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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 12540 | . . 3 ⊢ 1 ∈ ℕ0 | |
2 | 1z 12645 | . . . . . 6 ⊢ 1 ∈ ℤ | |
3 | fzsn 13603 | . . . . . 6 ⊢ (1 ∈ ℤ → (1...1) = {1}) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ (1...1) = {1} |
5 | 4 | eqcomi 2744 | . . . 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 25466 | . . 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 2831 | . . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓 ∈ (ℝ ↑m {1})) |
12 | reex 11244 | . . . . . . . . . . . . 13 ⊢ ℝ ∈ V | |
13 | snex 5442 | . . . . . . . . . . . . 13 ⊢ {1} ∈ V | |
14 | 12, 13 | elmap 8910 | . . . . . . . . . . . 12 ⊢ (𝑓 ∈ (ℝ ↑m {1}) ↔ 𝑓:{1}⟶ℝ) |
15 | 11, 14 | bitri 275 | . . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝑋 ↔ 𝑓:{1}⟶ℝ) |
16 | id 22 | . . . . . . . . . . . 12 ⊢ (𝑓:{1}⟶ℝ → 𝑓:{1}⟶ℝ) | |
17 | 1ex 11255 | . . . . . . . . . . . . . 14 ⊢ 1 ∈ V | |
18 | 17 | snid 4667 | . . . . . . . . . . . . 13 ⊢ 1 ∈ {1} |
19 | 18 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑓:{1}⟶ℝ → 1 ∈ {1}) |
20 | 16, 19 | ffvelcdmd 7105 | . . . . . . . . . . 11 ⊢ (𝑓:{1}⟶ℝ → (𝑓‘1) ∈ ℝ) |
21 | 15, 20 | sylbi 217 | . . . . . . . . . 10 ⊢ (𝑓 ∈ 𝑋 → (𝑓‘1) ∈ ℝ) |
22 | 21 | adantr 480 | . . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑓‘1) ∈ ℝ) |
23 | 7 | eleq2i 2831 | . . . . . . . . . . . 12 ⊢ (𝑔 ∈ 𝑋 ↔ 𝑔 ∈ (ℝ ↑m {1})) |
24 | 12, 13 | elmap 8910 | . . . . . . . . . . . 12 ⊢ (𝑔 ∈ (ℝ ↑m {1}) ↔ 𝑔:{1}⟶ℝ) |
25 | 23, 24 | bitri 275 | . . . . . . . . . . 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 | ffvelcdmd 7105 | . . . . . . . . . . 11 ⊢ (𝑔:{1}⟶ℝ → (𝑔‘1) ∈ ℝ) |
29 | 25, 28 | sylbi 217 | . . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑋 → (𝑔‘1) ∈ ℝ) |
30 | 29 | adantl 481 | . . . . . . . . 9 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑔‘1) ∈ ℝ) |
31 | 22, 30 | resubcld 11689 | . . . . . . . 8 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((𝑓‘1) − (𝑔‘1)) ∈ ℝ) |
32 | 31 | resqcld 14162 | . . . . . . 7 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℝ) |
33 | 32 | recnd 11287 | . . . . . 6 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (((𝑓‘1) − (𝑔‘1))↑2) ∈ ℂ) |
34 | fveq2 6907 | . . . . . . . . 9 ⊢ (𝑘 = 1 → (𝑓‘𝑘) = (𝑓‘1)) | |
35 | fveq2 6907 | . . . . . . . . 9 ⊢ (𝑘 = 1 → (𝑔‘𝑘) = (𝑔‘1)) | |
36 | 34, 35 | oveq12d 7449 | . . . . . . . 8 ⊢ (𝑘 = 1 → ((𝑓‘𝑘) − (𝑔‘𝑘)) = ((𝑓‘1) − (𝑔‘1))) |
37 | 36 | oveq1d 7446 | . . . . . . 7 ⊢ (𝑘 = 1 → (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘1) − (𝑔‘1))↑2)) |
38 | 37 | sumsn 15779 | . . . . . 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 587 | . . . . 5 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2) = (((𝑓‘1) − (𝑔‘1))↑2)) |
40 | 39 | fveq2d 6911 | . . . 4 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (√‘Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)) = (√‘(((𝑓‘1) − (𝑔‘1))↑2))) |
41 | 31 | absred 15452 | . . . 4 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (abs‘((𝑓‘1) − (𝑔‘1))) = (√‘(((𝑓‘1) − (𝑔‘1))↑2))) |
42 | 40, 41 | eqtr4d 2778 | . . 3 ⊢ ((𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (√‘Σ𝑘 ∈ {1} (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)) = (abs‘((𝑓‘1) − (𝑔‘1)))) |
43 | 42 | mpoeq3ia 7511 | . 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 2763 | 1 ⊢ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (abs‘((𝑓‘1) − (𝑔‘1)))) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2106 {csn 4631 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 ↑m cmap 8865 ℂcc 11151 ℝcr 11152 1c1 11154 − cmin 11490 2c2 12319 ℕ0cn0 12524 ℤcz 12611 ...cfz 13544 ↑cexp 14099 √csqrt 15269 abscabs 15270 Σcsu 15719 distcds 17307 𝔼hilcehl 25432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-drng 20748 df-field 20749 df-staf 20857 df-srng 20858 df-lmod 20877 df-lss 20948 df-sra 21190 df-rgmod 21191 df-cnfld 21383 df-refld 21641 df-dsmm 21770 df-frlm 21785 df-nm 24611 df-tng 24613 df-tcph 25217 df-rrx 25433 df-ehl 25434 |
This theorem is referenced by: ehl1eudisval 25469 |
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