Proof of Theorem sspmval
| Step | Hyp | Ref
| Expression |
| 1 | | sspm.h |
. . . . . . . 8
⊢ 𝐻 = (SubSp‘𝑈) |
| 2 | 1 | sspnv 30745 |
. . . . . . 7
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec) |
| 3 | | neg1cn 12380 |
. . . . . . . . 9
⊢ -1 ∈
ℂ |
| 4 | | sspm.y |
. . . . . . . . . 10
⊢ 𝑌 = (BaseSet‘𝑊) |
| 5 | | eqid 2737 |
. . . . . . . . . 10
⊢ (
·𝑠OLD ‘𝑊) = ( ·𝑠OLD
‘𝑊) |
| 6 | 4, 5 | nvscl 30645 |
. . . . . . . . 9
⊢ ((𝑊 ∈ NrmCVec ∧ -1 ∈
ℂ ∧ 𝐵 ∈
𝑌) → (-1(
·𝑠OLD ‘𝑊)𝐵) ∈ 𝑌) |
| 7 | 3, 6 | mp3an2 1451 |
. . . . . . . 8
⊢ ((𝑊 ∈ NrmCVec ∧ 𝐵 ∈ 𝑌) → (-1(
·𝑠OLD ‘𝑊)𝐵) ∈ 𝑌) |
| 8 | 7 | ex 412 |
. . . . . . 7
⊢ (𝑊 ∈ NrmCVec → (𝐵 ∈ 𝑌 → (-1(
·𝑠OLD ‘𝑊)𝐵) ∈ 𝑌)) |
| 9 | 2, 8 | syl 17 |
. . . . . 6
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝐵 ∈ 𝑌 → (-1(
·𝑠OLD ‘𝑊)𝐵) ∈ 𝑌)) |
| 10 | 9 | anim2d 612 |
. . . . 5
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌) → (𝐴 ∈ 𝑌 ∧ (-1(
·𝑠OLD ‘𝑊)𝐵) ∈ 𝑌))) |
| 11 | 10 | imp 406 |
. . . 4
⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌)) → (𝐴 ∈ 𝑌 ∧ (-1(
·𝑠OLD ‘𝑊)𝐵) ∈ 𝑌)) |
| 12 | | eqid 2737 |
. . . . 5
⊢ (
+𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) |
| 13 | | eqid 2737 |
. . . . 5
⊢ (
+𝑣 ‘𝑊) = ( +𝑣 ‘𝑊) |
| 14 | 4, 12, 13, 1 | sspgval 30748 |
. . . 4
⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ 𝑌 ∧ (-1(
·𝑠OLD ‘𝑊)𝐵) ∈ 𝑌)) → (𝐴( +𝑣 ‘𝑊)(-1(
·𝑠OLD ‘𝑊)𝐵)) = (𝐴( +𝑣 ‘𝑈)(-1(
·𝑠OLD ‘𝑊)𝐵))) |
| 15 | 11, 14 | syldan 591 |
. . 3
⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌)) → (𝐴( +𝑣 ‘𝑊)(-1(
·𝑠OLD ‘𝑊)𝐵)) = (𝐴( +𝑣 ‘𝑈)(-1(
·𝑠OLD ‘𝑊)𝐵))) |
| 16 | | eqid 2737 |
. . . . . . 7
⊢ (
·𝑠OLD ‘𝑈) = ( ·𝑠OLD
‘𝑈) |
| 17 | 4, 16, 5, 1 | sspsval 30750 |
. . . . . 6
⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (-1 ∈ ℂ ∧ 𝐵 ∈ 𝑌)) → (-1(
·𝑠OLD ‘𝑊)𝐵) = (-1(
·𝑠OLD ‘𝑈)𝐵)) |
| 18 | 3, 17 | mpanr1 703 |
. . . . 5
⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝐵 ∈ 𝑌) → (-1(
·𝑠OLD ‘𝑊)𝐵) = (-1(
·𝑠OLD ‘𝑈)𝐵)) |
| 19 | 18 | adantrl 716 |
. . . 4
⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌)) → (-1(
·𝑠OLD ‘𝑊)𝐵) = (-1(
·𝑠OLD ‘𝑈)𝐵)) |
| 20 | 19 | oveq2d 7447 |
. . 3
⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌)) → (𝐴( +𝑣 ‘𝑈)(-1(
