Theorem istlm 23680

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 469	. 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 3963	. . . . 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 277	. . 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 6888	. . . . . 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 2818	. . . 4 ⊢ (𝑤 = 𝑊 → ((Scalar‘𝑤) ∈ TopRing ↔ 𝐹 ∈ TopRing))
11		fveq2 6888	. . . . . 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 6892	. . . . . . . 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 6888	. . . . . . . 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 7423	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) = (𝐾 ×t 𝐽))
21	20, 19	oveq12d 7423	. . . . 5 ⊢ (𝑤 = 𝑊 → (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)) = ((𝐾 ×t 𝐽) Cn 𝐽))
22	13, 21	eleq12d 2827	. . . 4 ⊢ (𝑤 = 𝑊 → (( ·sf ‘𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)) ↔ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
23	10, 22	anbi12d 631	. . 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 23657	. . 3 ⊢ TopMod = {𝑤 ∈ (TopMnd ∩ LMod) ∣ ((Scalar‘𝑤) ∈ TopRing ∧ ( ·sf ‘𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)))}
25	23, 24	elrab2 3685	. 2 ⊢ (𝑊 ∈ TopMod ↔ (𝑊 ∈ (TopMnd ∩ LMod) ∧ (𝐹 ∈ TopRing ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))))
26	1, 6, 25	3bitr4ri 303	1 ⊢ (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ∩ cin 3946  ‘cfv 6540  (class class class)co 7405  Scalarcsca 17196  TopOpenctopn 17363  LModclmod 20463   ·_sf cscaf 20464   Cn ccn 22719   ×_t ctx 23055  TopMndctmd 23565  TopRingctrg 23651  TopModctlm 23653

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-ov 7408  df-tlm 23657

This theorem is referenced by:  vscacn  23681  tlmtmd  23682  tlmlmod  23684  tlmtrg  23685  nlmtlm  24202