ehl2eudis - Metamath Proof Explorer

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Theorem ehl2eudis 25758

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 (0..^2))
ehl2eudis.d	⊢ 𝐷 = (dist‘𝐸)

**Assertion**
Ref	Expression
ehl2eudis	⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘((((𝑓‘0) − (𝑔‘0))↑2) + (((𝑓‘1) − (𝑔‘1))↑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 12636	. . 3 ⊢ 2 ∈ ℕ₀
2		fzo0to2pr 13898	. . . . 5 ⊢ (0..^2) = {0, 1}
3	2	eqcomi 2772	. . . 4 ⊢ {0, 1} = (0..^2)
4		ehl2eudis.e	. . . 4 ⊢ 𝐸 = (𝔼_hil‘2)
5		ehl2eudis.x	. . . . 5 ⊢ 𝑋 = (ℝ ↑_m (0..^2))
6	3	oveq2i 7441	. . . . 5 ⊢ (ℝ ↑_m {0, 1}) = (ℝ ↑_m (0..^2))
7	5, 6	eqtr4i 2789	. . . 4 ⊢ 𝑋 = (ℝ ↑_m {0, 1})
8		ehl2eudis.d	. . . 4 ⊢ 𝐷 = (dist‘𝐸)
9	3, 4, 7, 8	ehleudis 25754	. . 3 ⊢ (2 ∈ ℕ₀ → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ {0, 1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))
10	1, 9	ax-mp 5	. 2 ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ {0, 1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))
11		fveq2 6896	. . . . . . 7 ⊢ (𝑘 = 0 → (𝑓‘𝑘) = (𝑓‘0))
12		fveq2 6896	. . . . . . 7 ⊢ (𝑘 = 0 → (𝑔‘𝑘) = (𝑔‘0))
13	11, 12	oveq12d 7448	. . . . . 6 ⊢ (𝑘 = 0 → ((𝑓‘𝑘) − (𝑔‘𝑘)) = ((𝑓‘0) − (𝑔‘0)))
14	13	oveq1d 7445	. . . . 5 ⊢ (𝑘 = 0 → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘0) − (𝑔‘0))↑2))
15		fveq2 6896	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑓‘𝑘) = (𝑓‘1))
16		fveq2 6896	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑔‘𝑘) = (𝑔‘1))
17	15, 16	oveq12d 7448	. . . . . 6 ⊢ (𝑘 = 1 → ((𝑓‘𝑘) − (𝑔‘𝑘)) = ((𝑓‘1) − (𝑔‘1)))
18	17	oveq1d 7445	. . . . 5 ⊢ (𝑘 = 1 → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘1) − (𝑔‘1))↑2))
19	7	eleq2i 2856	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓 ∈ (ℝ ↑_m {0, 1}))
20		reex 11304	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
21		prex 5400	. . . . . . . . . . . . 13 ⊢ {0, 1} ∈ V
22	20, 21	elmap 8913	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (ℝ ↑_m {0, 1}) ↔ 𝑓:{0, 1}⟶ℝ)
23	19, 22	bitri 278	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓:{0, 1}⟶ℝ)
24		id 23	. . . . . . . . . . . 12 ⊢ (𝑓:{0, 1}⟶ℝ → 𝑓:{0, 1}⟶ℝ)
25		0elpr01 11314	. . . . . . . . . . . . 13 ⊢ 0 ∈ {0, 1}
26	25	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑓:{0, 1}⟶ℝ → 0 ∈ {0, 1})
27	24, 26	ffvelcdmd 7096	. . . . . . . . . . 11 ⊢ (𝑓:{0, 1}⟶ℝ → (𝑓‘0) ∈ ℝ)
28	23, 27	sylbi 220	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝑋 → (𝑓‘0) ∈ ℝ)
29	28	adantr 486	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑓‘0) ∈ ℝ)
30	7	eleq2i 2856	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ 𝑋 ↔ 𝑔 ∈ (ℝ ↑_m {0, 1}))
31	20, 21	elmap 8913	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (ℝ ↑_m {0, 1}) ↔ 𝑔:{0, 1}⟶ℝ)
32	30, 31	bitri 278	. . . . . . . . . . 11 ⊢ (𝑔 ∈ 𝑋 ↔ 𝑔:{0, 1}⟶ℝ)
33		id 23	. . . . . . . . . . . 12 ⊢ (𝑔:{0, 1}⟶ℝ → 𝑔:{0, 1}⟶ℝ)
34	25	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑔:{0, 1}⟶ℝ → 0 ∈ {0, 1})
35	33, 34	ffvelcdmd 7096	. . . . . . . . . . 11 ⊢ (𝑔:{0, 1}⟶ℝ → (𝑔‘0) ∈ ℝ)
