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Theorem nmparlem 24747

Description: Lemma for nmpar 24748. (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)
nmpar.h	⊢ , = (·_𝑖‘𝑊)
nmpar.f	⊢ 𝐹 = (Scalar‘𝑊)
nmpar.k	⊢ 𝐾 = (Base‘𝐹)
nmpar.1	⊢ (𝜑 → 𝑊 ∈ ℂPreHil)
nmpar.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
nmpar.3	⊢ (𝜑 → 𝐵 ∈ 𝑉)

**Assertion**
Ref	Expression
nmparlem	⊢ (𝜑 → (((𝑁‘(𝐴 + 𝐵))↑2) + ((𝑁‘(𝐴 − 𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))

**Proof of Theorem nmparlem**
Step	Hyp	Ref	Expression
1		nmpar.h	. . . . 5 ⊢ , = (·_𝑖‘𝑊)
2		nmpar.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
3		nmpar.p	. . . . 5 ⊢ + = (+_g‘𝑊)
4		nmpar.1	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ ℂPreHil)
5		nmpar.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		nmpar.3	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
7	1, 2, 3, 4, 5, 6, 5, 6	cph2di 24715	. . . 4 ⊢ (𝜑 → ((𝐴 + 𝐵) , (𝐴 + 𝐵)) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴))))
8		nmpar.m	. . . . 5 ⊢ − = (-_g‘𝑊)
9	1, 2, 8, 4, 5, 6, 5, 6	cph2subdi 24718	. . . 4 ⊢ (𝜑 → ((𝐴 − 𝐵) , (𝐴 − 𝐵)) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) − ((𝐴 , 𝐵) + (𝐵 , 𝐴))))
10	7, 9	oveq12d 7423	. . 3 ⊢ (𝜑 → (((𝐴 + 𝐵) , (𝐴 + 𝐵)) + ((𝐴 − 𝐵) , (𝐴 − 𝐵))) = ((((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴))) + (((𝐴 , 𝐴) + (𝐵 , 𝐵)) − ((𝐴 , 𝐵) + (𝐵 , 𝐴)))))
11		cphclm 24697	. . . . . . 7 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod)
12	4, 11	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℂMod)
13		nmpar.f	. . . . . . 7 ⊢ 𝐹 = (Scalar‘𝑊)
14		nmpar.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝐹)
15	13, 14	clmsscn 24586	. . . . . 6 ⊢ (𝑊 ∈ ℂMod → 𝐾 ⊆ ℂ)
16	12, 15	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾 ⊆ ℂ)
17		cphphl 24679	. . . . . . . 8 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)
18	4, 17	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ PreHil)
19	13, 1, 2, 14	ipcl 21177	. . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 , 𝐴) ∈ 𝐾)
20	18, 5, 5, 19	syl3anc 1371	. . . . . 6 ⊢ (𝜑 → (𝐴 , 𝐴) ∈ 𝐾)
21	13, 1, 2, 14	ipcl 21177	. . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐵 , 𝐵) ∈ 𝐾)
22	18, 6, 6, 21	syl3anc 1371	. . . . . 6 ⊢ (𝜑 → (𝐵 , 𝐵) ∈ 𝐾)
23	13, 14	clmacl 24591	. . . . . 6 ⊢ ((𝑊 ∈ ℂMod ∧ (𝐴 , 𝐴) ∈ 𝐾 ∧ (𝐵 , 𝐵) ∈ 𝐾) → ((𝐴 , 𝐴) + (𝐵 , 𝐵)) ∈ 𝐾)
24	12, 20, 22, 23	syl3anc 1371	. . . . 5 ⊢ (𝜑 → ((𝐴 , 𝐴) + (𝐵 , 𝐵)) ∈ 𝐾)
25	16, 24	sseldd 3982	. . . 4 ⊢ (𝜑 → ((𝐴 , 𝐴) + (𝐵 , 𝐵)) ∈ ℂ)
26	13, 1, 2, 14	ipcl 21177	. . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 , 𝐵) ∈ 𝐾)
27	18, 5, 6, 26	syl3anc 1371	. . . . . 6 ⊢ (𝜑 → (𝐴 , 𝐵) ∈ 𝐾)
28	13, 1, 2, 14	ipcl 21177	. . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐵 , 𝐴) ∈ 𝐾)
29	18, 6, 5, 28	syl3anc 1371	. . . . . 6 ⊢ (𝜑 → (𝐵 , 𝐴) ∈ 𝐾)
30	13, 14	clmacl 24591	. . . . . 6 ⊢ ((𝑊 ∈ ℂMod ∧ (𝐴 , 𝐵) ∈ 𝐾 ∧ (𝐵 , 𝐴) ∈ 𝐾) → ((𝐴 , 𝐵) + (𝐵 , 𝐴)) ∈ 𝐾)
31	12, 27, 29, 30	syl3anc 1371	. . . . 5 ⊢ (𝜑 → ((𝐴 , 𝐵) + (𝐵 , 𝐴)) ∈ 𝐾)
32	16, 31	sseldd 3982	. . . 4 ⊢ (𝜑 → ((𝐴 , 𝐵) + (𝐵 , 𝐴)) ∈ ℂ)
33	25, 32, 25	ppncand 11607	. . 3 ⊢ (𝜑 → ((((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴))) + (((𝐴 , 𝐴) + (𝐵 , 𝐵)) − ((𝐴 , 𝐵) + (𝐵 , 𝐴)))) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐴) + (𝐵 , 𝐵))))
