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Theorem nmpar 24990

Description: A subcomplex pre-Hilbert space satisfies the parallelogram law. (Contributed by Mario Carneiro, 7-Oct-2015.)

**Hypotheses**
Ref	Expression
nmpar.v	⊢ 𝑉 = (Base‘𝑊)
nmpar.p	⊢ + = (+_g‘𝑊)
nmpar.m	⊢ − = (-_g‘𝑊)
nmpar.n	⊢ 𝑁 = (norm‘𝑊)

**Assertion**
Ref	Expression
nmpar	⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (((𝑁‘(𝐴 + 𝐵))↑2) + ((𝑁‘(𝐴 − 𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))

**Proof of Theorem nmpar**
Step	Hyp	Ref	Expression
1		nmpar.v	. 2 ⊢ 𝑉 = (Base‘𝑊)
2		nmpar.p	. 2 ⊢ + = (+_g‘𝑊)
3		nmpar.m	. 2 ⊢ − = (-_g‘𝑊)
4		nmpar.n	. 2 ⊢ 𝑁 = (norm‘𝑊)
5		eqid 2730	. 2 ⊢ (·_𝑖‘𝑊) = (·_𝑖‘𝑊)
6		eqid 2730	. 2 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
7		eqid 2730	. 2 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
8		simp1 1134	. 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝑊 ∈ ℂPreHil)
9		simp2 1135	. 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉)
10		simp3 1136	. 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉)
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	nmparlem 24989	1 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (((𝑁‘(𝐴 + 𝐵))↑2) + ((𝑁‘(𝐴 − 𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ‘cfv 6544 (class class class)co 7413 + caddc 11117 · cmul 11119 2c2 12273 ↑cexp 14033 Basecbs 17150 +_gcplusg 17203 Scalarcsca 17206 ·_𝑖cip 17208 -_gcsg 18859 normcnm 24307 ℂPreHilccph 24916

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 ax-addf 11193 ax-mulf 11194

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 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-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 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 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8215 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9441 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-div 11878 df-nn 12219 df-2 12281 df-3 12282 df-4 12283 df-5 12284 df-6 12285 df-7 12286 df-8 12287 df-9 12288 df-n0 12479 df-z 12565 df-dec 12684 df-uz 12829 df-rp 12981 df-fz 13491 df-seq 13973 df-exp 14034 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-starv 17218 df-sca 17219 df-vsca 17220 df-ip 17221 df-tset 17222 df-ple 17223 df-ds 17225 df-unif 17226 df-0g 17393 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18707 df-grp 18860 df-minusg 18861 df-sbg 18862 df-subg 19041 df-ghm 19130 df-cmn 19693 df-abl 19694 df-mgp 20031 df-rng 20049 df-ur 20078 df-ring 20131 df-cring 20132 df-oppr 20227 df-dvdsr 20250 df-unit 20251 df-rhm 20365 df-subrg 20461 df-drng 20504 df-staf 20598 df-srng 20599 df-lmod 20618 df-lmhm 20779 df-lvec 20860 df-sra 20932 df-rgmod 20933 df-cnfld 21147 df-phl 21400 df-nlm 24317 df-clm 24812 df-cph 24918

This theorem is referenced by: minveclem2 25176

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