36	32, 35	sylbi 220	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑋 → (𝑔‘0) ∈ ℝ)
37	36	adantl 487	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑔‘0) ∈ ℝ)
38	29, 37	resubcld 11757	. . . . . . . 8 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((𝑓‘0) − (𝑔‘0)) ∈ ℝ)
39	38	resqcld 14281	. . . . . . 7 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘0) − (𝑔‘0))↑2) ∈ ℝ)
40	39	recnd 11350	. . . . . 6 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘0) − (𝑔‘0))↑2) ∈ ℂ)
41		1elpr01 11317	. . . . . . . . . . . . 13 ⊢ 1 ∈ {0, 1}
42	41	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑓:{0, 1}⟶ℝ → 1 ∈ {0, 1})
43	24, 42	ffvelcdmd 7096	. . . . . . . . . . 11 ⊢ (𝑓:{0, 1}⟶ℝ → (𝑓‘1) ∈ ℝ)
44	23, 43	sylbi 220	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝑋 → (𝑓‘1) ∈ ℝ)
45	44	adantr 486	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑓‘1) ∈ ℝ)
46	41	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑔:{0, 1}⟶ℝ → 1 ∈ {0, 1})
47	33, 46	ffvelcdmd 7096	. . . . . . . . . . 11 ⊢ (𝑔:{0, 1}⟶ℝ → (𝑔‘1) ∈ ℝ)
48	32, 47	sylbi 220	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑋 → (𝑔‘1) ∈ ℝ)
49	48	adantl 487	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑔‘1) ∈ ℝ)
50	45, 49	resubcld 11757	. . . . . . . 8 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((𝑓‘1) − (𝑔‘1)) ∈ ℝ)
51	50	resqcld 14281	. . . . . . 7 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℝ)
52	51	recnd 11350	. . . . . 6 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℂ)
53	40, 52	jca 521	. . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((((𝑓‘0) − (𝑔‘0))↑2) ∈ ℂ ∧ (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℂ))
54		c0ex 11313	. . . . . . 7 ⊢ 0 ∈ V
55		1ex 11316	. . . . . . 7 ⊢ 1 ∈ V
56	54, 55	pm3.2i 476	. . . . . 6 ⊢ (0 ∈ V ∧ 1 ∈ V)
57	56	a1i 11	. . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (0 ∈ V ∧ 1 ∈ V))
58		0ne1 12427	. . . . . 6 ⊢ 0 ≠ 1
59	58	a1i 11	. . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → 0 ≠ 1)
60	14, 18, 53, 57, 59	sumpr 15929	. . . 4 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → Σ𝑘 ∈ {0, 1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = ((((𝑓‘0) − (𝑔‘0))↑2) + (((𝑓‘1) − (𝑔‘1))↑2)))
61	60	fveq2d 6900	. . 3 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (√‘Σ𝑘 ∈ {0, 1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)) = (√‘((((𝑓‘0) − (𝑔‘0))↑2) + (((𝑓‘1) − (𝑔‘1))↑2))))
62	61	mpoeq3ia 7508	. 2 ⊢ (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ {0, 1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘((((𝑓‘0) − (𝑔‘0))↑2) + (((𝑓‘1) − (𝑔‘1))↑2))))
63	10, 62	eqtri 2786	1 ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘((((𝑓‘0) − (𝑔‘0))↑2) + (((𝑓‘1) − (𝑔‘1))↑2))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∧ wa 401 = wceq 1569 ∈ wcel 2146 ≠ wne 2959 Vcvv 3454 {cpr 4590 ⟶wf 6543 ‘cfv 6547 (class class class)co 7430 ∈ cmpo 7432 ↑_m cmap 8861 ℂcc 11211 ℝcr 11212 0cc0 11213 1c1 11214 + caddc 11216 − cmin 11554 2c2 12410 ℕ₀cn0 12619 ..^cfzo 13801 ↑cexp 14217 √csqrt 15415 Σcsu 15868 distcds 17452 𝔼_hilcehl 25720

