ehl2eudis - Metamath Proof Explorer

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Theorem ehl2eudis 25402

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 12448	. . 3 ⊢ 2 ∈ ℕ₀
2		fz12pr 13529	. . . . 5 ⊢ (1...2) = {1, 2}
3	2	eqcomi 2746	. . . 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 25398	. . 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 6835	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑓‘𝑘) = (𝑓‘1))
10		fveq2 6835	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑔‘𝑘) = (𝑔‘1))
11	9, 10	oveq12d 7379	. . . . . 6 ⊢ (𝑘 = 1 → ((𝑓‘𝑘) − (𝑔‘𝑘)) = ((𝑓‘1) − (𝑔‘1)))
12	11	oveq1d 7376	. . . . 5 ⊢ (𝑘 = 1 → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘1) − (𝑔‘1))↑2))
13		fveq2 6835	. . . . . . 7 ⊢ (𝑘 = 2 → (𝑓‘𝑘) = (𝑓‘2))
14		fveq2 6835	. . . . . . 7 ⊢ (𝑘 = 2 → (𝑔‘𝑘) = (𝑔‘2))
15	13, 14	oveq12d 7379	. . . . . 6 ⊢ (𝑘 = 2 → ((𝑓‘𝑘) − (𝑔‘𝑘)) = ((𝑓‘2) − (𝑔‘2)))
16	15	oveq1d 7376	. . . . 5 ⊢ (𝑘 = 2 → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘2) − (𝑔‘2))↑2))
17	5	eleq2i 2829	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓 ∈ (ℝ ↑_m {1, 2}))
18		reex 11123	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
19		prex 5376	. . . . . . . . . . . . 13 ⊢ {1, 2} ∈ V
20	18, 19	elmap 8813	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (ℝ ↑_m {1, 2}) ↔ 𝑓:{1, 2}⟶ℝ)
21	17, 20	bitri 275	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓:{1, 2}⟶ℝ)
22		id 22	. . . . . . . . . . . 12 ⊢ (𝑓:{1, 2}⟶ℝ → 𝑓:{1, 2}⟶ℝ)
23		1ex 11134	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
24	23	prid1 4707	. . . . . . . . . . . . 13 ⊢ 1 ∈ {1, 2}
25	24	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑓:{1, 2}⟶ℝ → 1 ∈ {1, 2})
26	22, 25	ffvelcdmd 7032	. . . . . . . . . . 11 ⊢ (𝑓:{1, 2}⟶ℝ → (𝑓‘1) ∈ ℝ)
27	21, 26	sylbi 217	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝑋 → (𝑓‘1) ∈ ℝ)
28	27	adantr 480	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑓‘1) ∈ ℝ)
29	5	eleq2i 2829	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ 𝑋 ↔ 𝑔 ∈ (ℝ ↑_m {1, 2}))
30	18, 19	elmap 8813	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (ℝ ↑_m {1, 2}) ↔ 𝑔:{1, 2}⟶ℝ)
31	29, 30	bitri 275	. . . . . . . . . . 11 ⊢ (𝑔 ∈ 𝑋 ↔ 𝑔:{1, 2}⟶ℝ)
32		id 22	. . . . . . . . . . . 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 7032	. . . . . . . . . . 11 ⊢ (𝑔:{1, 2}⟶ℝ → (𝑔‘1) ∈ ℝ)
35	31, 34	sylbi 217	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑋 → (𝑔‘1) ∈ ℝ)
36	35	adantl 481	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑔‘1) ∈ ℝ)
37	28, 36	resubcld 11572	. . . . . . . 8 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((𝑓‘1) − (𝑔‘1)) ∈ ℝ)
38	37	resqcld 14081	. . . . . . 7 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℝ)
39	38	recnd 11167	. . . . . 6 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℂ)
40		2ex 12252	. . . . . . . . . . . . . 14 ⊢ 2 ∈ V
41	40	prid2 4708	. . . . . . . . . . . . 13 ⊢ 2 ∈ {1, 2}
42	41	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑓:{1, 2}⟶ℝ → 2 ∈ {1, 2})
43	22, 42	ffvelcdmd 7032	. . . . . . . . . . 11 ⊢ (𝑓:{1, 2}⟶ℝ → (𝑓‘2) ∈ ℝ)
44	21, 43	sylbi 217	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝑋 → (𝑓‘2) ∈ ℝ)
45	44	adantr 480	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑓‘2) ∈ ℝ)
46	41	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑔:{1, 2}⟶ℝ → 2 ∈ {1, 2})
47	32, 46	ffvelcdmd 7032	. . . . . . . . . . 11 ⊢ (𝑔:{1, 2}⟶ℝ → (𝑔‘2) ∈ ℝ)
48	31, 47	sylbi 217	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑋 → (𝑔‘2) ∈ ℝ)
49	48	adantl 481	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑔‘2) ∈ ℝ)
50	45, 49	resubcld 11572	. . . . . . . 8 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((𝑓‘2) − (𝑔‘2)) ∈ ℝ)
51	50	resqcld 14081	. . . . . . 7 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘2) − (𝑔‘2))↑2) ∈ ℝ)
52	51	recnd 11167	. . . . . 6 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘2) − (𝑔‘2))↑2) ∈ ℂ)
53	39, 52	jca 511	. . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((((𝑓‘1) − (𝑔‘1))↑2) ∈ ℂ ∧ (((𝑓‘2) − (𝑔‘2))↑2) ∈ ℂ))
54	23, 40	pm3.2i 470	. . . . . 6 ⊢ (1 ∈ V ∧ 2 ∈ V)
55	54	a1i 11	. . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (1 ∈ V ∧ 2 ∈ V))
56		1ne2 12378	. . . . . 6 ⊢ 1 ≠ 2
57	56	a1i 11	. . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → 1 ≠ 2)
58	12, 16, 53, 55, 57	sumpr 15704	. . . 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 6839	. . 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 7439	. 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 2760	1 ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘((((𝑓‘1) − (𝑔‘1))↑2) + (((𝑓‘2) − (𝑔‘2))↑2))))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 Vcvv 3430 {cpr 4570 ⟶wf 6489 ‘cfv 6493 (class class class)co 7361 ∈ cmpo 7363 ↑_m cmap 8767 ℂcc 11030 ℝcr 11031 1c1 11033 + caddc 11035 − cmin 11371 2c2 12230 ℕ₀cn0 12431 ...cfz 13455 ↑cexp 14017 √csqrt 15189 Σcsu 15642 distcds 17223 𝔼_hilcehl 25364

