Theorem lcvexch

Description: Subspaces satisfy the exchange axiom. Lemma 7.5 of MaedaMaeda p. 31. ( cvexchi analog.) TODO: combine some lemmas. (Contributed by NM, 10-Jan-2015)

	Ref	Expression
Hypotheses	lcvexch.s	⊢ 𝑆 = ( LSubSp ‘ 𝑊 )
	lcvexch.p	⊢ ⊕ = ( LSSum ‘ 𝑊 )
	lcvexch.c	⊢ 𝐶 = ( ⋖L ‘ 𝑊 )
	lcvexch.w	⊢ ( 𝜑 → 𝑊 ∈ LMod )
	lcvexch.t	⊢ ( 𝜑 → 𝑇 ∈ 𝑆 )
	lcvexch.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑆 )
Assertion	lcvexch	⊢ ( 𝜑 → ( ( 𝑇 ∩ 𝑈 ) 𝐶 𝑈 ↔ 𝑇 𝐶 ( 𝑇 ⊕ 𝑈 ) ) )

Proof

Step	Hyp	Ref	Expression
1		lcvexch.s	⊢ 𝑆 = ( LSubSp ‘ 𝑊 )
2		lcvexch.p	⊢ ⊕ = ( LSSum ‘ 𝑊 )
3		lcvexch.c	⊢ 𝐶 = ( ⋖L ‘ 𝑊 )
4		lcvexch.w	⊢ ( 𝜑 → 𝑊 ∈ LMod )
5		lcvexch.t	⊢ ( 𝜑 → 𝑇 ∈ 𝑆 )
6		lcvexch.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑆 )
7	4	adantr	⊢ ( ( 𝜑 ∧ ( 𝑇 ∩ 𝑈 ) 𝐶 𝑈 ) → 𝑊 ∈ LMod )
8	5	adantr	⊢ ( ( 𝜑 ∧ ( 𝑇 ∩ 𝑈 ) 𝐶 𝑈 ) → 𝑇 ∈ 𝑆 )
9	6	adantr	⊢ ( ( 𝜑 ∧ ( 𝑇 ∩ 𝑈 ) 𝐶 𝑈 ) → 𝑈 ∈ 𝑆 )
10		simpr	⊢ ( ( 𝜑 ∧ ( 𝑇 ∩ 𝑈 ) 𝐶 𝑈 ) → ( 𝑇 ∩ 𝑈 ) 𝐶 𝑈 )
11	1 2 3 7 8 9 10	lcvexchlem5	⊢ ( ( 𝜑 ∧ ( 𝑇 ∩ 𝑈 ) 𝐶 𝑈 ) → 𝑇 𝐶 ( 𝑇 ⊕ 𝑈 ) )
12	4	adantr	⊢ ( ( 𝜑 ∧ 𝑇 𝐶 ( 𝑇 ⊕ 𝑈 ) ) → 𝑊 ∈ LMod )
13	5	adantr	⊢ ( ( 𝜑 ∧ 𝑇 𝐶 ( 𝑇 ⊕ 𝑈 ) ) → 𝑇 ∈ 𝑆 )
14	6	adantr	⊢ ( ( 𝜑 ∧ 𝑇 𝐶 ( 𝑇 ⊕ 𝑈 ) ) → 𝑈 ∈ 𝑆 )
15		simpr	⊢ ( ( 𝜑 ∧ 𝑇 𝐶 ( 𝑇 ⊕ 𝑈 ) ) → 𝑇 𝐶 ( 𝑇 ⊕ 𝑈 ) )
16	1 2 3 12 13 14 15	lcvexchlem4	⊢ ( ( 𝜑 ∧ 𝑇 𝐶 ( 𝑇 ⊕ 𝑈 ) ) → ( 𝑇 ∩ 𝑈 ) 𝐶 𝑈 )
17	11 16	impbida	⊢ ( 𝜑 → ( ( 𝑇 ∩ 𝑈 ) 𝐶 𝑈 ↔ 𝑇 𝐶 ( 𝑇 ⊕ 𝑈 ) ) )