ehl1eudis - Metamath Proof Explorer

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Theorem ehl1eudis 25768

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 {0})
ehl1eudis.d	⊢ 𝐷 = (dist‘𝐸)

**Assertion**
Ref	Expression
ehl1eudis	⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (abs‘((𝑓‘0) − (𝑔‘0))))

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 12646	. . 3 ⊢ 1 ∈ ℕ₀
2		fzo01 13906	. . . . 5 ⊢ (0..^1) = {0}
3	2	eqcomi 2772	. . . 4 ⊢ {0} = (0..^1)
4		ehl1eudis.e	. . . 4 ⊢ 𝐸 = (𝔼_hil‘1)
5		ehl1eudis.x	. . . 4 ⊢ 𝑋 = (ℝ ↑_m {0})
6		ehl1eudis.d	. . . 4 ⊢ 𝐷 = (dist‘𝐸)
7	3, 4, 5, 6	ehleudis 25766	. . 3 ⊢ (1 ∈ ℕ₀ → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ {0} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))
8	1, 7	ax-mp 5	. 2 ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ {0} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))
9		0z 12728	. . . . . 6 ⊢ 0 ∈ ℤ
10	5	eleq2i 2856	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓 ∈ (ℝ ↑_m {0}))
11		reex 11315	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
12		snex 5401	. . . . . . . . . . . . 13 ⊢ {0} ∈ V
13	11, 12	elmap 8924	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (ℝ ↑_m {0}) ↔ 𝑓:{0}⟶ℝ)
14	10, 13	bitri 278	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓:{0}⟶ℝ)
15		id 23	. . . . . . . . . . . 12 ⊢ (𝑓:{0}⟶ℝ → 𝑓:{0}⟶ℝ)
16		c0ex 11324	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
17	16	snid 4623	. . . . . . . . . . . . 13 ⊢ 0 ∈ {0}
18	17	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑓:{0}⟶ℝ → 0 ∈ {0})
19	15, 18	ffvelcdmd 7106	. . . . . . . . . . 11 ⊢ (𝑓:{0}⟶ℝ → (𝑓‘0) ∈ ℝ)
20	14, 19	sylbi 220	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝑋 → (𝑓‘0) ∈ ℝ)
21	20	adantr 486	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑓‘0) ∈ ℝ)
22	5	eleq2i 2856	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ 𝑋 ↔ 𝑔 ∈ (ℝ ↑_m {0}))
23	11, 12	elmap 8924	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (ℝ ↑_m {0}) ↔ 𝑔:{0}⟶ℝ)
24	22, 23	bitri 278	. . . . . . . . . . 11 ⊢ (𝑔 ∈ 𝑋 ↔ 𝑔:{0}⟶ℝ)
25		id 23	. . . . . . . . . . . 12 ⊢ (𝑔:{0}⟶ℝ → 𝑔:{0}⟶ℝ)
26	17	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑔:{0}⟶ℝ → 0 ∈ {0})
27	25, 26	ffvelcdmd 7106	. . . . . . . . . . 11 ⊢ (𝑔:{0}⟶ℝ → (𝑔‘0) ∈ ℝ)
28	24, 27	sylbi 220	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑋 → (𝑔‘0) ∈ ℝ)
29	28	adantl 487	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑔‘0) ∈ ℝ)
30	21, 29	resubcld 11768	. . . . . . . 8 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((𝑓‘0) − (𝑔‘0)) ∈ ℝ)
31	30	resqcld 14292	. . . . . . 7 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘0) − (𝑔‘0))↑2) ∈ ℝ)
32	31	recnd 11361	. . . . . 6 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘0) − (𝑔‘0))↑2) ∈ ℂ)
33		fveq2 6905	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑓‘𝑘) = (𝑓‘0))
34		fveq2 6905	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑔‘𝑘) = (𝑔‘0))
35	33, 34	oveq12d 7458	. . . . . . . 8 ⊢ (𝑘 = 0 → ((𝑓‘𝑘) − (𝑔‘𝑘)) = ((𝑓‘0) − (𝑔‘0)))
36	35	oveq1d 7455	. . . . . . 7 ⊢ (𝑘 = 0 → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘0) − (𝑔‘0))↑2))
37	36	sumsn 15938	. . . . . 6 ⊢ ((0 ∈ ℤ ∧ (((𝑓‘0) − (𝑔‘0))↑2) ∈ ℂ) → Σ𝑘 ∈ {0} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘0) − (𝑔‘0))↑2))
38	9, 32, 37	sylancr 599	. . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → Σ𝑘 ∈ {0} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘0) − (𝑔‘0))↑2))
39	38	fveq2d 6909	. . . 4 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (√‘Σ𝑘 ∈ {0} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)) = (√‘(((𝑓‘0) − (𝑔‘0))↑2)))
40	30	absred 15610	. . . 4 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (abs‘((𝑓‘0) − (𝑔‘0))) = (√‘(((𝑓‘0) − (𝑔‘0))↑2)))
41	39, 40	eqtr4d 2801	. . 3 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (√‘Σ𝑘 ∈ {0} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)) = (abs‘((𝑓‘0) − (𝑔‘0))))
42	41	mpoeq3ia 7518	. 2 ⊢ (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ {0} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (abs‘((𝑓‘0) − (𝑔‘0))))
43	8, 42	eqtri 2786	1 ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (abs‘((𝑓‘0) − (𝑔‘0))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∧ wa 401 = wceq 1570 ∈ wcel 2147 {csn 4588 ⟶wf 6551 ‘cfv 6555 (class class class)co 7440 ∈ cmpo 7442 ↑_m cmap 8872 ℂcc 11222 ℝcr 11223 0cc0 11224 1c1 11225 − cmin 11565 2c2 12421 ℕ₀cn0 12630 ℤcz 12717 ..^cfzo 13812 ↑cexp 14228 √csqrt 15426 abscabs 15427 Σcsu 15879 distcds 17463 𝔼_hilcehl 25732