·𝑠OLD ‘𝑊)𝐵)) = (𝐴( +𝑣 ‘𝑈)(-1(
·𝑠OLD ‘𝑈)𝐵))) |
| 21 | 15, 20 | eqtrd 2777 |
. 2
⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌)) → (𝐴( +𝑣 ‘𝑊)(-1(
·𝑠OLD ‘𝑊)𝐵)) = (𝐴( +𝑣 ‘𝑈)(-1(
·𝑠OLD ‘𝑈)𝐵))) |
| 22 | | sspm.l |
. . . . 5
⊢ 𝐿 = ( −𝑣
‘𝑊) |
| 23 | 4, 13, 5, 22 | nvmval 30661 |
. . . 4
⊢ ((𝑊 ∈ NrmCVec ∧ 𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌) → (𝐴𝐿𝐵) = (𝐴( +𝑣 ‘𝑊)(-1(
·𝑠OLD ‘𝑊)𝐵))) |
| 24 | 23 | 3expb 1121 |
. . 3
⊢ ((𝑊 ∈ NrmCVec ∧ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌)) → (𝐴𝐿𝐵) = (𝐴( +𝑣 ‘𝑊)(-1(
·𝑠OLD ‘𝑊)𝐵))) |
| 25 | 2, 24 | sylan 580 |
. 2
⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌)) → (𝐴𝐿𝐵) = (𝐴( +𝑣 ‘𝑊)(-1(
·𝑠OLD ‘𝑊)𝐵))) |
| 26 | | eqid 2737 |
. . . . . . 7
⊢
(BaseSet‘𝑈) =
(BaseSet‘𝑈) |
| 27 | 26, 4, 1 | sspba 30746 |
. . . . . 6
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑌 ⊆ (BaseSet‘𝑈)) |
| 28 | 27 | sseld 3982 |
. . . . 5
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝐴 ∈ 𝑌 → 𝐴 ∈ (BaseSet‘𝑈))) |
| 29 | 27 | sseld 3982 |
. . . . 5
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝐵 ∈ 𝑌 → 𝐵 ∈ (BaseSet‘𝑈))) |
| 30 | 28, 29 | anim12d 609 |
. . . 4
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌) → (𝐴 ∈ (BaseSet‘𝑈) ∧ 𝐵 ∈ (BaseSet‘𝑈)))) |
| 31 | 30 | imp 406 |
. . 3
⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌)) → (𝐴 ∈ (BaseSet‘𝑈) ∧ 𝐵 ∈ (BaseSet‘𝑈))) |
| 32 | | sspm.m |
. . . . . 6
⊢ 𝑀 = ( −𝑣
‘𝑈) |
| 33 | 26, 12, 16, 32 | nvmval 30661 |
. . . . 5
⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ (BaseSet‘𝑈) ∧ 𝐵 ∈ (BaseSet‘𝑈)) → (𝐴𝑀𝐵) = (𝐴( +𝑣 ‘𝑈)(-1(
·𝑠OLD ‘𝑈)𝐵))) |
| 34 | 33 | 3expb 1121 |
. . . 4
⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ (BaseSet‘𝑈) ∧ 𝐵 ∈ (BaseSet‘𝑈))) → (𝐴𝑀𝐵) = (𝐴( +𝑣 ‘𝑈)(-1(
·𝑠OLD ‘𝑈)𝐵))) |
| 35 | 34 | adantlr 715 |
. . 3
⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ (BaseSet‘𝑈) ∧ 𝐵 ∈ (BaseSet‘𝑈))) → (𝐴𝑀𝐵) = (𝐴( +𝑣 ‘𝑈)(-1(
·𝑠OLD ‘𝑈)𝐵))) |
| 36 | 31, 35 | syldan 591 |
. 2
⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌)) → (𝐴𝑀𝐵) = (𝐴( +𝑣 ‘𝑈)(-1(
·𝑠OLD ‘𝑈)𝐵))) |
| 37 | 21, 25, 36 | 3eqtr4d 2787 |
1
⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌)) → (𝐴𝐿𝐵) = (𝐴𝑀𝐵)) |