ehl2eudis - Metamath Proof Explorer

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Theorem ehl2eudis 25590

Description: The Euclidean distance function in a real Euclidean space of dimension 2. (Contributed by AV, 16-Jan-2023.)

**Hypotheses**
Ref	Expression
ehl2eudis.e	⊢ 𝐸 = (𝔼_hil‘2)
ehl2eudis.x	⊢ 𝑋 = (ℝ ↑_m {1, 2})
ehl2eudis.d	⊢ 𝐷 = (dist‘𝐸)

**Assertion**
Ref	Expression
ehl2eudis	⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘((((𝑓‘1) − (𝑔‘1))↑2) + (((𝑓‘2) − (𝑔‘2))↑2))))

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

Proof of Theorem ehl2eudis Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2nn0 12525	. . 3 ⊢ 2 ∈ ℕ₀
2		fz12pr 13614	. . . . 5 ⊢ (1...2) = {1, 2}
3	2	eqcomi 2772	. . . 4 ⊢ {1, 2} = (1...2)
4		ehl2eudis.e	. . . 4 ⊢ 𝐸 = (𝔼_hil‘2)
5		ehl2eudis.x	. . . 4 ⊢ 𝑋 = (ℝ ↑_m {1, 2})
6		ehl2eudis.d	. . . 4 ⊢ 𝐷 = (dist‘𝐸)
7	3, 4, 5, 6	ehleudis 25586	. . 3 ⊢ (2 ∈ ℕ₀ → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ {1, 2} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))
8	1, 7	ax-mp 5	. 2 ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ {1, 2} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))
9		fveq2 6881	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑓‘𝑘) = (𝑓‘1))
10		fveq2 6881	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑔‘𝑘) = (𝑔‘1))
11	9, 10	oveq12d 7428	. . . . . 6 ⊢ (𝑘 = 1 → ((𝑓‘𝑘) − (𝑔‘𝑘)) = ((𝑓‘1) − (𝑔‘1)))
12	11	oveq1d 7425	. . . . 5 ⊢ (𝑘 = 1 → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘1) − (𝑔‘1))↑2))
13		fveq2 6881	. . . . . . 7 ⊢ (𝑘 = 2 → (𝑓‘𝑘) = (𝑓‘2))
14		fveq2 6881	. . . . . . 7 ⊢ (𝑘 = 2 → (𝑔‘𝑘) = (𝑔‘2))
15	13, 14	oveq12d 7428	. . . . . 6 ⊢ (𝑘 = 2 → ((𝑓‘𝑘) − (𝑔‘𝑘)) = ((𝑓‘2) − (𝑔‘2)))
16	15	oveq1d 7425	. . . . 5 ⊢ (𝑘 = 2 → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘2) − (𝑔‘2))↑2))
17	5	eleq2i 2855	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓 ∈ (ℝ ↑_m {1, 2}))
18		reex 11195	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
19		prex 5409	. . . . . . . . . . . . 13 ⊢ {1, 2} ∈ V
20	18, 19	elmap 8865	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (ℝ ↑_m {1, 2}) ↔ 𝑓:{1, 2}⟶ℝ)
21	17, 20	bitri 278	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓:{1, 2}⟶ℝ)
22		id 23	. . . . . . . . . . . 12 ⊢ (𝑓:{1, 2}⟶ℝ → 𝑓:{1, 2}⟶ℝ)
23		1ex 11207	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
24	23	prid1 4728	. . . . . . . . . . . . 13 ⊢ 1 ∈ {1, 2}
25	24	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑓:{1, 2}⟶ℝ → 1 ∈ {1, 2})
26	22, 25	ffvelcdmd 7080	. . . . . . . . . . 11 ⊢ (𝑓:{1, 2}⟶ℝ → (𝑓‘1) ∈ ℝ)
27	21, 26	sylbi 220	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝑋 → (𝑓‘1) ∈ ℝ)
28	27	adantr 485	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑓‘1) ∈ ℝ)
29	5	eleq2i 2855	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ 𝑋 ↔ 𝑔 ∈ (ℝ ↑_m {1, 2}))
30	18, 19	elmap 8865	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (ℝ ↑_m {1, 2}) ↔ 𝑔:{1, 2}⟶ℝ)
31	29, 30	bitri 278	. . . . . . . . . . 11 ⊢ (𝑔 ∈ 𝑋 ↔ 𝑔:{1, 2}⟶ℝ)
32		id 23	. . . . . . . . . . . 12 ⊢ (𝑔:{1, 2}⟶ℝ → 𝑔:{1, 2}⟶ℝ)
33	24	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑔:{1, 2}⟶ℝ → 1 ∈ {1, 2})
34	32, 33	ffvelcdmd 7080	. . . . . . . . . . 11 ⊢ (𝑔:{1, 2}⟶ℝ → (𝑔‘1) ∈ ℝ)
35	31, 34	sylbi 220	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑋 → (𝑔‘1) ∈ ℝ)
36	35	adantl 486	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑔‘1) ∈ ℝ)
37	28, 36	resubcld 11646	. . . . . . . 8 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((𝑓‘1) − (𝑔‘1)) ∈ ℝ)
38	37	resqcld 14166	. . . . . . 7 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℝ)
39	38	recnd 11241	. . . . . 6 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℂ)
40		2ex 12322	. . . . . . . . . . . . . 14 ⊢ 2 ∈ V
41	40	prid2 4729	. . . . . . . . . . . . 13 ⊢ 2 ∈ {1, 2}
42	41	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑓:{1, 2}⟶ℝ → 2 ∈ {1, 2})
43	22, 42	ffvelcdmd 7080	. . . . . . . . . . 11 ⊢ (𝑓:{1, 2}⟶ℝ → (𝑓‘2) ∈ ℝ)
44	21, 43	sylbi 220	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝑋 → (𝑓‘2) ∈ ℝ)
45	44	adantr 485	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑓‘2) ∈ ℝ)
46	41	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑔:{1, 2}⟶ℝ → 2 ∈ {1, 2})
47	32, 46	ffvelcdmd 7080	. . . . . . . . . . 11 ⊢ (𝑔:{1, 2}⟶ℝ → (𝑔‘2) ∈ ℝ)
48	31, 47	sylbi 220	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑋 → (𝑔‘2) ∈ ℝ)
49	48	adantl 486	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑔‘2) ∈ ℝ)
50	45, 49	resubcld 11646	. . . . . . . 8 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((𝑓‘2) − (𝑔‘2)) ∈ ℝ)
51	50	resqcld 14166	. . . . . . 7 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘2) − (𝑔‘2))↑2) ∈ ℝ)
52	51	recnd 11241	. . . . . 6 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘2) − (𝑔‘2))↑2) ∈ ℂ)
53	39, 52	jca 520	. . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((((𝑓‘1) − (𝑔‘1))↑2) ∈ ℂ ∧ (((𝑓‘2) − (𝑔‘2))↑2) ∈ ℂ))
54	23, 40	pm3.2i 475	. . . . . 6 ⊢ (1 ∈ V ∧ 2 ∈ V)
55	54	a1i 11	. . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (1 ∈ V ∧ 2 ∈ V))
56		1ne2 12455	. . . . . 6 ⊢ 1 ≠ 2
57	56	a1i 11	. . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → 1 ≠ 2)
58	12, 16, 53, 55, 57	sumpr 15804	. . . 4 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → Σ𝑘 ∈ {1, 2} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = ((((𝑓‘1) − (𝑔‘1))↑2) + (((𝑓‘2) − (𝑔‘2))↑2)))
59	58	fveq2d 6885	. . 3 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (√‘Σ𝑘 ∈ {1, 2} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)) = (√‘((((𝑓‘1) − (𝑔‘1))↑2) + (((𝑓‘2) − (𝑔‘2))↑2))))
60	59	mpoeq3ia 7488	. 2 ⊢ (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ {1, 2} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘((((𝑓‘1) − (𝑔‘1))↑2) + (((𝑓‘2) − (𝑔‘2))↑2))))
61	8, 60	eqtri 2786	1 ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘((((𝑓‘1) − (𝑔‘1))↑2) + (((𝑓‘2) − (𝑔‘2))↑2))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∧ wa 400 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 Vcvv 3455 {cpr 4591 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ∈ cmpo 7412 ↑_m cmap 8820 ℂcc 11102 ℝcr 11103 1c1 11105 + caddc 11107 − cmin 11445 2c2 12299 ℕ₀cn0 12508 ...cfz 13539 ↑cexp 14102 √csqrt 15289 Σcsu 15742 distcds 17323 𝔼_hilcehl 25552

