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Theorem cphpyth 25094

Description: The pythagorean theorem for a subcomplex pre-Hilbert space. The square of the norm of the sum of two orthogonal vectors (i.e., whose inner product is 0) is the sum of the squares of their norms. Problem 2 in [Kreyszig] p. 135. This is Metamath 100 proof #4. (Contributed by NM, 17-Apr-2008.) (Revised by SN, 22-Sep-2024.)

**Hypotheses**
Ref	Expression
cphpyth.v	⊢ 𝑉 = (Base‘𝑊)
cphpyth.h	⊢ , = (·_𝑖‘𝑊)
cphpyth.p	⊢ + = (+_g‘𝑊)
cphpyth.n	⊢ 𝑁 = (norm‘𝑊)
cphpyth.w	⊢ (𝜑 → 𝑊 ∈ ℂPreHil)
cphpyth.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
cphpyth.b	⊢ (𝜑 → 𝐵 ∈ 𝑉)

**Assertion**
Ref	Expression
cphpyth	⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝑁‘(𝐴 + 𝐵))↑2) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))

**Proof of Theorem cphpyth**
Step	Hyp	Ref	Expression
1		cphpyth.h	. . . . 5 ⊢ , = (·_𝑖‘𝑊)
2		cphpyth.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
3		cphpyth.p	. . . . 5 ⊢ + = (+_g‘𝑊)
4		cphpyth.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ ℂPreHil)
5		cphpyth.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		cphpyth.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
7	1, 2, 3, 4, 5, 6, 5, 6	cph2di 25085	. . . 4 ⊢ (𝜑 → ((𝐴 + 𝐵) , (𝐴 + 𝐵)) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴))))
8	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝐴 + 𝐵) , (𝐴 + 𝐵)) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴))))
9		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → (𝐴 , 𝐵) = 0)
10	1, 2	cphorthcom 25079	. . . . . . . 8 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐴 , 𝐵) = 0 ↔ (𝐵 , 𝐴) = 0))
11	4, 5, 6, 10	syl3anc 1368	. . . . . . 7 ⊢ (𝜑 → ((𝐴 , 𝐵) = 0 ↔ (𝐵 , 𝐴) = 0))
12	11	biimpa 476	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → (𝐵 , 𝐴) = 0)
13	9, 12	oveq12d 7422	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝐴 , 𝐵) + (𝐵 , 𝐴)) = (0 + 0))
14		00id 11390	. . . . 5 ⊢ (0 + 0) = 0
15	13, 14	eqtrdi 2782	. . . 4 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝐴 , 𝐵) + (𝐵 , 𝐴)) = 0)
16	15	oveq2d 7420	. . 3 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴))) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + 0))
17	2, 1	cphipcl 25069	. . . . . . 7 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 , 𝐴) ∈ ℂ)
18	4, 5, 5, 17	syl3anc 1368	. . . . . 6 ⊢ (𝜑 → (𝐴 , 𝐴) ∈ ℂ)
19	2, 1	cphipcl 25069	. . . . . . 7 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐵 , 𝐵) ∈ ℂ)
20	4, 6, 6, 19	syl3anc 1368	. . . . . 6 ⊢ (𝜑 → (𝐵 , 𝐵) ∈ ℂ)
21	18, 20	addcld 11234	. . . . 5 ⊢ (𝜑 → ((𝐴 , 𝐴) + (𝐵 , 𝐵)) ∈ ℂ)
22	21	addridd 11415	. . . 4 ⊢ (𝜑 → (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + 0) = ((𝐴 , 𝐴) + (𝐵 , 𝐵)))
23	22	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + 0) = ((𝐴 , 𝐴) + (𝐵 , 𝐵)))
24	8, 16, 23	3eqtrd 2770	. 2 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝐴 + 𝐵) , (𝐴 + 𝐵)) = ((𝐴 , 𝐴) + (𝐵 , 𝐵)))
25		cphngp 25051	. . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp)
26		ngpgrp 24458	. . . . . 6 ⊢ (𝑊 ∈ NrmGrp → 𝑊 ∈ Grp)
27	4, 25, 26	3syl 18	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ Grp)
28	2, 3, 27, 5, 6	grpcld 18874	. . . 4 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ 𝑉)
29		cphpyth.n	. . . . 5 ⊢ 𝑁 = (norm‘𝑊)
30	2, 1, 29	nmsq 25072	. . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 + 𝐵) ∈ 𝑉) → ((𝑁‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) , (𝐴 + 𝐵)))
31	4, 28, 30	syl2anc 583	. . 3 ⊢ (𝜑 → ((𝑁‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) , (𝐴 + 𝐵)))
32	31	adantr 480	. 2 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝑁‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) , (𝐴 + 𝐵)))
33	2, 1, 29	nmsq 25072	. . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → ((𝑁‘𝐴)↑2) = (𝐴 , 𝐴))
34	4, 5, 33	syl2anc 583	. . . 4 ⊢ (𝜑 → ((𝑁‘𝐴)↑2) = (𝐴 , 𝐴))
35	2, 1, 29	nmsq 25072	. . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐵 ∈ 𝑉) → ((𝑁‘𝐵)↑2) = (𝐵 , 𝐵))
36	4, 6, 35	syl2anc 583	. . . 4 ⊢ (𝜑 → ((𝑁‘𝐵)↑2) = (𝐵 , 𝐵))
37	34, 36	oveq12d 7422	. . 3 ⊢ (𝜑 → (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)) = ((𝐴 , 𝐴) + (𝐵 , 𝐵)))
38	37	adantr 480	. 2 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)) = ((𝐴 , 𝐴) + (𝐵 , 𝐵)))
39	24, 32, 38	3eqtr4d 2776	1 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝑁‘(𝐴 + 𝐵))↑2) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ‘cfv 6536 (class class class)co 7404 ℂcc 11107 0cc0 11109 + caddc 11112 2c2 12268 ↑cexp 14029 Basecbs 17150 +_gcplusg 17203 ·_𝑖cip 17208 Grpcgrp 18860 normcnm 24435 NrmGrpcngp 24436 ℂPreHilccph 25044

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8209 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-rp 12978 df-fz 13488 df-seq 13970 df-exp 14030 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 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 18570 df-sgrp 18649 df-mnd 18665 df-mhm 18710 df-grp 18863 df-minusg 18864 df-subg 19047 df-ghm 19136 df-cmn 19699 df-abl 19700 df-mgp 20037 df-rng 20055 df-ur 20084 df-ring 20137 df-cring 20138 df-oppr 20233 df-dvdsr 20256 df-unit 20257 df-rhm 20371 df-subrg 20468 df-drng 20586 df-staf 20685 df-srng 20686 df-lmod 20705 df-lmhm 20867 df-lvec 20948 df-sra 21018 df-rgmod 21019 df-cnfld 21236 df-phl 21514 df-ngp 24442 df-nlm 24445 df-clm 24940 df-cph 25046

This theorem is referenced by: (None)

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