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Metamath Proof Explorer

Theorem padd4N

Description: Rearrangement of 4 terms in a projective subspace sum. (Contributed by NM, 14-Jan-2012) (New usage is discouraged.)

Ref Expression
Hypotheses paddass.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
paddass.p ⊢ + = ( +𝑃 ‘ 𝐾 )
Assertion padd4N ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( 𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴 ) ) → ( ( 𝑋 + 𝑌 ) + ( 𝑍 + 𝑊 ) ) = ( ( 𝑋 + 𝑍 ) + ( 𝑌 + 𝑊 ) ) )

Proof

Step Hyp Ref Expression
1 paddass.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
2 paddass.p ⊢ + = ( +𝑃 ‘ 𝐾 )
3 simp1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( 𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴 ) ) → 𝐾 ∈ HL )
4 simp2r ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( 𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴 ) ) → 𝑌 ⊆ 𝐴 )
5 simp3l ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( 𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴 ) ) → 𝑍 ⊆ 𝐴 )
6 simp3r ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( 𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴 ) ) → 𝑊 ⊆ 𝐴 )
7 1 2 padd12N ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴 ) ) → ( 𝑌 + ( 𝑍 + 𝑊 ) ) = ( 𝑍 + ( 𝑌 + 𝑊 ) ) )
8 3 4 5 6 7 syl13anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( 𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴 ) ) → ( 𝑌 + ( 𝑍 + 𝑊 ) ) = ( 𝑍 + ( 𝑌 + 𝑊 ) ) )
9 8 oveq2d ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( 𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴 ) ) → ( 𝑋 + ( 𝑌 + ( 𝑍 + 𝑊 ) ) ) = ( 𝑋 + ( 𝑍 + ( 𝑌 + 𝑊 ) ) ) )
10 simp2l ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( 𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴 ) ) → 𝑋 ⊆ 𝐴 )
11 1 2 paddssat ⊢ ( ( 𝐾 ∈ HL ∧ 𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴 ) → ( 𝑍 + 𝑊 ) ⊆ 𝐴 )
12 3 5 6 11 syl3anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( 𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴 ) ) → ( 𝑍 + 𝑊 ) ⊆ 𝐴 )
13 1 2 paddass ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ ( 𝑍 + 𝑊 ) ⊆ 𝐴 ) ) → ( ( 𝑋 + 𝑌 ) + ( 𝑍 + 𝑊 ) ) = ( 𝑋 + ( 𝑌 + ( 𝑍 + 𝑊 ) ) ) )
14 3 10 4 12 13 syl13anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( 𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴 ) ) → ( ( 𝑋 + 𝑌 ) + ( 𝑍 + 𝑊 ) ) = ( 𝑋 + ( 𝑌 + ( 𝑍 + 𝑊 ) ) ) )
15 1 2 paddssat ⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴 ) → ( 𝑌 + 𝑊 ) ⊆ 𝐴 )
16 3 4 6 15 syl3anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( 𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴 ) ) → ( 𝑌 + 𝑊 ) ⊆ 𝐴 )
17 1 2 paddass ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ∧ ( 𝑌 + 𝑊 ) ⊆ 𝐴 ) ) → ( ( 𝑋 + 𝑍 ) + ( 𝑌 + 𝑊 ) ) = ( 𝑋 + ( 𝑍 + ( 𝑌 + 𝑊 ) ) ) )
18 3 10 5 16 17 syl13anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( 𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴 ) ) → ( ( 𝑋 + 𝑍 ) + ( 𝑌 + 𝑊 ) ) = ( 𝑋 + ( 𝑍 + ( 𝑌 + 𝑊 ) ) ) )
19 9 14 18 3eqtr4d ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ ( 𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴 ) ) → ( ( 𝑋 + 𝑌 ) + ( 𝑍 + 𝑊 ) ) = ( ( 𝑋 + 𝑍 ) + ( 𝑌 + 𝑊 ) ) )