rrxdsfival - Metamath Proof Explorer

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Theorem rrxdsfival 25642

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 25640	. . . . 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 7433	. . 3 ⊢ (𝐼 ∈ Fin → (𝐹𝐷𝐺) = (𝐹(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2)))𝐺))
7	6	3ad2ant1 1151	. 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 6881	. . . . . . . 8 ⊢ (𝑥 = 𝐹 → (𝑥‘𝑘) = (𝐹‘𝑘))
10		fveq1 6881	. . . . . . . 8 ⊢ (𝑦 = 𝐺 → (𝑦‘𝑘) = (𝐺‘𝑘))
11	9, 10	oveqan12d 7435	. . . . . . 7 ⊢ ((𝑥 = 𝐹 ∧ 𝑦 = 𝐺) → ((𝑥‘𝑘) − (𝑦‘𝑘)) = ((𝐹‘𝑘) − (𝐺‘𝑘)))
12	11	oveq1d 7431	. . . . . 6 ⊢ ((𝑥 = 𝐹 ∧ 𝑦 = 𝐺) → (((𝑥‘𝑘) − (𝑦‘𝑘))↑2) = (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
13	12	sumeq2sdv 15792	. . . . 5 ⊢ ((𝑥 = 𝐹 ∧ 𝑦 = 𝐺) → Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2) = Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
14	13	fveq2d 6886	. . . 4 ⊢ ((𝑥 = 𝐹 ∧ 𝑦 = 𝐺) → (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2)) = (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
15	14	adantl 487	. . 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 1155	. . 3 ⊢ ((𝐼 ∈ Fin ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐹 ∈ 𝑋)
17		simp3 1156	. . 3 ⊢ ((𝐼 ∈ Fin ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐺 ∈ 𝑋)
18		fvexd 6897	. . 3 ⊢ ((𝐼 ∈ Fin ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2)) ∈ V)
19	8, 15, 16, 17, 18	ovmpod 7568	. 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 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 Vcvv 3453 ‘cfv 6537 (class class class)co 7416 ∈ cmpo 7418 ↑_m cmap 8829 Fincfn 8955 ℝcr 11126 − cmin 11468 2c2 12322 ↑cexp 14127 √csqrt 15322 Σcsu 15775 distcds 17355 ℝ^crrx 25612

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-inf2 9623 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206 ax-mulf 11207

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 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 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7681 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8162 df-tpos 8227 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-er 8699 df-map 8831 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-sup 9415 df-oi 9485 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-uz 12891 df-rp 13045 df-fz 13564 df-fzo 13712 df-seq 14068 df-exp 14128 df-hash 14397 df-cj 15188 df-re 15189 df-im 15190 df-sqrt 15324 df-abs 15325 df-clim 15577 df-sum 15776 df-struct 17243 df-sets 17260 df-slot 17278 df-ndx 17290 df-base 17306 df-ress 17327 df-plusg 17359 df-mulr 17360 df-starv 17361 df-sca 17362 df-vsca 17363 df-ip 17364 df-tset 17365 df-ple 17366 df-ds 17368 df-unif 17369 df-hom 17370 df-cco 17371 df-0g 17530 df-gsum 17531 df-prds 17536 df-pws 17538 df-mgm 18734 df-sgrp 18823 df-mnd 18839 df-mhm 18892 df-grp 19061 df-minusg 19062 df-sbg 19063 df-subg 19247 df-ghm 19342 df-cntz 19445 df-cmn 19910 df-abl 19911 df-mgp 20275 df-rng 20289 df-ur 20322 df-ring 20375 df-cring 20376 df-oppr 20479 df-dvdsr 20499 df-unit 20500 df-invr 20530 df-dvr 20543 df-rhm 20614 df-subrng 20709 df-subrg 20733 df-drng 20893 df-field 20894 df-staf 21006 df-srng 21007 df-lmod 21047 df-lss 21117 df-sra 21358 df-rgmod 21359 df-cnfld 21587 df-refld 21819 df-dsmm 21946 df-frlm 21961 df-nm 24809 df-tng 24811 df-tcph 25398 df-rrx 25614

This theorem is used by: ehleudisval 25648

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