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181 ax-pre-sup 11182 ax-addf 11183 ax-mulf 11184

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-tpos 8218 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-sup 9398 df-oi 9468 df-card 9930 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-div 11876 df-nn 12238 df-2 12307 df-3 12308 df-4 12309 df-5 12310 df-6 12311 df-7 12312 df-8 12313 df-9 12314 df-n0 12509 df-z 12596 df-dec 12716 df-uz 12867 df-rp 13021 df-fz 13540 df-fzo 13688 df-seq 14043 df-exp 14103 df-hash 14372 df-cj 15155 df-re 15156 df-im 15157 df-sqrt 15291 df-abs 15292 df-clim 15544 df-sum 15743 df-struct 17211 df-sets 17228 df-slot 17246 df-ndx 17258 df-base 17274 df-ress 17295 df-plusg 17327 df-mulr 17328 df-starv 17329 df-sca 17330 df-vsca 17331 df-ip 17332 df-tset 17333 df-ple 17334 df-ds 17336 df-unif 17337 df-hom 17338 df-cco 17339 df-0g 17498 df-gsum 17499 df-prds 17504 df-pws 17506 df-mgm 18702 df-sgrp 18781 df-mnd 18797 df-mhm 18845 df-grp 19007 df-minusg 19008 df-sbg 19009 df-subg 19193 df-ghm 19288 df-cntz 19391 df-cmn 19856 df-abl 19857 df-mgp 20221 df-rng 20235 df-ur 20268 df-ring 20321 df-cring 20322 df-oppr 20424 df-dvdsr 20444 df-unit 20445 df-invr 20475 df-dvr 20488 df-rhm 20559 df-subrng 20654 df-subrg 20678 df-drng 20838 df-field 20839 df-staf 20951 df-srng 20952 df-lmod 20992 df-lss 21062 df-sra 21303 df-rgmod 21304 df-cnfld 21532 df-refld 21764 df-dsmm 21891 df-frlm 21906 df-nm 24748 df-tng 24750 df-tcph 25337 df-rrx 25553 df-ehl 25554

This theorem is used by: ehl2eudisval 25591

W3C validator