ehl2eudis - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ehl2eudis		Structured version Visualization version GIF version

Theorem ehl2eudis 25380

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 12420	. . 3 ⊢ 2 ∈ ℕ₀
2		fz12pr 13499	. . . . 5 ⊢ (1...2) = {1, 2}
3	2	eqcomi 2745	. . . 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 25376	. . 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 6834	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑓‘𝑘) = (𝑓‘1))
10		fveq2 6834	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑔‘𝑘) = (𝑔‘1))
11	9, 10	oveq12d 7376	. . . . . 6 ⊢ (𝑘 = 1 → ((𝑓‘𝑘) − (𝑔‘𝑘)) = ((𝑓‘1) − (𝑔‘1)))
12	11	oveq1d 7373	. . . . 5 ⊢ (𝑘 = 1 → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘1) − (𝑔‘1))↑2))
13		fveq2 6834	. . . . . . 7 ⊢ (𝑘 = 2 → (𝑓‘𝑘) = (𝑓‘2))
14		fveq2 6834	. . . . . . 7 ⊢ (𝑘 = 2 → (𝑔‘𝑘) = (𝑔‘2))
15	13, 14	oveq12d 7376	. . . . . 6 ⊢ (𝑘 = 2 → ((𝑓‘𝑘) − (𝑔‘𝑘)) = ((𝑓‘2) − (𝑔‘2)))
16	15	oveq1d 7373	. . . . 5 ⊢ (𝑘 = 2 → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘2) − (𝑔‘2))↑2))
17	5	eleq2i 2828	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓 ∈ (ℝ ↑_m {1, 2}))
18		reex 11119	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
19		prex 5382	. . . . . . . . . . . . 13 ⊢ {1, 2} ∈ V
20	18, 19	elmap 8811	. . . . . . . . . . . 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 11130	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
24	23	prid1 4719	. . . . . . . . . . . . 13 ⊢ 1 ∈ {1, 2}
25	24	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑓:{1, 2}⟶ℝ → 1 ∈ {1, 2})
26	22, 25	ffvelcdmd 7030	. . . . . . . . . . 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 2828	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ 𝑋 ↔ 𝑔 ∈ (ℝ ↑_m {1, 2}))
30	18, 19	elmap 8811	. . . . . . . . . . . 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 7030	. . . . . . . . . . 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 11567	. . . . . . . 8 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((𝑓‘1) − (𝑔‘1)) ∈ ℝ)
38	37	resqcld 14050	. . . . . . 7 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℝ)
39	38	recnd 11162	. . . . . 6 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℂ)
40		2ex 12224	. . . . . . . . . . . . . 14 ⊢ 2 ∈ V
41	40	prid2 4720	. . . . . . . . . . . . 13 ⊢ 2 ∈ {1, 2}
42	41	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑓:{1, 2}⟶ℝ → 2 ∈ {1, 2})
43	22, 42	ffvelcdmd 7030	. . . . . . . . . . 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 7030	. . . . . . . . . . 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 11567	. . . . . . . 8 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((𝑓‘2) − (𝑔‘2)) ∈ ℝ)
51	50	resqcld 14050	. . . . . . 7 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘2) − (𝑔‘2))↑2) ∈ ℝ)
52	51	recnd 11162	. . . . . 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 12350	. . . . . 6 ⊢ 1 ≠ 2
57	56	a1i 11	. . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → 1 ≠ 2)
58	12, 16, 53, 55, 57	sumpr 15673	. . . 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 6838	. . 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 7436	. 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 2759	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 1541 ∈ wcel 2113 ≠ wne 2932 Vcvv 3440 {cpr 4582 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ∈ cmpo 7360 ↑_m cmap 8765 ℂcc 11026 ℝcr 11027 1c1 11029 + caddc 11031 − cmin 11366 2c2 12202 ℕ₀cn0 12403 ...cfz 13425 ↑cexp 13986 √csqrt 15158 Σcsu 15611 distcds 17188 𝔼_hilcehl 25342

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9552 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107 ax-mulf 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-tpos 8168 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8767 df-ixp 8838 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-fsupp 9267 df-sup 9347 df-oi 9417 df-card 9853 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12610 df-uz 12754 df-rp 12908 df-fz 13426 df-fzo 13573 df-seq 13927 df-exp 13987 df-hash 14256 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-clim 15413 df-sum 15612 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-0g 17363 df-gsum 17364 df-prds 17369 df-pws 17371 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-grp 18868 df-minusg 18869 df-sbg 18870 df-subg 19055 df-ghm 19144 df-cntz 19248 df-cmn 19713 df-abl 19714 df-mgp 20078 df-rng 20090 df-ur 20119 df-ring 20172 df-cring 20173 df-oppr 20275 df-dvdsr 20295 df-unit 20296 df-invr 20326 df-dvr 20339 df-rhm 20410 df-subrng 20481 df-subrg 20505 df-drng 20666 df-field 20667 df-staf 20774 df-srng 20775 df-lmod 20815 df-lss 20885 df-sra 21127 df-rgmod 21128 df-cnfld 21312 df-refld 21562 df-dsmm 21689 df-frlm 21704 df-nm 24528 df-tng 24530 df-tcph 25127 df-rrx 25343 df-ehl 25344

This theorem is referenced by: ehl2eudisval 25381

W3C validator