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Theorem phpar2 30076

Description: The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
isph.1	⊢ 𝑋 = (BaseSet‘𝑈)
isph.2	⊢ 𝐺 = ( +_𝑣 ‘𝑈)
isph.3	⊢ 𝑀 = ( −_𝑣 ‘𝑈)
isph.6	⊢ 𝑁 = (norm_CV‘𝑈)

**Assertion**
Ref	Expression
phpar2	⊢ ((𝑈 ∈ CPreHil_OLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))

Proof of Theorem phpar2 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isph.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
2		isph.2	. . . . 5 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
3		isph.3	. . . . 5 ⊢ 𝑀 = ( −_𝑣 ‘𝑈)
4		isph.6	. . . . 5 ⊢ 𝑁 = (norm_CV‘𝑈)
5	1, 2, 3, 4	isph 30075	. . . 4 ⊢ (𝑈 ∈ CPreHil_OLD ↔ (𝑈 ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))
6	5	simprbi 498	. . 3 ⊢ (𝑈 ∈ CPreHil_OLD → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))
7	6	3ad2ant1 1134	. 2 ⊢ ((𝑈 ∈ CPreHil_OLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))
8		fvoveq1 7432	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺𝑦)) = (𝑁‘(𝐴𝐺𝑦)))
9	8	oveq1d 7424	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺𝑦))↑2) = ((𝑁‘(𝐴𝐺𝑦))↑2))
10		fvoveq1 7432	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑁‘(𝑥𝑀𝑦)) = (𝑁‘(𝐴𝑀𝑦)))
11	10	oveq1d 7424	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑁‘(𝑥𝑀𝑦))↑2) = ((𝑁‘(𝐴𝑀𝑦))↑2))
12	9, 11	oveq12d 7427	. . . . 5 ⊢ (𝑥 = 𝐴 → (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝑀𝑦))↑2)))
13		fveq2 6892	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑁‘𝑥) = (𝑁‘𝐴))
14	13	oveq1d 7424	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑁‘𝑥)↑2) = ((𝑁‘𝐴)↑2))
15	14	oveq1d 7424	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2)))
16	15	oveq2d 7425	. . . . 5 ⊢ (𝑥 = 𝐴 → (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2))))
17	12, 16	eqeq12d 2749	. . . 4 ⊢ (𝑥 = 𝐴 → ((((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))) ↔ (((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝑀𝑦))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2)))))
18		oveq2 7417	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
19	18	fveq2d 6896	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺𝑦)) = (𝑁‘(𝐴𝐺𝐵)))
20	19	oveq1d 7424	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺𝑦))↑2) = ((𝑁‘(𝐴𝐺𝐵))↑2))
21		oveq2 7417	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴𝑀𝑦) = (𝐴𝑀𝐵))
22	21	fveq2d 6896	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑁‘(𝐴𝑀𝑦)) = (𝑁‘(𝐴𝑀𝐵)))
23	22	oveq1d 7424	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑁‘(𝐴𝑀𝑦))↑2) = ((𝑁‘(𝐴𝑀𝐵))↑2))
24	20, 23	oveq12d 7427	. . . . 5 ⊢ (𝑦 = 𝐵 → (((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝑀𝑦))↑2)) = (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)))
25		fveq2 6892	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑁‘𝑦) = (𝑁‘𝐵))
26	25	oveq1d 7424	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝑁‘𝑦)↑2) = ((𝑁‘𝐵)↑2))
27	26	oveq2d 7425	. . . . . 6 ⊢ (𝑦 = 𝐵 → (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2)) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))
28	27	oveq2d 7425	. . . . 5 ⊢ (𝑦 = 𝐵 → (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2))) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))
29	24, 28	eqeq12d 2749	. . . 4 ⊢ (𝑦 = 𝐵 → ((((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝑀𝑦))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2))) ↔ (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))))
30	17, 29	rspc2v 3623	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))))
31	30	3adant1 1131	. 2 ⊢ ((𝑈 ∈ CPreHil_OLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))))
32	7, 31	mpd 15	1 ⊢ ((𝑈 ∈ CPreHil_OLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ‘cfv 6544 (class class class)co 7409 + caddc 11113 · cmul 11115 2c2 12267 ↑cexp 14027 NrmCVeccnv 29837 +_𝑣 cpv 29838 BaseSetcba 29839 −_𝑣 cnsb 29842 norm_CVcnmcv 29843 CPreHil_OLDccphlo 30065

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-ltxr 11253 df-sub 11446 df-neg 11447 df-grpo 29746 df-gid 29747 df-ginv 29748 df-gdiv 29749 df-ablo 29798 df-vc 29812 df-nv 29845 df-va 29848 df-ba 29849 df-sm 29850 df-0v 29851 df-vs 29852 df-nmcv 29853 df-ph 30066

This theorem is referenced by: minvecolem2 30128 hlpar2 30149

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