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Mirrors > Home > MPE Home > Th. List > ehl1eudis | Structured version Visualization version GIF version |
Description: The Euclidean distance function in a real Euclidean space of dimension 1. (Contributed by AV, 16-Jan-2023.) |
Ref | Expression |
---|---|
ehl1eudis.e | ⊢ 𝐸 = (𝔼hil‘1) |
ehl1eudis.x | ⊢ 𝑋 = (ℝ ↑m {1}) |
ehl1eudis.d | ⊢ 𝐷 = (dist‘𝐸) |
Ref | Expression |
---|---|
ehl1eudis | ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (abs‘((𝑓‘1) − (𝑔‘1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn0 11907 | . . 3 ⊢ 1 ∈ ℕ0 | |
2 | 1z 12006 | . . . . . 6 ⊢ 1 ∈ ℤ | |
3 | fzsn 12946 | . . . . . 6 ⊢ (1 ∈ ℤ → (1...1) = {1}) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ (1...1) = {1} |
5 | 4 | eqcomi 2829 | . . . 4 ⊢ {1} = (1...1) |
6 | ehl1eudis.e | . . . 4 ⊢ 𝐸 = (𝔼hil‘1) | |
7 | ehl1eudis.x | . . . 4 ⊢ 𝑋 = (ℝ ↑m {1}) | |
8 | ehl1eudis.d | . . . 4 ⊢ 𝐷 = (dist‘𝐸) | |
9 | 5, 6, 7, 8 | ehleudis 24014 | . . 3 ⊢ (1 ∈ ℕ0 → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) |
10 | 1, 9 | ax-mp 5 | . 2 ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) |
11 | 7 | eleq2i 2903 | . . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓 ∈ (ℝ ↑m {1})) |
12 | reex 10621 | . . . . . . . . . . . . 13 ⊢ ℝ ∈ V | |
13 | snex 5325 | . . . . . . . . . . . . 13 ⊢ {1} ∈ V | |
14 | 12, 13 | elmap 8428 | . . . . . . . . . . . 12 ⊢ (𝑓 ∈ (ℝ ↑m {1}) ↔ 𝑓:{1}⟶ℝ) |
15 | 11, 14 | bitri 277 | . . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓:{1}⟶ℝ) |
16 | id 22 | . . . . . . . . . . . 12 ⊢ (𝑓:{1}⟶ℝ → 𝑓:{1}⟶ℝ) | |
17 | 1ex 10630 | . . . . . . . . . . . . . 14 ⊢ 1 ∈ V | |
18 | 17 | snid 4594 | . . . . . . . . . . . . 13 ⊢ 1 ∈ {1} |
19 | 18 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑓:{1}⟶ℝ → 1 ∈ {1}) |
20 | 16, 19 | ffvelrnd 6845 | . . . . . . . . . . 11 ⊢ (𝑓:{1}⟶ℝ → (𝑓‘1) ∈ ℝ) |
21 | 15, 20 | sylbi 219 | . . . . . . . . . 10 ⊢ (𝑓 ∈ 𝑋 → (𝑓‘1) ∈ ℝ) |
22 | 21 | adantr 483 | . . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑓‘1) ∈ ℝ) |
23 | 7 | eleq2i 2903 | . . . . . . . . . . . 12 ⊢ (𝑔 ∈ 𝑋 ↔ 𝑔 ∈ (ℝ ↑m {1})) |
24 | 12, 13 | elmap 8428 | . . . . . . . . . . . 12 ⊢ (𝑔 ∈ (ℝ ↑m {1}) ↔ 𝑔:{1}⟶ℝ) |
25 | 23, 24 | bitri 277 | . . . . . . . . . . 11 ⊢ (𝑔 ∈ 𝑋 ↔ 𝑔:{1}⟶ℝ) |
26 | id 22 | . . . . . . . . . . . 12 ⊢ (𝑔:{1}⟶ℝ → 𝑔:{1}⟶ℝ) | |
27 | 18 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑔:{1}⟶ℝ → 1 ∈ {1}) |
28 | 26, 27 | ffvelrnd 6845 | . . . . . . . . . . 11 ⊢ (𝑔:{1}⟶ℝ → (𝑔‘1) ∈ ℝ) |
29 | 25, 28 | sylbi 219 | . . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑋 → (𝑔‘1) ∈ ℝ) |
30 | 29 | adantl 484 | . . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑔‘1) ∈ ℝ) |
31 | 22, 30 | resubcld 11061 | . . . . . . . 8 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((𝑓‘1) − (𝑔‘1)) ∈ ℝ) |
32 | 31 | resqcld 13608 | . . . . . . 7 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℝ) |
33 | 32 | recnd 10662 | . . . . . 6 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℂ) |
34 | fveq2 6663 | . . . . . . . . 9 ⊢ (𝑘 = 1 → (𝑓‘𝑘) = (𝑓‘1)) | |
35 | fveq2 6663 | . . . . . . . . 9 ⊢ (𝑘 = 1 → (𝑔‘𝑘) = (𝑔‘1)) | |