34	10, 33	eqtrd 2772	. 2 ⊢ (𝜑 → (((𝐴 + 𝐵) , (𝐴 + 𝐵)) + ((𝐴 − 𝐵) , (𝐴 − 𝐵))) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐴) + (𝐵 , 𝐵))))
35		cphlmod 24682	. . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod)
36	4, 35	syl 17	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod)
37	2, 3	lmodvacl 20478	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 + 𝐵) ∈ 𝑉)
38	36, 5, 6, 37	syl3anc 1371	. . . 4 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ 𝑉)
39		nmpar.n	. . . . 5 ⊢ 𝑁 = (norm‘𝑊)
40	2, 1, 39	nmsq 24702	. . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 + 𝐵) ∈ 𝑉) → ((𝑁‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) , (𝐴 + 𝐵)))
41	4, 38, 40	syl2anc 584	. . 3 ⊢ (𝜑 → ((𝑁‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) , (𝐴 + 𝐵)))
42	2, 8	lmodvsubcl 20509	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) ∈ 𝑉)
43	36, 5, 6, 42	syl3anc 1371	. . . 4 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ 𝑉)
44	2, 1, 39	nmsq 24702	. . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 − 𝐵) ∈ 𝑉) → ((𝑁‘(𝐴 − 𝐵))↑2) = ((𝐴 − 𝐵) , (𝐴 − 𝐵)))
45	4, 43, 44	syl2anc 584	. . 3 ⊢ (𝜑 → ((𝑁‘(𝐴 − 𝐵))↑2) = ((𝐴 − 𝐵) , (𝐴 − 𝐵)))
46	41, 45	oveq12d 7423	. 2 ⊢ (𝜑 → (((𝑁‘(𝐴 + 𝐵))↑2) + ((𝑁‘(𝐴 − 𝐵))↑2)) = (((𝐴 + 𝐵) , (𝐴 + 𝐵)) + ((𝐴 − 𝐵) , (𝐴 − 𝐵))))
47	2, 1, 39	nmsq 24702	. . . . . 6 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → ((𝑁‘𝐴)↑2) = (𝐴 , 𝐴))
48	4, 5, 47	syl2anc 584	. . . . 5 ⊢ (𝜑 → ((𝑁‘𝐴)↑2) = (𝐴 , 𝐴))
49	2, 1, 39	nmsq 24702	. . . . . 6 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐵 ∈ 𝑉) → ((𝑁‘𝐵)↑2) = (𝐵 , 𝐵))
50	4, 6, 49	syl2anc 584	. . . . 5 ⊢ (𝜑 → ((𝑁‘𝐵)↑2) = (𝐵 , 𝐵))
51	48, 50	oveq12d 7423	. . . 4 ⊢ (𝜑 → (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)) = ((𝐴 , 𝐴) + (𝐵 , 𝐵)))
52	51	oveq2d 7421	. . 3 ⊢ (𝜑 → (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))) = (2 · ((𝐴 , 𝐴) + (𝐵 , 𝐵))))
53	25	2timesd 12451	. . 3 ⊢ (𝜑 → (2 · ((𝐴 , 𝐴) + (𝐵 , 𝐵))) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐴) + (𝐵 , 𝐵))))
54	52, 53	eqtrd 2772	. 2 ⊢ (𝜑 → (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐴) + (𝐵 , 𝐵))))
55	34, 46, 54	3eqtr4d 2782	1 ⊢ (𝜑 → (((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 = wceq 1541 ∈ wcel 2106 ⊆ wss 3947 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 + caddc 11109 · cmul 11111 − cmin 11440 2c2 12263 ↑cexp 14023 Basecbs 17140 +_gcplusg 17193 Scalarcsca 17196 ·_𝑖cip 17198 -_gcsg 18817 LModclmod 20463 PreHilcphl 21168 normcnm 24076 ℂModcclm 24569 ℂPreHilccph 24674

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-rp 12971 df-fz 13481 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-grp 18818 df-minusg 18819 df-sbg 18820 df-subg 18997 df-ghm 19084 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-cring 20052 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-rnghom 20243 df-drng 20309 df-subrg 20353 df-staf 20445 df-srng 20446 df-lmod 20465 df-lmhm 20625 df-lvec 20706 df-sra 20777 df-rgmod 20778 df-cnfld 20937 df-phl 21170 df-nlm 24086 df-clm 24570 df-cph 24676

This theorem is referenced by: nmpar 24748

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