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7761 ax-inf2 9655 ax-cnex 11269 ax-resscn 11270 ax-1cn 11271 ax-icn 11272 ax-addcl 11273 ax-addrcl 11274 ax-mulcl 11275 ax-mulrcl 11276 ax-mulcom 11277 ax-addass 11278 ax-mulass 11279 ax-distr 11280 ax-i2m1 11281 ax-1ne0 11282 ax-1rid 11283 ax-rnegex 11284 ax-rrecex 11285 ax-cnre 11286 ax-pre-lttri 11287 ax-pre-lttrn 11288 ax-pre-ltadd 11289 ax-pre-mulgt0 11290 ax-pre-sup 11291 ax-addf 11292 ax-mulf 11293

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 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 6311 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6501 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-isom 6556 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7703 df-om 7888 df-1st 8011 df-2nd 8012 df-supp 8191 df-tpos 8257 df-frecs 8313 df-wrecs 8344 df-recs 8393 df-rdg 8432 df-1o 8490 df-2o 8491 df-er 8731 df-map 8863 df-ixp 8940 df-en 8988 df-dom 8989 df-sdom 8990 df-fin 8991 df-fsupp 9368 df-sup 9447 df-oi 9517 df-card 10033 df-pnf 11358 df-mnf 11359 df-xr 11360 df-ltxr 11361 df-le 11362 df-sub 11556 df-neg 11557 df-div 11987 df-nn 12349 df-2 12418 df-3 12419 df-4 12420 df-5 12421 df-6 12422 df-7 12423 df-8 12424 df-9 12425 df-n0 12620 df-z 12707 df-dec 12828 df-uz 12979 df-rp 13134 df-fz 13653 df-fzo 13802 df-seq 14158 df-exp 14218 df-hash 14488 df-cj 15281 df-re 15282 df-im 15283 df-sqrt 15417 df-abs 15418 df-clim 15670 df-sum 15869 df-struct 17340 df-sets 17357 df-slot 17375 df-ndx 17387 df-base 17403 df-ress 17424 df-plusg 17456 df-mulr 17457 df-starv 17458 df-sca 17459 df-vsca 17460 df-ip 17461 df-tset 17462 df-ple 17463 df-ds 17465 df-unif 17466 df-hom 17467 df-cco 17468 df-0g 17627 df-gsum 17628 df-prds 17633 df-pws 17635 df-mgm 18831 df-sgrp 18923 df-mnd 18939 df-mhm 18993 df-grp 19162 df-minusg 19163 df-sbg 19164 df-subg 19348 df-ghm 19443 df-cntz 19546 df-cmn 20011 df-abl 20012 df-mgp 20376 df-rng 20390 df-ur 20423 df-ring 20476 df-cring 20477 df-oppr 20582 df-dvdsr 20602 df-unit 20603 df-invr 20633 df-dvr 20646 df-rhm 20717 df-subrng 20813 df-subrg 20837 df-drng 20997 df-field 20998 df-staf 21111 df-srng 21112 df-lmod 21152 df-lss 21222 df-sra 21463 df-rgmod 21464 df-cnfld 21694 df-refld 21926 df-dsmm 22053 df-frlm 22068 df-nm 24916 df-tng 24918 df-tcph 25505 df-rrx 25721 df-ehl 25722

This theorem is used by: ehl2eudisval 25759

W3C validator