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Metamath Proof Explorer

Theorem lcvbr2

Description: The covers relation for a left vector space (or a left module). ( cvbr2 analog.) (Contributed by NM, 9-Jan-2015)

Ref Expression
Hypotheses lcvfbr.s ⊢ 𝑆 = ( LSubSp ‘ 𝑊 )
lcvfbr.c ⊢ 𝐶 = ( ⋖L ‘ 𝑊 )
lcvfbr.w ⊢ ( 𝜑 → 𝑊 ∈ 𝑋 )
lcvfbr.t ⊢ ( 𝜑 → 𝑇 ∈ 𝑆 )
lcvfbr.u ⊢ ( 𝜑 → 𝑈 ∈ 𝑆 )
Assertion lcvbr2 ( 𝜑 → ( 𝑇 𝐶 𝑈 ↔ ( 𝑇 ⊊ 𝑈 ∧ ∀ 𝑠 ∈ 𝑆 ( ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈 ) → 𝑠 = 𝑈 ) ) ) )

Proof

Step Hyp Ref Expression
1 lcvfbr.s ⊢ 𝑆 = ( LSubSp ‘ 𝑊 )
2 lcvfbr.c ⊢ 𝐶 = ( ⋖L ‘ 𝑊 )
3 lcvfbr.w ⊢ ( 𝜑 → 𝑊 ∈ 𝑋 )
4 lcvfbr.t ⊢ ( 𝜑 → 𝑇 ∈ 𝑆 )
5 lcvfbr.u ⊢ ( 𝜑 → 𝑈 ∈ 𝑆 )
6 1 2 3 4 5 lcvbr ⊢ ( 𝜑 → ( 𝑇 𝐶 𝑈 ↔ ( 𝑇 ⊊ 𝑈 ∧ ¬ ∃ 𝑠 ∈ 𝑆 ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈 ) ) ) )
7 iman ⊢ ( ( ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈 ) → 𝑠 = 𝑈 ) ↔ ¬ ( ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈 ) ∧ ¬ 𝑠 = 𝑈 ) )
8 anass ⊢ ( ( ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈 ) ∧ ¬ 𝑠 = 𝑈 ) ↔ ( 𝑇 ⊊ 𝑠 ∧ ( 𝑠 ⊆ 𝑈 ∧ ¬ 𝑠 = 𝑈 ) ) )
9 dfpss2 ⊢ ( 𝑠 ⊊ 𝑈 ↔ ( 𝑠 ⊆ 𝑈 ∧ ¬ 𝑠 = 𝑈 ) )
10 9 anbi2i ⊢ ( ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈 ) ↔ ( 𝑇 ⊊ 𝑠 ∧ ( 𝑠 ⊆ 𝑈 ∧ ¬ 𝑠 = 𝑈 ) ) )
11 8 10 bitr4i ⊢ ( ( ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈 ) ∧ ¬ 𝑠 = 𝑈 ) ↔ ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈 ) )
12 7 11 xchbinx ⊢ ( ( ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈 ) → 𝑠 = 𝑈 ) ↔ ¬ ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈 ) )
13 12 ralbii ⊢ ( ∀ 𝑠 ∈ 𝑆 ( ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈 ) → 𝑠 = 𝑈 ) ↔ ∀ 𝑠 ∈ 𝑆 ¬ ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈 ) )
14 ralnex ⊢ ( ∀ 𝑠 ∈ 𝑆 ¬ ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈 ) ↔ ¬ ∃ 𝑠 ∈ 𝑆 ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈 ) )
15 13 14 bitri ⊢ ( ∀ 𝑠 ∈ 𝑆 ( ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈 ) → 𝑠 = 𝑈 ) ↔ ¬ ∃ 𝑠 ∈ 𝑆 ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈 ) )
16 15 anbi2i ⊢ ( ( 𝑇 ⊊ 𝑈 ∧ ∀ 𝑠 ∈ 𝑆 ( ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈 ) → 𝑠 = 𝑈 ) ) ↔ ( 𝑇 ⊊ 𝑈 ∧ ¬ ∃ 𝑠 ∈ 𝑆 ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈 ) ) )
17 6 16 bitr4di ⊢ ( 𝜑 → ( 𝑇 𝐶 𝑈 ↔ ( 𝑇 ⊊ 𝑈 ∧ ∀ 𝑠 ∈ 𝑆 ( ( 𝑇 ⊊ 𝑠 ∧ 𝑠 ⊆ 𝑈 ) → 𝑠 = 𝑈 ) ) ) )