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 5303 ax-pr 5371 ax-un 7683 ax-inf2 9556 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-pre-sup 11110 ax-addf 11111 ax-mulf 11112

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 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-om 7812 df-1st 7936 df-2nd 7937 df-supp 8105 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-sup 9349 df-oi 9419 df-card 9857 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-z 12519 df-dec 12639 df-uz 12783 df-rp 12937 df-fz 13456 df-fzo 13603 df-seq 13958 df-exp 14018 df-hash 14287 df-cj 15055 df-re 15056 df-im 15057 df-sqrt 15191 df-abs 15192 df-clim 15444 df-sum 15643 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-starv 17229 df-sca 17230 df-vsca 17231 df-ip 17232 df-tset 17233 df-ple 17234 df-ds 17236 df-unif 17237 df-hom 17238 df-cco 17239 df-0g 17398 df-gsum 17399 df-prds 17404 df-pws 17406 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-mhm 18745 df-grp 18906 df-minusg 18907 df-sbg 18908 df-subg 19093 df-ghm 19182 df-cntz 19286 df-cmn 19751 df-abl 19752 df-mgp 20116 df-rng 20128 df-ur 20157 df-ring 20210 df-cring 20211 df-oppr 20311 df-dvdsr 20331 df-unit 20332 df-invr 20362 df-dvr 20375 df-rhm 20446 df-subrng 20517 df-subrg 20541 df-drng 20702 df-field 20703 df-staf 20810 df-srng 20811 df-lmod 20851 df-lss 20921 df-sra 21163 df-rgmod 21164 df-cnfld 21348 df-refld 21598 df-dsmm 21725 df-frlm 21740 df-nm 24560 df-tng 24562 df-tcph 25149 df-rrx 25365 df-ehl 25366

This theorem is referenced by: ehl2eudisval 25403

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