Proof of Theorem blometi
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | blometi.u | . . . . 5
⊢ 𝑈 ∈ NrmCVec | 
| 2 |  | blometi.1 | . . . . . 6
⊢ 𝑋 = (BaseSet‘𝑈) | 
| 3 |  | eqid 2737 | . . . . . 6
⊢ (
−𝑣 ‘𝑈) = ( −𝑣
‘𝑈) | 
| 4 | 2, 3 | nvmcl 30665 | . . . . 5
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → (𝑃( −𝑣 ‘𝑈)𝑄) ∈ 𝑋) | 
| 5 | 1, 4 | mp3an1 1450 | . . . 4
⊢ ((𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → (𝑃( −𝑣 ‘𝑈)𝑄) ∈ 𝑋) | 
| 6 |  | eqid 2737 | . . . . 5
⊢
(normCV‘𝑈) = (normCV‘𝑈) | 
| 7 |  | eqid 2737 | . . . . 5
⊢
(normCV‘𝑊) = (normCV‘𝑊) | 
| 8 |  | blometi.6 | . . . . 5
⊢ 𝑁 = (𝑈 normOpOLD 𝑊) | 
| 9 |  | blometi.7 | . . . . 5
⊢ 𝐵 = (𝑈 BLnOp 𝑊) | 
| 10 |  | blometi.w | . . . . 5
⊢ 𝑊 ∈ NrmCVec | 
| 11 | 2, 6, 7, 8, 9, 1, 10 | nmblolbi 30819 | . . . 4
⊢ ((𝑇 ∈ 𝐵 ∧ (𝑃( −𝑣 ‘𝑈)𝑄) ∈ 𝑋) → ((normCV‘𝑊)‘(𝑇‘(𝑃( −𝑣 ‘𝑈)𝑄))) ≤ ((𝑁‘𝑇) · ((normCV‘𝑈)‘(𝑃( −𝑣 ‘𝑈)𝑄)))) | 
| 12 | 5, 11 | sylan2 593 | . . 3
⊢ ((𝑇 ∈ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋)) → ((normCV‘𝑊)‘(𝑇‘(𝑃( −𝑣 ‘𝑈)𝑄))) ≤ ((𝑁‘𝑇) · ((normCV‘𝑈)‘(𝑃( −𝑣 ‘𝑈)𝑄)))) | 
| 13 | 12 | 3impb 1115 | . 2
⊢ ((𝑇 ∈ 𝐵 ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → ((normCV‘𝑊)‘(𝑇‘(𝑃( −𝑣 ‘𝑈)𝑄))) ≤ ((𝑁‘𝑇) · ((normCV‘𝑈)‘(𝑃( −𝑣 ‘𝑈)𝑄)))) | 
| 14 |  | blometi.2 | . . . . . . . 8
⊢ 𝑌 = (BaseSet‘𝑊) | 
| 15 | 2, 14, 9 | blof 30804 | . . . . . . 7
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → 𝑇:𝑋⟶𝑌) | 
| 16 | 1, 10, 15 | mp3an12 1453 | . . . . . 6
⊢ (𝑇 ∈ 𝐵 → 𝑇:𝑋⟶𝑌) | 
| 17 | 16 | ffvelcdmda 7104 | . . . . 5
⊢ ((𝑇 ∈ 𝐵 ∧ 𝑃 ∈ 𝑋) → (𝑇‘𝑃) ∈ 𝑌) | 
| 18 | 17 | 3adant3 1133 | . . . 4
⊢ ((𝑇 ∈ 𝐵 ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → (𝑇‘𝑃) ∈ 𝑌) | 
| 19 | 16 | ffvelcdmda 7104 | . . . . 5
⊢ ((𝑇 ∈ 𝐵 ∧ 𝑄 ∈ 𝑋) → (𝑇‘𝑄) ∈ 𝑌) | 
| 20 | 19 | 3adant2 1132 | . . . 4
⊢ ((𝑇 ∈ 𝐵 ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → (𝑇‘𝑄) ∈ 𝑌) | 
| 21 |  | eqid 2737 | . . . . . 6
⊢ (
−𝑣 ‘𝑊) = ( −𝑣
‘𝑊) | 
| 22 |  | blometi.d | . . . . . 6
⊢ 𝐷 = (IndMet‘𝑊) | 
| 23 | 14, 21, 7, 22 | imsdval 30705 | . . . . 5
⊢ ((𝑊 ∈ NrmCVec ∧ (𝑇‘𝑃) ∈ 𝑌 ∧ (𝑇‘𝑄) ∈ 𝑌) → ((𝑇‘𝑃)𝐷(𝑇‘𝑄)) = ((normCV‘𝑊)‘((𝑇‘𝑃)( −𝑣 ‘𝑊)(𝑇‘𝑄)))) | 
| 24 | 10, 23 | mp3an1 1450 | . . . 4
⊢ (((𝑇‘𝑃) ∈ 𝑌 ∧ (𝑇‘𝑄) ∈ 𝑌) → ((𝑇‘𝑃)𝐷(𝑇‘𝑄)) = ((normCV‘𝑊)‘((𝑇‘𝑃)( −𝑣 ‘𝑊)(𝑇‘𝑄)))) | 
| 25 | 18, 20, 24 | syl2anc 584 | . . 3
