Step | Hyp | Ref
| Expression |
1 | | lnoval.7 |
. 2
⊢ 𝐿 = (𝑈 LnOp 𝑊) |
2 | | fveq2 6674 |
. . . . . 6
⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈)) |
3 | | lnoval.1 |
. . . . . 6
⊢ 𝑋 = (BaseSet‘𝑈) |
4 | 2, 3 | eqtr4di 2791 |
. . . . 5
⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋) |
5 | 4 | oveq2d 7186 |
. . . 4
⊢ (𝑢 = 𝑈 → ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢)) = ((BaseSet‘𝑤) ↑m 𝑋)) |
6 | | fveq2 6674 |
. . . . . . . . . 10
⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = ( +𝑣
‘𝑈)) |
7 | | lnoval.3 |
. . . . . . . . . 10
⊢ 𝐺 = ( +𝑣
‘𝑈) |
8 | 6, 7 | eqtr4di 2791 |
. . . . . . . . 9
⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = 𝐺) |
9 | | fveq2 6674 |
. . . . . . . . . . 11
⊢ (𝑢 = 𝑈 → (
·𝑠OLD ‘𝑢) = ( ·𝑠OLD
‘𝑈)) |
10 | | lnoval.5 |
. . . . . . . . . . 11
⊢ 𝑅 = (
·𝑠OLD ‘𝑈) |
11 | 9, 10 | eqtr4di 2791 |
. . . . . . . . . 10
⊢ (𝑢 = 𝑈 → (
·𝑠OLD ‘𝑢) = 𝑅) |
12 | 11 | oveqd 7187 |
. . . . . . . . 9
⊢ (𝑢 = 𝑈 → (𝑥( ·𝑠OLD
‘𝑢)𝑦) = (𝑥𝑅𝑦)) |
13 | | eqidd 2739 |
. . . . . . . . 9
⊢ (𝑢 = 𝑈 → 𝑧 = 𝑧) |
14 | 8, 12, 13 | oveq123d 7191 |
. . . . . . . 8
⊢ (𝑢 = 𝑈 → ((𝑥( ·𝑠OLD
‘𝑢)𝑦)( +𝑣 ‘𝑢)𝑧) = ((𝑥𝑅𝑦)𝐺𝑧)) |
15 | 14 | fveqeq2d 6682 |
. . . . . . 7
⊢ (𝑢 = 𝑈 → ((𝑡‘((𝑥( ·𝑠OLD
‘𝑢)𝑦)( +𝑣 ‘𝑢)𝑧)) = ((𝑥( ·𝑠OLD
‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧)) ↔ (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD
‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧)))) |
16 | 4, 15 | raleqbidv 3304 |
. . . . . 6
⊢ (𝑢 = 𝑈 → (∀𝑧 ∈ (BaseSet‘𝑢)(𝑡‘((𝑥( ·𝑠OLD
‘𝑢)𝑦)( +𝑣 ‘𝑢)𝑧)) = ((𝑥( ·𝑠OLD
‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧)) ↔ ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD
‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧)))) |
17 | 4, 16 | raleqbidv 3304 |
. . . . 5
⊢ (𝑢 = 𝑈 → (∀𝑦 ∈ (BaseSet‘𝑢)∀𝑧 ∈ (BaseSet‘𝑢)(𝑡‘((𝑥( ·𝑠OLD
‘𝑢)𝑦)( +𝑣 ‘𝑢)𝑧)) = ((𝑥( ·𝑠OLD
‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧)) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD
‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧)))) |
18 | 17 | ralbidv 3109 |
. . . 4
⊢ (𝑢 = 𝑈 → (∀𝑥 ∈ ℂ ∀𝑦 ∈ (BaseSet‘𝑢)∀𝑧 ∈ (BaseSet‘𝑢)(𝑡‘((𝑥( ·𝑠OLD
‘𝑢)𝑦)( +𝑣 ‘𝑢)𝑧)) = ((𝑥( ·𝑠OLD
‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧)) ↔ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD
‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧)))) |
19 | 5, 18 | rabeqbidv 3387 |
. . 3
⊢ (𝑢 = 𝑈 → {𝑡 ∈ ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢)) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (BaseSet‘𝑢)∀𝑧 ∈ (BaseSet‘𝑢)(𝑡‘((𝑥( ·𝑠OLD
‘𝑢)𝑦)( +𝑣 ‘𝑢)𝑧)) = ((𝑥( ·𝑠OLD
‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧))} = {𝑡 ∈ ((BaseSet‘𝑤) ↑m 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD
‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧))}) |
