmtrg  24198  nlmtlm  24715