ehl2eudis - Metamath Proof Explorer

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Theorem ehl2eudis 25743

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 12623	. . 3 ⊢ 2 ∈ ℕ₀
2		fz12pr 13715	. . . . 5 ⊢ (1...2) = {1, 2}
3	2	eqcomi 2770	. . . 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 25739	. . 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 6885	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑓‘𝑘) = (𝑓‘1))
10		fveq2 6885	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑔‘𝑘) = (𝑔‘1))
11	9, 10	oveq12d 7438	. . . . . 6 ⊢ (𝑘 = 1 → ((𝑓‘𝑘) − (𝑔‘𝑘)) = ((𝑓‘1) − (𝑔‘1)))
12	11	oveq1d 7435	. . . . 5 ⊢ (𝑘 = 1 → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘1) − (𝑔‘1))↑2))
13		fveq2 6885	. . . . . . 7 ⊢ (𝑘 = 2 → (𝑓‘𝑘) = (𝑓‘2))
14		fveq2 6885	. . . . . . 7 ⊢ (𝑘 = 2 → (𝑔‘𝑘) = (𝑔‘2))
15	13, 14	oveq12d 7438	. . . . . 6 ⊢ (𝑘 = 2 → ((𝑓‘𝑘) − (𝑔‘𝑘)) = ((𝑓‘2) − (𝑔‘2)))
16	15	oveq1d 7435	. . . . 5 ⊢ (𝑘 = 2 → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘2) − (𝑔‘2))↑2))
17	5	eleq2i 2853	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓 ∈ (ℝ ↑_m {1, 2}))
18		reex 11291	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
19		prex 5396	. . . . . . . . . . . . 13 ⊢ {1, 2} ∈ V
20	18, 19	elmap 8899	. . . . . . . . . . . 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 11303	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
24	23	prid1 4723	. . . . . . . . . . . . 13 ⊢ 1 ∈ {1, 2}
25	24	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑓:{1, 2}⟶ℝ → 1 ∈ {1, 2})
26	22, 25	ffvelcdmd 7085	. . . . . . . . . . 11 ⊢ (𝑓:{1, 2}⟶ℝ → (𝑓‘1) ∈ ℝ)
27	21, 26	sylbi 220	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝑋 → (𝑓‘1) ∈ ℝ)
28	27	adantr 486	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑓‘1) ∈ ℝ)
29	5	eleq2i 2853	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ 𝑋 ↔ 𝑔 ∈ (ℝ ↑_m {1, 2}))
30	18, 19	elmap 8899	. . . . . . . . . . . 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 7085	. . . . . . . . . . 11 ⊢ (𝑔:{1, 2}⟶ℝ → (𝑔‘1) ∈ ℝ)
35	31, 34	sylbi 220	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑋 → (𝑔‘1) ∈ ℝ)
36	35	adantl 487	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑔‘1) ∈ ℝ)
37	28, 36	resubcld 11744	. . . . . . . 8 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((𝑓‘1) − (𝑔‘1)) ∈ ℝ)
38	37	resqcld 14268	. . . . . . 7 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℝ)
39	38	recnd 11337	. . . . . 6 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℂ)
40		2ex 12420	. . . . . . . . . . . . . 14 ⊢ 2 ∈ V
41	40	prid2 4724	. . . . . . . . . . . . 13 ⊢ 2 ∈ {1, 2}
42	41	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑓:{1, 2}⟶ℝ → 2 ∈ {1, 2})
43	22, 42	ffvelcdmd 7085	. . . . . . . . . . 11 ⊢ (𝑓:{1, 2}⟶ℝ → (𝑓‘2) ∈ ℝ)
44	21, 43	sylbi 220	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝑋 → (𝑓‘2) ∈ ℝ)
45	44	adantr 486	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑓‘2) ∈ ℝ)
46	41	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑔:{1, 2}⟶ℝ → 2 ∈ {1, 2})
47	32, 46	ffvelcdmd 7085	. . . . . . . . . . 11 ⊢ (𝑔:{1, 2}⟶ℝ → (𝑔‘2) ∈ ℝ)
48	31, 47	sylbi 220	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑋 → (𝑔‘2) ∈ ℝ)
49	48	adantl 487	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑔‘2) ∈ ℝ)
50	45, 49	resubcld 11744	. . . . . . . 8 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((𝑓‘2) − (𝑔‘2)) ∈ ℝ)
51	50	resqcld 14268	. . . . . . 7 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘2) − (𝑔‘2))↑2) ∈ ℝ)
52	51	recnd 11337	. . . . . 6 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘2) − (𝑔‘2))↑2) ∈ ℂ)
53	39, 52	jca 521	. . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((((𝑓‘1) − (𝑔‘1))↑2) ∈ ℂ ∧ (((𝑓‘2) − (𝑔‘2))↑2) ∈ ℂ))
54	23, 40	pm3.2i 476	. . . . . 6 ⊢ (1 ∈ V ∧ 2 ∈ V)
55	54	a1i 11	. . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (1 ∈ V ∧ 2 ∈ V))
56		1ne2 12553	. . . . . 6 ⊢ 1 ≠ 2
57	56	a1i 11	. . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → 1 ≠ 2)
58	12, 16, 53, 55, 57	sumpr 15914	. . . 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 6889	. . 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 7498	. 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 2784	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 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 Vcvv 3451 {cpr 4586 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ∈ cmpo 7422 ↑_m cmap 8847 ℂcc 11198 ℝcr 11199 1c1 11201 + caddc 11203 − cmin 11541 2c2 12397 ℕ₀cn0 12606 ...cfz 13639 ↑cexp 14204 √csqrt 15400 Σcsu 15853 distcds 17437 𝔼_hilcehl 25705

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279 ax-mulf 11280

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 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-tpos 8243 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-er 8717 df-map 8849 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-sup 9434 df-oi 9504 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-rp 13121 df-fz 13640 df-fzo 13789 df-seq 14145 df-exp 14205 df-hash 14475 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-clim 15655 df-sum 15854 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-0g 17612 df-gsum 17613 df-prds 17618 df-pws 17620 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-mhm 18978 df-grp 19147 df-minusg 19148 df-sbg 19149 df-subg 19333 df-ghm 19428 df-cntz 19531 df-cmn 19996 df-abl 19997 df-mgp 20361 df-rng 20375 df-ur 20408 df-ring 20461 df-cring 20462 df-oppr 20567 df-dvdsr 20587 df-unit 20588 df-invr 20618 df-dvr 20631 df-rhm 20702 df-subrng 20798 df-subrg 20822 df-drng 20982 df-field 20983 df-staf 21096 df-srng 21097 df-lmod 21137 df-lss 21207 df-sra 21448 df-rgmod 21449 df-cnfld 21679 df-refld 21911 df-dsmm 22038 df-frlm 22053 df-nm 24901 df-tng 24903 df-tcph 25490 df-rrx 25706 df-ehl 25707

This theorem is used by: ehl2eudisval 25744

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