rrxdsfival - Metamath Proof Explorer

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Theorem rrxdsfival 25583

Description: The value of the Euclidean distance function in a generalized real Euclidean space of finite dimension. (Contributed by AV, 15-Jan-2023.)

**Hypotheses**
Ref	Expression
rrxdsfival.1	⊢ 𝑋 = (ℝ ↑_m 𝐼)
rrxdsfival.d	⊢ 𝐷 = (dist‘(ℝ^‘𝐼))

**Assertion**
Ref	Expression
rrxdsfival	⊢ ((𝐼 ∈ Fin ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))

Distinct variable groups: 𝑘,𝐼 𝑘,𝐹 𝑘,𝐺 𝑘,𝑋 Allowed substitution hint: 𝐷(𝑘)

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

Step	Hyp	Ref	Expression
1		rrxdsfival.d	. . . . 5 ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))
2		eqid 2762	. . . . . 6 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼)
3		rrxdsfival.1	. . . . . 6 ⊢ 𝑋 = (ℝ ↑_m 𝐼)
4	2, 3	rrxdsfi 25581	. . . . 5 ⊢ (𝐼 ∈ Fin → (dist‘(ℝ^‘𝐼)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2))))
5	1, 4	eqtrid 2809	. . . 4 ⊢ (𝐼 ∈ Fin → 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2))))
6	5	oveqd 7429	. . 3 ⊢ (𝐼 ∈ Fin → (𝐹𝐷𝐺) = (𝐹(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2)))𝐺))
7	6	3ad2ant1 1150	. 2 ⊢ ((𝐼 ∈ Fin ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (𝐹(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2)))𝐺))
8		eqidd 2763	. . 3 ⊢ ((𝐼 ∈ Fin ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2))))
9		fveq1 6880	. . . . . . . 8 ⊢ (𝑥 = 𝐹 → (𝑥‘𝑘) = (𝐹‘𝑘))
10		fveq1 6880	. . . . . . . 8 ⊢ (𝑦 = 𝐺 → (𝑦‘𝑘) = (𝐺‘𝑘))
11	9, 10	oveqan12d 7431	. . . . . . 7 ⊢ ((𝑥 = 𝐹 ∧ 𝑦 = 𝐺) → ((𝑥‘𝑘) − (𝑦‘𝑘)) = ((𝐹‘𝑘) − (𝐺‘𝑘)))
12	11	oveq1d 7427	. . . . . 6 ⊢ ((𝑥 = 𝐹 ∧ 𝑦 = 𝐺) → (((𝑥‘𝑘) − (𝑦‘𝑘))↑2) = (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
13	12	sumeq2sdv 15761	. . . . 5 ⊢ ((𝑥 = 𝐹 ∧ 𝑦 = 𝐺) → Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2) = Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
14	13	fveq2d 6885	. . . 4 ⊢ ((𝑥 = 𝐹 ∧ 𝑦 = 𝐺) → (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2)) = (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
15	14	adantl 486	. . 3 ⊢ (((𝐼 ∈ Fin ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑥 = 𝐹 ∧ 𝑦 = 𝐺)) → (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2)) = (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
16		simp2 1154	. . 3 ⊢ ((𝐼 ∈ Fin ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐹 ∈ 𝑋)
17		simp3 1155	. . 3 ⊢ ((𝐼 ∈ Fin ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐺 ∈ 𝑋)
18		fvexd 6896	. . 3 ⊢ ((𝐼 ∈ Fin ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2)) ∈ V)
19	8, 15, 16, 17, 18	ovmpod 7564	. 2 ⊢ ((𝐼 ∈ Fin ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2)))𝐺) = (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
20	7, 19	eqtrd 2797	1 ⊢ ((𝐼 ∈ Fin ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 400 ∧ w3a 1102 = wceq 1569 ∈ wcel 2142 Vcvv 3454 ‘cfv 6536 (class class class)co 7412 ∈ cmpo 7414 ↑_m cmap 8822 Fincfn 8941 ℝcr 11105 − cmin 11447 2c2 12301 ↑cexp 14104 √csqrt 15291 Σcsu 15744 distcds 17325 ℝ^crrx 25553

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-inf2 9608 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 proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-se 5614 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8155 df-tpos 8220 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-er 8692 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9320 df-sup 9400 df-oi 9470 df-card 9932 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12869 df-rp 13023 df-fz 13542 df-fzo 13690 df-seq 14045 df-exp 14105 df-hash 14374 df-cj 15157 df-re 15158 df-im 15159 df-sqrt 15293 df-abs 15294 df-clim 15546 df-sum 15745 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-starv 17331 df-sca 17332 df-vsca 17333 df-ip 17334 df-tset 17335 df-ple 17336 df-ds 17338 df-unif 17339 df-hom 17340 df-cco 17341 df-0g 17500 df-gsum 17501 df-prds 17506 df-pws 17508 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-mhm 18847 df-grp 19009 df-minusg 19010 df-sbg 19011 df-subg 19195 df-ghm 19290 df-cntz 19393 df-cmn 19858 df-abl 19859 df-mgp 20223 df-rng 20237 df-ur 20270 df-ring 20323 df-cring 20324 df-oppr 20426 df-dvdsr 20446 df-unit 20447 df-invr 20477 df-dvr 20490 df-rhm 20561 df-subrng 20656 df-subrg 20680 df-drng 20840 df-field 20841 df-staf 20953 df-srng 20954 df-lmod 20994 df-lss 21064 df-sra 21305 df-rgmod 21306 df-cnfld 21534 df-refld 21766 df-dsmm 21893 df-frlm 21908 df-nm 24750 df-tng 24752 df-tcph 25339 df-rrx 25555

This theorem is used by: ehleudisval 25589

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