Step |
Hyp |
Ref |
Expression |
1 |
|
spansnsh |
⊢ ( 𝐴 ∈ ℋ → ( span ‘ { 𝐴 } ) ∈ Sℋ ) |
2 |
1
|
adantr |
⊢ ( ( 𝐴 ∈ ℋ ∧ ( 𝐴 +ℎ 𝐵 ) ∈ ( span ‘ { 𝐴 } ) ) → ( span ‘ { 𝐴 } ) ∈ Sℋ ) |
3 |
|
simpr |
⊢ ( ( 𝐴 ∈ ℋ ∧ ( 𝐴 +ℎ 𝐵 ) ∈ ( span ‘ { 𝐴 } ) ) → ( 𝐴 +ℎ 𝐵 ) ∈ ( span ‘ { 𝐴 } ) ) |
4 |
|
spansnid |
⊢ ( 𝐴 ∈ ℋ → 𝐴 ∈ ( span ‘ { 𝐴 } ) ) |
5 |
4
|
adantr |
⊢ ( ( 𝐴 ∈ ℋ ∧ ( 𝐴 +ℎ 𝐵 ) ∈ ( span ‘ { 𝐴 } ) ) → 𝐴 ∈ ( span ‘ { 𝐴 } ) ) |
6 |
|
shsubcl |
⊢ ( ( ( span ‘ { 𝐴 } ) ∈ Sℋ ∧ ( 𝐴 +ℎ 𝐵 ) ∈ ( span ‘ { 𝐴 } ) ∧ 𝐴 ∈ ( span ‘ { 𝐴 } ) ) → ( ( 𝐴 +ℎ 𝐵 ) −ℎ 𝐴 ) ∈ ( span ‘ { 𝐴 } ) ) |
7 |
2 3 5 6
|
syl3anc |
⊢ ( ( 𝐴 ∈ ℋ ∧ ( 𝐴 +ℎ 𝐵 ) ∈ ( span ‘ { 𝐴 } ) ) → ( ( 𝐴 +ℎ 𝐵 ) −ℎ 𝐴 ) ∈ ( span ‘ { 𝐴 } ) ) |
8 |
7
|
ex |
⊢ ( 𝐴 ∈ ℋ → ( ( 𝐴 +ℎ 𝐵 ) ∈ ( span ‘ { 𝐴 } ) → ( ( 𝐴 +ℎ 𝐵 ) −ℎ 𝐴 ) ∈ ( span ‘ { 𝐴 } ) ) ) |
9 |
8
|
adantr |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐵 ) ∈ ( span ‘ { 𝐴 } ) → ( ( 𝐴 +ℎ 𝐵 ) −ℎ 𝐴 ) ∈ ( span ‘ { 𝐴 } ) ) ) |
10 |
|
hvpncan2 |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐵 ) −ℎ 𝐴 ) = 𝐵 ) |
11 |
10
|
eleq1d |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( ( 𝐴 +ℎ 𝐵 ) −ℎ 𝐴 ) ∈ ( span ‘ { 𝐴 } ) ↔ 𝐵 ∈ ( span ‘ { 𝐴 } ) ) ) |
12 |
9 11
|
sylibd |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐵 ) ∈ ( span ‘ { 𝐴 } ) → 𝐵 ∈ ( span ‘ { 𝐴 } ) ) ) |
13 |
|
shaddcl |
⊢ ( ( ( span ‘ { 𝐴 } ) ∈ Sℋ ∧ 𝐴 ∈ ( span ‘ { 𝐴 } ) ∧ 𝐵 ∈ ( span ‘ { 𝐴 } ) ) → ( 𝐴 +ℎ 𝐵 ) ∈ ( span ‘ { 𝐴 } ) ) |
14 |
13
|
3expia |
⊢ ( ( ( span ‘ { 𝐴 } ) ∈ Sℋ ∧ 𝐴 ∈ ( span ‘ { 𝐴 } ) ) → ( 𝐵 ∈ ( span ‘ { 𝐴 } ) → ( 𝐴 +ℎ 𝐵 ) ∈ ( span ‘ { 𝐴 } ) ) ) |
15 |
1 4 14
|
syl2anc |
⊢ ( 𝐴 ∈ ℋ → ( 𝐵 ∈ ( span ‘ { 𝐴 } ) → ( 𝐴 +ℎ 𝐵 ) ∈ ( span ‘ { 𝐴 } ) ) ) |
16 |
15
|
adantr |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐵 ∈ ( span ‘ { 𝐴 } ) → ( 𝐴 +ℎ 𝐵 ) ∈ ( span ‘ { 𝐴 } ) ) ) |
17 |
12 16
|
impbid |
⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐵 ) ∈ ( span ‘ { 𝐴 } ) ↔ 𝐵 ∈ ( span ‘ { 𝐴 } ) ) ) |