⊢ ((𝑇 ∈ 𝐵 ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → ((𝑇‘𝑃)𝐷(𝑇‘𝑄)) = ((normCV‘𝑊)‘((𝑇‘𝑃)( −𝑣 ‘𝑊)(𝑇‘𝑄)))) | 
| 26 |  | eqid 2737 | . . . . . . 7
⊢ (𝑈 LnOp 𝑊) = (𝑈 LnOp 𝑊) | 
| 27 | 26, 9 | bloln 30803 | . . . . . 6
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → 𝑇 ∈ (𝑈 LnOp 𝑊)) | 
| 28 | 1, 10, 27 | mp3an12 1453 | . . . . 5
⊢ (𝑇 ∈ 𝐵 → 𝑇 ∈ (𝑈 LnOp 𝑊)) | 
| 29 | 2, 3, 21, 26 | lnosub 30778 | . . . . . . . 8
⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ (𝑈 LnOp 𝑊)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋)) → (𝑇‘(𝑃( −𝑣 ‘𝑈)𝑄)) = ((𝑇‘𝑃)( −𝑣 ‘𝑊)(𝑇‘𝑄))) | 
| 30 | 1, 29 | mp3anl1 1457 | . . . . . . 7
⊢ (((𝑊 ∈ NrmCVec ∧ 𝑇 ∈ (𝑈 LnOp 𝑊)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋)) → (𝑇‘(𝑃( −𝑣 ‘𝑈)𝑄)) = ((𝑇‘𝑃)( −𝑣 ‘𝑊)(𝑇‘𝑄))) | 
| 31 | 10, 30 | mpanl1 700 | . . . . . 6
⊢ ((𝑇 ∈ (𝑈 LnOp 𝑊) ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋)) → (𝑇‘(𝑃( −𝑣 ‘𝑈)𝑄)) = ((𝑇‘𝑃)( −𝑣 ‘𝑊)(𝑇‘𝑄))) | 
| 32 | 31 | 3impb 1115 | . . . . 5
⊢ ((𝑇 ∈ (𝑈 LnOp 𝑊) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → (𝑇‘(𝑃( −𝑣 ‘𝑈)𝑄)) = ((𝑇‘𝑃)( −𝑣 ‘𝑊)(𝑇‘𝑄))) | 
| 33 | 28, 32 | syl3an1 1164 | . . . 4
⊢ ((𝑇 ∈ 𝐵 ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → (𝑇‘(𝑃( −𝑣 ‘𝑈)𝑄)) = ((𝑇‘𝑃)( −𝑣 ‘𝑊)(𝑇‘𝑄))) | 
| 34 | 33 | fveq2d 6910 | . . 3
⊢ ((𝑇 ∈ 𝐵 ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → ((normCV‘𝑊)‘(𝑇‘(𝑃( −𝑣 ‘𝑈)𝑄))) = ((normCV‘𝑊)‘((𝑇‘𝑃)( −𝑣 ‘𝑊)(𝑇‘𝑄)))) | 
| 35 | 25, 34 | eqtr4d 2780 | . 2
⊢ ((𝑇 ∈ 𝐵 ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → ((𝑇‘𝑃)𝐷(𝑇‘𝑄)) = ((normCV‘𝑊)‘(𝑇‘(𝑃( −𝑣 ‘𝑈)𝑄)))) | 
| 36 |  | blometi.8 | . . . . . 6
⊢ 𝐶 = (IndMet‘𝑈) | 
| 37 | 2, 3, 6, 36 | imsdval 30705 | . . . . 5
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → (𝑃𝐶𝑄) = ((normCV‘𝑈)‘(𝑃( −𝑣 ‘𝑈)𝑄))) | 
| 38 | 1, 37 | mp3an1 1450 | . . . 4
⊢ ((𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → (𝑃𝐶𝑄) = ((normCV‘𝑈)‘(𝑃( −𝑣 ‘𝑈)𝑄))) | 
| 39 | 38 | 3adant1 1131 | . . 3
⊢ ((𝑇 ∈ 𝐵 ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → (𝑃𝐶𝑄) = ((normCV‘𝑈)‘(𝑃( −𝑣 ‘𝑈)𝑄))) | 
| 40 | 39 | oveq2d 7447 | . 2
⊢ ((𝑇 ∈ 𝐵 ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → ((𝑁‘𝑇) · (𝑃𝐶𝑄)) = ((𝑁‘𝑇) · ((normCV‘𝑈)‘(𝑃( −𝑣 ‘𝑈)𝑄)))) | 
| 41 | 13, 35, 40 | 3brtr4d 5175 | 1
⊢ ((𝑇 ∈ 𝐵 ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → ((𝑇‘𝑃)𝐷(𝑇‘𝑄)) ≤ ((𝑁‘𝑇) · (𝑃𝐶𝑄))) |