36 | 34, 35 | oveq12d 7167 | . . . . . . . 8 ⊢ (𝑘 = 1 → ((𝑓‘𝑘) − (𝑔‘𝑘)) = ((𝑓‘1) − (𝑔‘1))) |
37 | 36 | oveq1d 7164 | . . . . . . 7 ⊢ (𝑘 = 1 → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘1) − (𝑔‘1))↑2)) |
38 | 37 | sumsn 15094 | . . . . . 6 ⊢ ((1 ∈ ℤ ∧ (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℂ) → Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘1) − (𝑔‘1))↑2)) |
39 | 2, 33, 38 | sylancr 589 | . . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘1) − (𝑔‘1))↑2)) |
40 | 39 | fveq2d 6667 | . . . 4 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (√‘Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)) = (√‘(((𝑓‘1) − (𝑔‘1))↑2))) |
41 | 31 | absred 14769 | . . . 4 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (abs‘((𝑓‘1) − (𝑔‘1))) = (√‘(((𝑓‘1) − (𝑔‘1))↑2))) |
42 | 40, 41 | eqtr4d 2858 | . . 3 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (√‘Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)) = (abs‘((𝑓‘1) − (𝑔‘1)))) |
43 | 42 | mpoeq3ia 7225 | . 2 ⊢ (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (abs‘((𝑓‘1) − (𝑔‘1)))) |
44 | 10, 43 | eqtri 2843 | 1 ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (abs‘((𝑓‘1) − (𝑔‘1)))) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1536 ∈ wcel 2113 {csn 4560 ⟶wf 6344 ‘cfv 6348 (class class class)co 7149 ∈ cmpo 7151 ↑m cmap 8399 ℂcc 10528 ℝcr 10529 1c1 10531 − cmin 10863 2c2 11686 ℕ0cn0 11891 ℤcz 11975 ...cfz 12889 ↑cexp 13426 √csqrt 14585 abscabs 14586 Σcsu 15035 distcds 16567 𝔼hilcehl 23980 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-inf2 9097 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 ax-addf 10609 ax-mulf 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-of 7402 df-om 7574 df-1st 7682 df-2nd 7683 df-supp 7824 df-tpos 7885 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-ixp 8455 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-fsupp 8827 df-sup 8899 df-oi 8967 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-rp 12384 df-fz 12890 df-fzo 13031 df-seq 13367 df-exp 13427 df-hash 13688 df-cj 14451 df-re 14452 df-im 14453 df-sqrt 14587 df-abs 14588 df-clim 14838 df-sum 15036 df-struct 16478 df-ndx 16479 df-slot 16480 df-base 16482 df-sets 16483 df-ress 16484 df-plusg 16571 df-mulr 16572 df-starv 16573 df-sca 16574 df-vsca 16575 df-ip 16576 df-tset 16577 df-ple 16578 df-ds 16580 df-unif 16581 df-hom 16582 df-cco 16583 df-0g 16708 df-gsum 16709 df-prds 16714 df-pws 16716 df-mgm 17845 df-sgrp 17894 df-mnd 17905 df-mhm 17949 df-grp 18099 df-minusg 18100 df-sbg 18101 df-subg 18269 df-ghm 18349 df-cntz 18440 df-cmn 18901 df-abl 18902 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19366 df-dvdsr 19384 df-unit 19385 df-invr 19415 df-dvr 19426 df-rnghom 19460 df-drng 19497 df-field 19498 df-subrg 19526 df-staf 19609 df-srng 19610 df-lmod 19629 df-lss 19697 df-sra 19937 df-rgmod 19938 df-cnfld 20539 df-refld 20742 df-dsmm 20869 df-frlm 20884 df-nm 23185 df-tng 23187 df-tcph 23766 df-rrx 23981 df-ehl 23982 |
This theorem is referenced by: ehl1eudisval 24017 |
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