ehl1eudis - Metamath Proof Explorer

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Theorem ehl1eudis 24024

Description: The Euclidean distance function in a real Euclidean space of dimension 1. (Contributed by AV, 16-Jan-2023.)

**Hypotheses**
Ref	Expression
ehl1eudis.e	⊢ 𝐸 = (𝔼_hil‘1)
ehl1eudis.x	⊢ 𝑋 = (ℝ ↑_m {1})
ehl1eudis.d	⊢ 𝐷 = (dist‘𝐸)

**Assertion**
Ref	Expression
ehl1eudis	⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (abs‘((𝑓‘1) − (𝑔‘1))))

Distinct variable group: 𝑓,𝑔 Allowed substitution hints: 𝐷(𝑓,𝑔) 𝐸(𝑓,𝑔) 𝑋(𝑓,𝑔)

Proof of Theorem ehl1eudis Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1nn0 11901	. . 3 ⊢ 1 ∈ ℕ₀
2		1z 12000	. . . . . 6 ⊢ 1 ∈ ℤ
3		fzsn 12944	. . . . . 6 ⊢ (1 ∈ ℤ → (1...1) = {1})
4	2, 3	ax-mp 5	. . . . 5 ⊢ (1...1) = {1}
5	4	eqcomi 2807	. . . 4 ⊢ {1} = (1...1)
6		ehl1eudis.e	. . . 4 ⊢ 𝐸 = (𝔼_hil‘1)
7		ehl1eudis.x	. . . 4 ⊢ 𝑋 = (ℝ ↑_m {1})
8		ehl1eudis.d	. . . 4 ⊢ 𝐷 = (dist‘𝐸)
9	5, 6, 7, 8	ehleudis 24022	. . 3 ⊢ (1 ∈ ℕ₀ → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))
10	1, 9	ax-mp 5	. 2 ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))
11	7	eleq2i 2881	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓 ∈ (ℝ ↑_m {1}))
12		reex 10617	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
13		snex 5297	. . . . . . . . . . . . 13 ⊢ {1} ∈ V
14	12, 13	elmap 8418	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (ℝ ↑_m {1}) ↔ 𝑓:{1}⟶ℝ)
15	11, 14	bitri 278	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓:{1}⟶ℝ)
16		id 22	. . . . . . . . . . . 12 ⊢ (𝑓:{1}⟶ℝ → 𝑓:{1}⟶ℝ)
17		1ex 10626	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
18	17	snid 4561	. . . . . . . . . . . . 13 ⊢ 1 ∈ {1}
19	18	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑓:{1}⟶ℝ → 1 ∈ {1})
20	16, 19	ffvelrnd 6829	. . . . . . . . . . 11 ⊢ (𝑓:{1}⟶ℝ → (𝑓‘1) ∈ ℝ)
21	15, 20	sylbi 220	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝑋 → (𝑓‘1) ∈ ℝ)
22	21	adantr 484	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑓‘1) ∈ ℝ)
23	7	eleq2i 2881	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ 𝑋 ↔ 𝑔 ∈ (ℝ ↑_m {1}))
24	12, 13	elmap 8418	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (ℝ ↑_m {1}) ↔ 𝑔:{1}⟶ℝ)
25	23, 24	bitri 278	. . . . . . . . . . 11 ⊢ (𝑔 ∈ 𝑋 ↔ 𝑔:{1}⟶ℝ)
26		id 22	. . . . . . . . . . . 12 ⊢ (𝑔:{1}⟶ℝ → 𝑔:{1}⟶ℝ)
27	18	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑔:{1}⟶ℝ → 1 ∈ {1})
28	26, 27	ffvelrnd 6829	. . . . . . . . . . 11 ⊢ (𝑔:{1}⟶ℝ → (𝑔‘1) ∈ ℝ)
29	25, 28	sylbi 220	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑋 → (𝑔‘1) ∈ ℝ)
30	29	adantl 485	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑔‘1) ∈ ℝ)
31	22, 30	resubcld 11057	. . . . . . . 8 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((𝑓‘1) − (𝑔‘1)) ∈ ℝ)
32	31	resqcld 13607	. . . . . . 7 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℝ)
33	32	recnd 10658	. . . . . 6 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℂ)
34		fveq2 6645	. . . . . . . . 9 ⊢ (𝑘 = 1 → (𝑓‘𝑘) = (𝑓‘1))
35		fveq2 6645	. . . . . . . . 9 ⊢ (𝑘 = 1 → (𝑔‘𝑘) = (𝑔‘1))
36	34, 35	oveq12d 7153	. . . . . . . 8 ⊢ (𝑘 = 1 → ((𝑓‘𝑘) − (𝑔‘𝑘)) = ((𝑓‘1) − (𝑔‘1)))
37	36	oveq1d 7150	. . . . . . 7 ⊢ (𝑘 = 1 → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘1) − (𝑔‘1))↑2))
38	37	sumsn 15093	. . . . . 6 ⊢ ((1 ∈ ℤ ∧ (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℂ) → Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘1) − (𝑔‘1))↑2))
39	2, 33, 38	sylancr 590	. . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘1) − (𝑔‘1))↑2))
40	39	fveq2d 6649	. . . 4 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (√‘Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)) = (√‘(((𝑓‘1) − (𝑔‘1))↑2)))
41	31	absred 14768	. . . 4 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (abs‘((𝑓‘1) − (𝑔‘1))) = (√‘(((𝑓‘1) − (𝑔‘1))↑2)))
42	40, 41	eqtr4d 2836	. . 3 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (√‘Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)) = (abs‘((𝑓‘1) − (𝑔‘1))))
43	42	mpoeq3ia 7211	. 2 ⊢ (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (abs‘((𝑓‘1) − (𝑔‘1))))
44	10, 43	eqtri 2821	1 ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (abs‘((𝑓‘1) − (𝑔‘1))))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 399 = wceq 1538 ∈ wcel 2111 {csn 4525 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 ↑_m cmap 8389 ℂcc 10524 ℝcr 10525 1c1 10527 − cmin 10859 2c2 11680 ℕ₀cn0 11885 ℤcz 11969 ...cfz 12885 ↑cexp 13425 √csqrt 14584 abscabs 14585 Σcsu 15034 distcds 16566 𝔼_hilcehl 23988

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-0g 16707 df-gsum 16708 df-prds 16713 df-pws 16715 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-ghm 18348 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-rnghom 19463 df-drng 19497 df-field 19498 df-subrg 19526 df-staf 19609 df-srng 19610 df-lmod 19629 df-lss 19697 df-sra 19937 df-rgmod 19938 df-cnfld 20092 df-refld 20294 df-dsmm 20421 df-frlm 20436 df-nm 23189 df-tng 23191 df-tcph 23774 df-rrx 23989 df-ehl 23990

This theorem is referenced by: ehl1eudisval 24025

W3C validator