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Metamath Proof Explorer

Theorem shless

Description: Subset implies subset of subspace sum. (Contributed by Mario Carneiro, 15-May-2014) (New usage is discouraged.)

Ref Expression
Assertion shless ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐴 +ℋ 𝐶 ) ⊆ ( 𝐵 +ℋ 𝐶 ) )

Proof

Step Hyp Ref Expression
1 ssrexv ⊢ ( 𝐴 ⊆ 𝐵 → ( ∃ 𝑦 ∈ 𝐴 ∃ 𝑧 ∈ 𝐶 𝑥 = ( 𝑦 +ℎ 𝑧 ) → ∃ 𝑦 ∈ 𝐵 ∃ 𝑧 ∈ 𝐶 𝑥 = ( 𝑦 +ℎ 𝑧 ) ) )
2 1 adantl ⊢ ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵 ) → ( ∃ 𝑦 ∈ 𝐴 ∃ 𝑧 ∈ 𝐶 𝑥 = ( 𝑦 +ℎ 𝑧 ) → ∃ 𝑦 ∈ 𝐵 ∃ 𝑧 ∈ 𝐶 𝑥 = ( 𝑦 +ℎ 𝑧 ) ) )
3 simpl1 ⊢ ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 ∈ Sℋ )
4 simpl3 ⊢ ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵 ) → 𝐶 ∈ Sℋ )
5 shsel ⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) → ( 𝑥 ∈ ( 𝐴 +ℋ 𝐶 ) ↔ ∃ 𝑦 ∈ 𝐴 ∃ 𝑧 ∈ 𝐶 𝑥 = ( 𝑦 +ℎ 𝑧 ) ) )
6 3 4 5 syl2anc ⊢ ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝑥 ∈ ( 𝐴 +ℋ 𝐶 ) ↔ ∃ 𝑦 ∈ 𝐴 ∃ 𝑧 ∈ 𝐶 𝑥 = ( 𝑦 +ℎ 𝑧 ) ) )
7 simpl2 ⊢ ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵 ) → 𝐵 ∈ Sℋ )
8 shsel ⊢ ( ( 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) → ( 𝑥 ∈ ( 𝐵 +ℋ 𝐶 ) ↔ ∃ 𝑦 ∈ 𝐵 ∃ 𝑧 ∈ 𝐶 𝑥 = ( 𝑦 +ℎ 𝑧 ) ) )
9 7 4 8 syl2anc ⊢ ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝑥 ∈ ( 𝐵 +ℋ 𝐶 ) ↔ ∃ 𝑦 ∈ 𝐵 ∃ 𝑧 ∈ 𝐶 𝑥 = ( 𝑦 +ℎ 𝑧 ) ) )
10 2 6 9 3imtr4d ⊢ ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝑥 ∈ ( 𝐴 +ℋ 𝐶 ) → 𝑥 ∈ ( 𝐵 +ℋ 𝐶 ) ) )
11 10 ssrdv ⊢ ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐴 +ℋ 𝐶 ) ⊆ ( 𝐵 +ℋ 𝐶 ) )