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-inf2 9666 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301 ax-pre-sup 11302 ax-addf 11303 ax-mulf 11304

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-isom 6564 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-of 7714 df-om 7899 df-1st 8022 df-2nd 8023 df-supp 8202 df-tpos 8268 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-1o 8501 df-er 8742 df-map 8874 df-ixp 8951 df-en 8999 df-dom 9000 df-sdom 9001 df-fin 9002 df-fsupp 9379 df-sup 9458 df-oi 9528 df-card 10044 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-div 11998 df-nn 12360 df-2 12429 df-3 12430 df-4 12431 df-5 12432 df-6 12433 df-7 12434 df-8 12435 df-9 12436 df-n0 12631 df-z 12718 df-dec 12839 df-uz 12990 df-rp 13145 df-fz 13664 df-fzo 13813 df-seq 14169 df-exp 14229 df-hash 14499 df-cj 15292 df-re 15293 df-im 15294 df-sqrt 15428 df-abs 15429 df-clim 15681 df-sum 15880 df-struct 17351 df-sets 17368 df-slot 17386 df-ndx 17398 df-base 17414 df-ress 17435 df-plusg 17467 df-mulr 17468 df-starv 17469 df-sca 17470 df-vsca 17471 df-ip 17472 df-tset 17473 df-ple 17474 df-ds 17476 df-unif 17477 df-hom 17478 df-cco 17479 df-0g 17638 df-gsum 17639 df-prds 17644 df-pws 17646 df-mgm 18842 df-sgrp 18934 df-mnd 18950 df-mhm 19004 df-grp 19173 df-minusg 19174 df-sbg 19175 df-subg 19359 df-ghm 19454 df-cntz 19558 df-cmn 20023 df-abl 20024 df-mgp 20388 df-rng 20402 df-ur 20435 df-ring 20488 df-cring 20489 df-oppr 20594 df-dvdsr 20614 df-unit 20615 df-invr 20645 df-dvr 20658 df-rhm 20729 df-subrng 20825 df-subrg 20849 df-drng 21009 df-field 21010 df-staf 21123 df-srng 21124 df-lmod 21164 df-lss 21234 df-sra 21475 df-rgmod 21476 df-cnfld 21706 df-refld 21938 df-dsmm 22065 df-frlm 22080 df-nm 24928 df-tng 24930 df-tcph 25517 df-rrx 25733 df-ehl 25734

This theorem is used by: ehl1eudisval 25769

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