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Metamath Proof Explorer

Theorem cdlemksv

Description: Part of proof of Lemma K of Crawley p. 118. Value of the sigma(p) function. (Contributed by NM, 26-Jun-2013)

Ref Expression
Hypotheses cdlemk.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdlemk.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemk.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemk.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemk.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemk.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemk.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
cdlemk.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdlemk.s ⊢ 𝑆 = ( 𝑓 ∈ 𝑇 ↦ ( ℩ 𝑖 ∈ 𝑇 ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑓 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝐹 ) ) ) ) ) )
Assertion cdlemksv ( 𝐺 ∈ 𝑇 → ( 𝑆 ‘ 𝐺 ) = ( ℩ 𝑖 ∈ 𝑇 ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) ) )

Proof

Step Hyp Ref Expression
1 cdlemk.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdlemk.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdlemk.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdlemk.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdlemk.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdlemk.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7 cdlemk.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8 cdlemk.m ⊢ ∧ = ( meet ‘ 𝐾 )
9 cdlemk.s ⊢ 𝑆 = ( 𝑓 ∈ 𝑇 ↦ ( ℩ 𝑖 ∈ 𝑇 ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑓 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝐹 ) ) ) ) ) )
10 fveq2 ⊢ ( 𝑓 = 𝐺 → ( 𝑅 ‘ 𝑓 ) = ( 𝑅 ‘ 𝐺 ) )
11 10 oveq2d ⊢ ( 𝑓 = 𝐺 → ( 𝑃 ∨ ( 𝑅 ‘ 𝑓 ) ) = ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) )
12 coeq1 ⊢ ( 𝑓 = 𝐺 → ( 𝑓 ∘ ◡ 𝐹 ) = ( 𝐺 ∘ ◡ 𝐹 ) )
13 12 fveq2d ⊢ ( 𝑓 = 𝐺 → ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝐹 ) ) = ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) )
14 13 oveq2d ⊢ ( 𝑓 = 𝐺 → ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝐹 ) ) ) = ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) )
15 11 14 oveq12d ⊢ ( 𝑓 = 𝐺 → ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑓 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝐹 ) ) ) ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) )
16 15 eqeq2d ⊢ ( 𝑓 = 𝐺 → ( ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑓 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝐹 ) ) ) ) ↔ ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) ) )
17 16 riotabidv ⊢ ( 𝑓 = 𝐺 → ( ℩ 𝑖 ∈ 𝑇 ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑓 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝐹 ) ) ) ) ) = ( ℩ 𝑖 ∈ 𝑇 ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) ) )
18 riotaex ⊢ ( ℩ 𝑖 ∈ 𝑇 ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) ) ∈ V
19 17 9 18 fvmpt ⊢ ( 𝐺 ∈ 𝑇 → ( 𝑆 ‘ 𝐺 ) = ( ℩ 𝑖 ∈ 𝑇 ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) ) )