Theorem istlm 22787

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 471	. 2 ⊢ (((𝑊 ∈ (TopMnd ∩ LMod) ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) ↔ (𝑊 ∈ (TopMnd ∩ LMod) ∧ (𝐹 ∈ TopRing ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))))
2		df-3an 1085	. . . 4 ⊢ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod) ∧ 𝐹 ∈ TopRing))
3		elin 4168	. . . . 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 280	. . 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 6664	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
8		istlm.f	. . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊)
9	7, 8	syl6eqr 2874	. . . . 5 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
10	9	eleq1d 2897	. . . 4 ⊢ (𝑤 = 𝑊 → ((Scalar‘𝑤) ∈ TopRing ↔ 𝐹 ∈ TopRing))
11		fveq2 6664	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( ·sf ‘𝑤) = ( ·sf ‘𝑊))
12		istlm.s	. . . . . 6 ⊢ · = ( ·sf ‘𝑊)
13	11, 12	syl6eqr 2874	. . . . 5 ⊢ (𝑤 = 𝑊 → ( ·sf ‘𝑤) = · )
14	9	fveq2d 6668	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (TopOpen‘(Scalar‘𝑤)) = (TopOpen‘𝐹))
15		istlm.k	. . . . . . . 8 ⊢ 𝐾 = (TopOpen‘𝐹)
16	14, 15	syl6eqr 2874	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (TopOpen‘(Scalar‘𝑤)) = 𝐾)
17		fveq2 6664	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (TopOpen‘𝑤) = (TopOpen‘𝑊))
18		istlm.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝑊)
19	17, 18	syl6eqr 2874	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (TopOpen‘𝑤) = 𝐽)
20	16, 19	oveq12d 7168	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) = (𝐾 ×t 𝐽))
21	20, 19	oveq12d 7168	. . . . 5 ⊢ (𝑤 = 𝑊 → (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)) = ((𝐾 ×t 𝐽) Cn 𝐽))
22	13, 21	eleq12d 2907	. . . 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 22764	. . 3 ⊢ TopMod = {𝑤 ∈ (TopMnd ∩ LMod) ∣ ((Scalar‘𝑤) ∈ TopRing ∧ ( ·sf ‘𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)))}
25	23, 24	elrab2 3682	. 2 ⊢ (𝑊 ∈ TopMod ↔ (𝑊 ∈ (TopMnd ∩ LMod) ∧ (𝐹 ∈ TopRing ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))))
26	1, 6, 25	3bitr4ri 306	1 ⊢ (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ∩ cin 3934  ‘cfv 6349  (class class class)co 7150  Scalarcsca 16562  TopOpenctopn 16689  LModclmod 19628   ·_sf cscaf 19629   Cn ccn 21826   ×_t ctx 22162  TopMndctmd 22672  TopRingctrg 22758  TopModctlm 22760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-iota 6308  df-fv 6357  df-ov 7153  df-tlm 22764

This theorem is referenced by:  vscacn  22788  tlmtmd  22789  tlmlmod  22791  tlmtrg  22792  nlmtlm  23297