20 | | fveq2 6674 |
. . . . . 6
⊢ (𝑤 = 𝑊 → (BaseSet‘𝑤) = (BaseSet‘𝑊)) |
21 | | lnoval.2 |
. . . . . 6
⊢ 𝑌 = (BaseSet‘𝑊) |
22 | 20, 21 | eqtr4di 2791 |
. . . . 5
⊢ (𝑤 = 𝑊 → (BaseSet‘𝑤) = 𝑌) |
23 | 22 | oveq1d 7185 |
. . . 4
⊢ (𝑤 = 𝑊 → ((BaseSet‘𝑤) ↑m 𝑋) = (𝑌 ↑m 𝑋)) |
24 | | fveq2 6674 |
. . . . . . . . 9
⊢ (𝑤 = 𝑊 → ( +𝑣 ‘𝑤) = ( +𝑣
‘𝑊)) |
25 | | lnoval.4 |
. . . . . . . . 9
⊢ 𝐻 = ( +𝑣
‘𝑊) |
26 | 24, 25 | eqtr4di 2791 |
. . . . . . . 8
⊢ (𝑤 = 𝑊 → ( +𝑣 ‘𝑤) = 𝐻) |
27 | | fveq2 6674 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑊 → (
·𝑠OLD ‘𝑤) = ( ·𝑠OLD
‘𝑊)) |
28 | | lnoval.6 |
. . . . . . . . . 10
⊢ 𝑆 = (
·𝑠OLD ‘𝑊) |
29 | 27, 28 | eqtr4di 2791 |
. . . . . . . . 9
⊢ (𝑤 = 𝑊 → (
·𝑠OLD ‘𝑤) = 𝑆) |
30 | 29 | oveqd 7187 |
. . . . . . . 8
⊢ (𝑤 = 𝑊 → (𝑥( ·𝑠OLD
‘𝑤)(𝑡‘𝑦)) = (𝑥𝑆(𝑡‘𝑦))) |
31 | | eqidd 2739 |
. . . . . . . 8
⊢ (𝑤 = 𝑊 → (𝑡‘𝑧) = (𝑡‘𝑧)) |
32 | 26, 30, 31 | oveq123d 7191 |
. . . . . . 7
⊢ (𝑤 = 𝑊 → ((𝑥( ·𝑠OLD
‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧)) = ((𝑥𝑆(𝑡‘𝑦))𝐻(𝑡‘𝑧))) |
33 | 32 | eqeq2d 2749 |
. . . . . 6
⊢ (𝑤 = 𝑊 → ((𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD
‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧)) ↔ (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡‘𝑦))𝐻(𝑡‘𝑧)))) |
34 | 33 | 2ralbidv 3111 |
. . . . 5
⊢ (𝑤 = 𝑊 → (∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD
‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧)) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡‘𝑦))𝐻(𝑡‘𝑧)))) |
35 | 34 | ralbidv 3109 |
. . . 4
⊢ (𝑤 = 𝑊 → (∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD
‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧)) ↔ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡‘𝑦))𝐻(𝑡‘𝑧)))) |
36 | 23, 35 | rabeqbidv 3387 |
. . 3
⊢ (𝑤 = 𝑊 → {𝑡 ∈ ((BaseSet‘𝑤) ↑m 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD
‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧))} = {𝑡 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡‘𝑦))𝐻(𝑡‘𝑧))}) |
37 | | df-lno 28679 |
. . 3
⊢ LnOp =
(𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ ((BaseSet‘𝑤) ↑m
(BaseSet‘𝑢)) ∣
∀𝑥 ∈ ℂ
∀𝑦 ∈
(BaseSet‘𝑢)∀𝑧 ∈ (BaseSet‘𝑢)(𝑡‘((𝑥( ·𝑠OLD
‘𝑢)𝑦)( +𝑣 ‘𝑢)𝑧)) = ((𝑥( ·𝑠OLD
‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧))}) |
38 | | ovex 7203 |
. . . 4
⊢ (𝑌 ↑m 𝑋) ∈ V |
39 | 38 | rabex 5200 |
. . 3
⊢ {𝑡 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡‘𝑦))𝐻(𝑡‘𝑧))} ∈ V |
40 | 19, 36, 37, 39 | ovmpo 7325 |
. 2
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 LnOp 𝑊) = {𝑡 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡‘𝑦))𝐻(𝑡‘𝑧))}) |
41 | 1, 40 | syl5eq 2785 |
1
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐿 = {𝑡 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡‘𝑦))𝐻(𝑡‘𝑧))}) |