| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | lnoval.7 | . 2
⊢ 𝐿 = (𝑈 LnOp 𝑊) | 
| 2 |  | fveq2 6906 | . . . . . 6
⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈)) | 
| 3 |  | lnoval.1 | . . . . . 6
⊢ 𝑋 = (BaseSet‘𝑈) | 
| 4 | 2, 3 | eqtr4di 2795 | . . . . 5
⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋) | 
| 5 | 4 | oveq2d 7447 | . . . 4
⊢ (𝑢 = 𝑈 → ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢)) = ((BaseSet‘𝑤) ↑m 𝑋)) | 
| 6 |  | fveq2 6906 | . . . . . . . . . 10
⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = ( +𝑣
‘𝑈)) | 
| 7 |  | lnoval.3 | . . . . . . . . . 10
⊢ 𝐺 = ( +𝑣
‘𝑈) | 
| 8 | 6, 7 | eqtr4di 2795 | . . . . . . . . 9
⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = 𝐺) | 
| 9 |  | fveq2 6906 | . . . . . . . . . . 11
⊢ (𝑢 = 𝑈 → (
·𝑠OLD ‘𝑢) = ( ·𝑠OLD
‘𝑈)) | 
| 10 |  | lnoval.5 | . . . . . . . . . . 11
⊢ 𝑅 = (
·𝑠OLD ‘𝑈) | 
| 11 | 9, 10 | eqtr4di 2795 | . . . . . . . . . 10
⊢ (𝑢 = 𝑈 → (
·𝑠OLD ‘𝑢) = 𝑅) | 
| 12 | 11 | oveqd 7448 | . . . . . . . . 9
⊢ (𝑢 = 𝑈 → (𝑥( ·𝑠OLD
‘𝑢)𝑦) = (𝑥𝑅𝑦)) | 
| 13 |  | eqidd 2738 | . . . . . . . . 9
⊢ (𝑢 = 𝑈 → 𝑧 = 𝑧) | 
| 14 | 8, 12, 13 | oveq123d 7452 | . . . . . . . 8
⊢ (𝑢 = 𝑈 → ((𝑥( ·𝑠OLD
‘𝑢)𝑦)( +𝑣 ‘𝑢)𝑧) = ((𝑥𝑅𝑦)𝐺𝑧)) | 
| 15 | 14 | fveqeq2d 6914 | . . . . . . 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 3346 | . . . . . 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 3346 | . . . . 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 3178 | . . . 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 3455 | . . 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 6906 | . . . . . 6
⊢ (𝑤 = 𝑊 → (BaseSet‘𝑤) = (BaseSet‘𝑊)) | 
| 21 |  | lnoval.2 | . . . . . 6
⊢ 𝑌 = (BaseSet‘𝑊) | 
| 22 | 20, 21 | eqtr4di 2795 | . . . . 5
⊢ (𝑤 = 𝑊 → (BaseSet‘𝑤) = 𝑌) | 
| 23 | 22 | oveq1d 7446 | . . . 4
⊢ (𝑤 = 𝑊 → ((BaseSet‘𝑤) ↑m 𝑋) = (𝑌 ↑m 𝑋)) | 
| 24 |  | fveq2 6906 | . . . . . . . . 9
⊢ (𝑤 = 𝑊 → ( +𝑣 ‘𝑤) = ( +𝑣
‘𝑊)) | 
| 25 |  | lnoval.4 | . . . . . . . . 9
⊢ 𝐻 = ( +𝑣
‘𝑊) | 
| 26 | 24, 25 | eqtr4di 2795 | . . . . . . . 8
⊢ (𝑤 = 𝑊 → ( +𝑣 ‘𝑤) = 𝐻) | 
| 27 |  | fveq2 6906 | . . . . . . . . . 10
⊢ (𝑤 = 𝑊 → (
·𝑠OLD ‘𝑤) = ( ·𝑠OLD
‘𝑊)) | 
| 28 |  | lnoval.6 | . . . . . . . . . 10
⊢ 𝑆 = (
·𝑠OLD ‘𝑊) | 
| 29 | 27, 28 | eqtr4di 2795 | . . . . . . . . 9
⊢ (𝑤 = 𝑊 → (
·𝑠OLD ‘𝑤) = 𝑆) | 
| 30 | 29 | oveqd 7448 | . . . . . . . 8
⊢ (𝑤 = 𝑊 → (𝑥( ·𝑠OLD
‘𝑤)(𝑡‘𝑦)) = (𝑥𝑆(𝑡‘𝑦))) | 
| 31 |  | eqidd 2738 | . . . . . . . 8
⊢ (𝑤 = 𝑊 → (𝑡‘𝑧) = (𝑡‘𝑧)) | 
| 32 | 26, 30, 31 | oveq123d 7452 | . . . . . . 7
⊢ (𝑤 = 𝑊 → ((𝑥( ·𝑠OLD
‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧)) = ((𝑥𝑆(𝑡‘𝑦))𝐻(𝑡‘𝑧))) | 
| 33 | 32 | eqeq2d 2748 | . . . . . 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 3221 | . . . . 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 3178 | . . . 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 3455 | . . 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 30763 | . . 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 7464 | . . . 4
⊢ (𝑌 ↑m 𝑋) ∈ V | 
| 39 | 38 | rabex 5339 | . . 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 7593 | . 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 | eqtrid 2789 | 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))}) |