Step | Hyp | Ref
| Expression |
1 | | lnoval.7 |
. 2
⊢ 𝐿 = (𝑈 LnOp 𝑊) |
2 | | fveq2 6191 |
. . . . . 6
⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈)) |
3 | | lnoval.1 |
. . . . . 6
⊢ 𝑋 = (BaseSet‘𝑈) |
4 | 2, 3 | syl6eqr 2674 |
. . . . 5
⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋) |
5 | 4 | oveq2d 6666 |
. . . 4
⊢ (𝑢 = 𝑈 → ((BaseSet‘𝑤) ↑𝑚
(BaseSet‘𝑢)) =
((BaseSet‘𝑤)
↑𝑚 𝑋)) |
6 | | fveq2 6191 |
. . . . . . . . . . 11
⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = ( +𝑣
‘𝑈)) |
7 | | lnoval.3 |
. . . . . . . . . . 11
⊢ 𝐺 = ( +𝑣
‘𝑈) |
8 | 6, 7 | syl6eqr 2674 |
. . . . . . . . . 10
⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = 𝐺) |
9 | | fveq2 6191 |
. . . . . . . . . . . 12
⊢ (𝑢 = 𝑈 → (
·𝑠OLD ‘𝑢) = ( ·𝑠OLD
‘𝑈)) |
10 | | lnoval.5 |
. . . . . . . . . . . 12
⊢ 𝑅 = (
·𝑠OLD ‘𝑈) |
11 | 9, 10 | syl6eqr 2674 |
. . . . . . . . . . 11
⊢ (𝑢 = 𝑈 → (
·𝑠OLD ‘𝑢) = 𝑅) |
12 | 11 | oveqd 6667 |
. . . . . . . . . 10
⊢ (𝑢 = 𝑈 → (𝑥( ·𝑠OLD
‘𝑢)𝑦) = (𝑥𝑅𝑦)) |
13 | | eqidd 2623 |
. . . . . . . . . 10
⊢ (𝑢 = 𝑈 → 𝑧 = 𝑧) |
14 | 8, 12, 13 | oveq123d 6671 |
. . . . . . . . 9
⊢ (𝑢 = 𝑈 → ((𝑥( ·𝑠OLD
‘𝑢)𝑦)( +𝑣 ‘𝑢)𝑧) = ((𝑥𝑅𝑦)𝐺𝑧)) |
15 | 14 | fveq2d 6195 |
. . . . . . . 8
⊢ (𝑢 = 𝑈 → (𝑡‘((𝑥( ·𝑠OLD
‘𝑢)𝑦)( +𝑣 ‘𝑢)𝑧)) = (𝑡‘((𝑥𝑅𝑦)𝐺𝑧))) |
16 | 15 | eqeq1d 2624 |
. . . . . . 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)))) |
17 | 4, 16 | raleqbidv 3152 |
. . . . . 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)))) |
18 | 4, 17 | raleqbidv 3152 |
. . . . 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)))) |
19 | 18 | ralbidv 2986 |
. . . 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)))) |
20 | 5, 19 | rabeqbidv 3195 |
. . 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))}) |
21 | | fveq2 6191 |
. . . . . 6
⊢ (𝑤 = 𝑊 → (BaseSet‘𝑤) = (BaseSet‘𝑊)) |
22 | | lnoval.2 |
. . . . . 6
⊢ 𝑌 = (BaseSet‘𝑊) |
23 | 21, 22 | syl6eqr 2674 |
. . . . 5
⊢ (𝑤 = 𝑊 → (BaseSet‘𝑤) = 𝑌) |
24 | 23 | oveq1d 6665 |
. . . 4
⊢ (𝑤 = 𝑊 → ((BaseSet‘𝑤) ↑𝑚 𝑋) = (𝑌 ↑𝑚 𝑋)) |
25 | | fveq2 6191 |
. . . . . . . . 9
⊢ (𝑤 = 𝑊 → ( +𝑣 ‘𝑤) = ( +𝑣
‘𝑊)) |
26 | | lnoval.4 |
. . . . . . . . 9
⊢ 𝐻 = ( +𝑣
‘𝑊) |
27 | 25, 26 | syl6eqr 2674 |
. . . . . . . 8
⊢ (𝑤 = 𝑊 → ( +𝑣 ‘𝑤) = 𝐻) |
28 | | fveq2 6191 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑊 → (
·𝑠OLD ‘𝑤) = ( ·𝑠OLD
‘𝑊)) |
29 | | lnoval.6 |
. . . . . . . . . 10
⊢ 𝑆 = (
·𝑠OLD ‘𝑊) |
30 | 28, 29 | syl6eqr 2674 |
. . . . . . . . 9
⊢ (𝑤 = 𝑊 → (
·𝑠OLD ‘𝑤) = 𝑆) |
31 | 30 | oveqd 6667 |
. . . . . . . 8
⊢ (𝑤 = 𝑊 → (𝑥( ·𝑠OLD
‘𝑤)(𝑡‘𝑦)) = (𝑥𝑆(𝑡‘𝑦))) |
32 | | eqidd 2623 |
. . . . . . . 8
⊢ (𝑤 = 𝑊 → (𝑡‘𝑧) = (𝑡‘𝑧)) |
33 | 27, 31, 32 | oveq123d 6671 |
. . . . . . 7
⊢ (𝑤 = 𝑊 → ((𝑥( ·𝑠OLD
‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧)) = ((𝑥𝑆(𝑡‘𝑦))𝐻(𝑡‘𝑧))) |
34 | 33 | eqeq2d 2632 |
. . . . . 6
⊢ (𝑤 = 𝑊 → ((𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD
‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧)) ↔ (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡‘𝑦))𝐻(𝑡‘𝑧)))) |
35 | 34 | 2ralbidv 2989 |
. . . . 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)))) |
36 | 35 | ralbidv 2986 |
. . . 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)))) |
37 | 24, 36 | rabeqbidv 3195 |
. . 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))}) |
38 | | df-lno 27599 |
. . 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))}) |
39 | | ovex 6678 |
. . . 4
⊢ (𝑌 ↑𝑚
𝑋) ∈
V |
40 | 39 | rabex 4813 |
. . 3
⊢ {𝑡 ∈ (𝑌 ↑𝑚 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡‘𝑦))𝐻(𝑡‘𝑧))} ∈ V |
41 | 20, 37, 38, 40 | ovmpt2 6796 |
. 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))}) |
42 | 1, 41 | syl5eq 2668 |
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))}) |