ehl1eudis - Metamath Proof Explorer

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Theorem ehl1eudis 25342

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 12513	. . 3 ⊢ 1 ∈ ℕ₀
2		1z 12617	. . . . . 6 ⊢ 1 ∈ ℤ
3		fzsn 13570	. . . . . 6 ⊢ (1 ∈ ℤ → (1...1) = {1})
4	2, 3	ax-mp 5	. . . . 5 ⊢ (1...1) = {1}
5	4	eqcomi 2737	. . . 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 25340	. . 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 2821	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓 ∈ (ℝ ↑_m {1}))
12		reex 11224	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
13		snex 5428	. . . . . . . . . . . . 13 ⊢ {1} ∈ V
14	12, 13	elmap 8884	. . . . . . . . . . . 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 11235	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
18	17	snid 4661	. . . . . . . . . . . . 13 ⊢ 1 ∈ {1}
19	18	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑓:{1}⟶ℝ → 1 ∈ {1})
20	16, 19	ffvelcdmd 7090	. . . . . . . . . . 11 ⊢ (𝑓:{1}⟶ℝ → (𝑓‘1) ∈ ℝ)
21	15, 20	sylbi 216	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝑋 → (𝑓‘1) ∈ ℝ)
22	21	adantr 480	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑓‘1) ∈ ℝ)
23	7	eleq2i 2821	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ 𝑋 ↔ 𝑔 ∈ (ℝ ↑_m {1}))
24	12, 13	elmap 8884	. . . . . . . . . . . 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 7090	. . . . . . . . . . 11 ⊢ (𝑔:{1}⟶ℝ → (𝑔‘1) ∈ ℝ)
29	25, 28	sylbi 216	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑋 → (𝑔‘1) ∈ ℝ)
30	29	adantl 481	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑔‘1) ∈ ℝ)
31	22, 30	resubcld 11667	. . . . . . . 8 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((𝑓‘1) − (𝑔‘1)) ∈ ℝ)
32	31	resqcld 14116	. . . . . . 7 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℝ)
33	32	recnd 11267	. . . . . 6 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℂ)
34		fveq2 6892	. . . . . . . . 9 ⊢ (𝑘 = 1 → (𝑓‘𝑘) = (𝑓‘1))
35		fveq2 6892	. . . . . . . . 9 ⊢ (𝑘 = 1 → (𝑔‘𝑘) = (𝑔‘1))
36	34, 35	oveq12d 7433	. . . . . . . 8 ⊢ (𝑘 = 1 → ((𝑓‘𝑘) − (𝑔‘𝑘)) = ((𝑓‘1) − (𝑔‘1)))
37	36	oveq1d 7430	. . . . . . 7 ⊢ (𝑘 = 1 → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘1) − (𝑔‘1))↑2))
38	37	sumsn 15719	. . . . . 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 586	. . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘1) − (𝑔‘1))↑2))
40	39	fveq2d 6896	. . . 4 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (√‘Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)) = (√‘(((𝑓‘1) − (𝑔‘1))↑2)))
41	31	absred 15390	. . . 4 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (abs‘((𝑓‘1) − (𝑔‘1))) = (√‘(((𝑓‘1) − (𝑔‘1))↑2)))
42	40, 41	eqtr4d 2771	. . 3 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (√‘Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)) = (abs‘((𝑓‘1) − (𝑔‘1))))
43	42	mpoeq3ia 7493	. 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 2756	1 ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (abs‘((𝑓‘1) − (𝑔‘1))))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1534 ∈ wcel 2099 {csn 4625 ⟶wf 6539 ‘cfv 6543 (class class class)co 7415 ∈ cmpo 7417 ↑_m cmap 8839 ℂcc 11131 ℝcr 11132 1c1 11134 − cmin 11469 2c2 12292 ℕ₀cn0 12497 ℤcz 12583 ...cfz 13511 ↑cexp 14053 √csqrt 15207 abscabs 15208 Σcsu 15659 distcds 17236 𝔼_hilcehl 25306

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 ax-addf 11212 ax-mulf 11213

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7680 df-om 7866 df-1st 7988 df-2nd 7989 df-supp 8161 df-tpos 8226 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-er 8719 df-map 8841 df-ixp 8911 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-fsupp 9381 df-sup 9460 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-rp 13002 df-fz 13512 df-fzo 13655 df-seq 13994 df-exp 14054 df-hash 14317 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-clim 15459 df-sum 15660 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-0g 17417 df-gsum 17418 df-prds 17423 df-pws 17425 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-mhm 18734 df-grp 18887 df-minusg 18888 df-sbg 18889 df-subg 19072 df-ghm 19162 df-cntz 19262 df-cmn 19731 df-abl 19732 df-mgp 20069 df-rng 20087 df-ur 20116 df-ring 20169 df-cring 20170 df-oppr 20267 df-dvdsr 20290 df-unit 20291 df-invr 20321 df-dvr 20334 df-rhm 20405 df-subrng 20477 df-subrg 20502 df-drng 20620 df-field 20621 df-staf 20719 df-srng 20720 df-lmod 20739 df-lss 20810 df-sra 21052 df-rgmod 21053 df-cnfld 21274 df-refld 21531 df-dsmm 21660 df-frlm 21675 df-nm 24485 df-tng 24487 df-tcph 25091 df-rrx 25307 df-ehl 25308

This theorem is referenced by: ehl1eudisval 25343

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