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| Description: The parallelogram law satisfied by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| hlpar.1 | ⊢ 𝑋 = (BaseSet‘𝑈) | 
| hlpar.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) | 
| hlpar.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | 
| hlpar.6 | ⊢ 𝑁 = (normCV‘𝑈) | 
| Ref | Expression | 
|---|---|
| hlpar | ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | hlph 30909 | . 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CPreHilOLD) | |
| 2 | hlpar.1 | . . 3 ⊢ 𝑋 = (BaseSet‘𝑈) | |
| 3 | hlpar.2 | . . 3 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
| 4 | hlpar.4 | . . 3 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
| 5 | hlpar.6 | . . 3 ⊢ 𝑁 = (normCV‘𝑈) | |
| 6 | 2, 3, 4, 5 | phpar 30844 | . 2 ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))) | 
| 7 | 1, 6 | syl3an1 1163 | 1 ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ‘cfv 6560 (class class class)co 7432 1c1 11157 + caddc 11159 · cmul 11161 -cneg 11494 2c2 12322 ↑cexp 14103 +𝑣 cpv 30605 BaseSetcba 30606 ·𝑠OLD cns 30607 normCVcnmcv 30610 CPreHilOLDccphlo 30832 CHilOLDchlo 30905 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-1st 8015 df-2nd 8016 df-vc 30579 df-nv 30612 df-va 30615 df-ba 30616 df-sm 30617 df-0v 30618 df-nmcv 30620 df-ph 30833 df-hlo 30906 | 
| This theorem is referenced by: (None) | 
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