ehl1eudis - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ehl1eudis		Structured version Visualization version GIF version

Theorem ehl1eudis 25401

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 12448	. . 3 ⊢ 1 ∈ ℕ₀
2		1z 12552	. . . . . 6 ⊢ 1 ∈ ℤ
3		fzsn 13515	. . . . . 6 ⊢ (1 ∈ ℤ → (1...1) = {1})
4	2, 3	ax-mp 5	. . . . 5 ⊢ (1...1) = {1}
5	4	eqcomi 2746	. . . 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 25399	. . 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 2829	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓 ∈ (ℝ ↑_m {1}))
12		reex 11124	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
13		snex 5378	. . . . . . . . . . . . 13 ⊢ {1} ∈ V
14	12, 13	elmap 8814	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (ℝ ↑_m {1}) ↔ 𝑓:{1}⟶ℝ)
15	11, 14	bitri 275	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓:{1}⟶ℝ)
16		id 22	. . . . . . . . . . . 12 ⊢ (𝑓:{1}⟶ℝ → 𝑓:{1}⟶ℝ)
17		1ex 11135	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
18	17	snid 4607	. . . . . . . . . . . . 13 ⊢ 1 ∈ {1}
19	18	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑓:{1}⟶ℝ → 1 ∈ {1})
20	16, 19	ffvelcdmd 7033	. . . . . . . . . . 11 ⊢ (𝑓:{1}⟶ℝ → (𝑓‘1) ∈ ℝ)
21	15, 20	sylbi 217	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝑋 → (𝑓‘1) ∈ ℝ)
22	21	adantr 480	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑓‘1) ∈ ℝ)
23	7	eleq2i 2829	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ 𝑋 ↔ 𝑔 ∈ (ℝ ↑_m {1}))
24	12, 13	elmap 8814	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (ℝ ↑_m {1}) ↔ 𝑔:{1}⟶ℝ)
25	23, 24	bitri 275	. . . . . . . . . . 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	ffvelcdmd 7033	. . . . . . . . . . 11 ⊢ (𝑔:{1}⟶ℝ → (𝑔‘1) ∈ ℝ)
29	25, 28	sylbi 217	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑋 → (𝑔‘1) ∈ ℝ)
30	29	adantl 481	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑔‘1) ∈ ℝ)
31	22, 30	resubcld 11573	. . . . . . . 8 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((𝑓‘1) − (𝑔‘1)) ∈ ℝ)
32	31	resqcld 14082	. . . . . . 7 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℝ)
33	32	recnd 11168	. . . . . 6 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℂ)
34		fveq2 6836	. . . . . . . . 9 ⊢ (𝑘 = 1 → (𝑓‘𝑘) = (𝑓‘1))
35		fveq2 6836	. . . . . . . . 9 ⊢ (𝑘 = 1 → (𝑔‘𝑘) = (𝑔‘1))
36	34, 35	oveq12d 7380	. . . . . . . 8 ⊢ (𝑘 = 1 → ((𝑓‘𝑘) − (𝑔‘𝑘)) = ((𝑓‘1) − (𝑔‘1)))
37	36	oveq1d 7377	. . . . . . 7 ⊢ (𝑘 = 1 → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘1) − (𝑔‘1))↑2))
38	37	sumsn 15703	. . . . . 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 588	. . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘1) − (𝑔‘1))↑2))
40	39	fveq2d 6840	. . . 4 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (√‘Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)) = (√‘(((𝑓‘1) − (𝑔‘1))↑2)))
41	31	absred 15374	. . . 4 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (abs‘((𝑓‘1) − (𝑔‘1))) = (√‘(((𝑓‘1) − (𝑔‘1))↑2)))
42	40, 41	eqtr4d 2775	. . 3 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (√‘Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)) = (abs‘((𝑓‘1) − (𝑔‘1))))
43	42	mpoeq3ia 7440	. 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 2760	1 ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (abs‘((𝑓‘1) − (𝑔‘1))))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1542 ∈ wcel 2114 {csn 4568 ⟶wf 6490 ‘cfv 6494 (class class class)co 7362 ∈ cmpo 7364 ↑_m cmap 8768 ℂcc 11031 ℝcr 11032 1c1 11034 − cmin 11372 2c2 12231 ℕ₀cn0 12432 ℤcz 12519 ...cfz 13456 ↑cexp 14018 √csqrt 15190 abscabs 15191 Σcsu 15643 distcds 17224 𝔼_hilcehl 25365

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-inf2 9557 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111 ax-addf 11112 ax-mulf 11113

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-se 5580 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7626 df-om 7813 df-1st 7937 df-2nd 7938 df-supp 8106 df-tpos 8171 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-er 8638 df-map 8770 df-ixp 8841 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-fsupp 9270 df-sup 9350 df-oi 9420 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-rp 12938 df-fz 13457 df-fzo 13604 df-seq 13959 df-exp 14019 df-hash 14288 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-clim 15445 df-sum 15644 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-0g 17399 df-gsum 17400 df-prds 17405 df-pws 17407 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18746 df-grp 18907 df-minusg 18908 df-sbg 18909 df-subg 19094 df-ghm 19183 df-cntz 19287 df-cmn 19752 df-abl 19753 df-mgp 20117 df-rng 20129 df-ur 20158 df-ring 20211 df-cring 20212 df-oppr 20312 df-dvdsr 20332 df-unit 20333 df-invr 20363 df-dvr 20376 df-rhm 20447 df-subrng 20518 df-subrg 20542 df-drng 20703 df-field 20704 df-staf 20811 df-srng 20812 df-lmod 20852 df-lss 20922 df-sra 21164 df-rgmod 21165 df-cnfld 21349 df-refld 21599 df-dsmm 21726 df-frlm 21741 df-nm 24561 df-tng 24563 df-tcph 25150 df-rrx 25366 df-ehl 25367

This theorem is referenced by: ehl1eudisval 25402

W3C validator