ehl1eudisval - Metamath Proof Explorer

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Theorem ehl1eudisval 25389

Description: The value of the Euclidean distance function in a real Euclidean space of dimension 1. (Contributed by AV, 16-Jan-2023.)

**Hypotheses**
Ref	Expression
ehl1eudis.e	⊢ 𝐸 = (𝔼_hil‘1)
ehl1eudis.x	⊢ 𝑋 = (ℝ ↑_m {1})
ehl1eudis.d	⊢ 𝐷 = (dist‘𝐸)

**Assertion**
Ref	Expression
ehl1eudisval	⊢ ((𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (abs‘((𝐹‘1) − (𝐺‘1))))

Proof of Theorem ehl1eudisval Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 6894	. . 3 ⊢ (𝑥 = 𝐹 → (𝑥‘1) = (𝐹‘1))
2	1	fvoveq1d 7440	. 2 ⊢ (𝑥 = 𝐹 → (abs‘((𝑥‘1) − (𝑦‘1))) = (abs‘((𝐹‘1) − (𝑦‘1))))
3		fveq1 6894	. . . 4 ⊢ (𝑦 = 𝐺 → (𝑦‘1) = (𝐺‘1))
4	3	oveq2d 7434	. . 3 ⊢ (𝑦 = 𝐺 → ((𝐹‘1) − (𝑦‘1)) = ((𝐹‘1) − (𝐺‘1)))
5	4	fveq2d 6899	. 2 ⊢ (𝑦 = 𝐺 → (abs‘((𝐹‘1) − (𝑦‘1))) = (abs‘((𝐹‘1) − (𝐺‘1))))
6		ehl1eudis.e	. . 3 ⊢ 𝐸 = (𝔼_hil‘1)
7		ehl1eudis.x	. . 3 ⊢ 𝑋 = (ℝ ↑_m {1})
8		ehl1eudis.d	. . 3 ⊢ 𝐷 = (dist‘𝐸)
9	6, 7, 8	ehl1eudis 25388	. 2 ⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (abs‘((𝑥‘1) − (𝑦‘1))))
10		fvex 6908	. 2 ⊢ (abs‘((𝐹‘1) − (𝐺‘1))) ∈ V
11	2, 5, 9, 10	ovmpo 7580	1 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (abs‘((𝐹‘1) − (𝐺‘1))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {csn 4630 ‘cfv 6548 (class class class)co 7418 ↑_m cmap 8844 ℝcr 11138 1c1 11140 − cmin 11475 abscabs 15215 distcds 17243 𝔼_hilcehl 25352

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218 ax-mulf 11219

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6306 df-ord 6373 df-on 6374 df-lim 6375 df-suc 6376 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7374 df-ov 7421 df-oprab 7422 df-mpo 7423 df-of 7684 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8165 df-tpos 8231 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-ixp 8916 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-fsupp 9387 df-sup 9466 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-rp 13008 df-fz 13518 df-fzo 13661 df-seq 14001 df-exp 14061 df-hash 14324 df-cj 15080 df-re 15081 df-im 15082 df-sqrt 15216 df-abs 15217 df-clim 15466 df-sum 15667 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17182 df-ress 17211 df-plusg 17247 df-mulr 17248 df-starv 17249 df-sca 17250 df-vsca 17251 df-ip 17252 df-tset 17253 df-ple 17254 df-ds 17256 df-unif 17257 df-hom 17258 df-cco 17259 df-0g 17424 df-gsum 17425 df-prds 17430 df-pws 17432 df-mgm 18601 df-sgrp 18680 df-mnd 18696 df-mhm 18741 df-grp 18899 df-minusg 18900 df-sbg 18901 df-subg 19084 df-ghm 19174 df-cntz 19277 df-cmn 19746 df-abl 19747 df-mgp 20084 df-rng 20102 df-ur 20131 df-ring 20184 df-cring 20185 df-oppr 20282 df-dvdsr 20305 df-unit 20306 df-invr 20336 df-dvr 20349 df-rhm 20420 df-subrng 20492 df-subrg 20517 df-drng 20635 df-field 20636 df-staf 20734 df-srng 20735 df-lmod 20754 df-lss 20825 df-sra 21067 df-rgmod 21068 df-cnfld 21294 df-refld 21551 df-dsmm 21680 df-frlm 21695 df-nm 24531 df-tng 24533 df-tcph 25137 df-rrx 25353 df-ehl 25354

This theorem is referenced by: (None)

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