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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrnprjdstle | Structured version Visualization version GIF version |
Description: The distance between two points in Euclidean space is greater than the distance between the projections onto one coordinate. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
rrnprjdstle.x | ⊢ (𝜑 → 𝑋 ∈ Fin) |
rrnprjdstle.f | ⊢ (𝜑 → 𝐹:𝑋⟶ℝ) |
rrnprjdstle.g | ⊢ (𝜑 → 𝐺:𝑋⟶ℝ) |
rrnprjdstle.i | ⊢ (𝜑 → 𝐼 ∈ 𝑋) |
rrnprjdstle.d | ⊢ 𝐷 = (dist‘(ℝ^‘𝑋)) |
Ref | Expression |
---|---|
rrnprjdstle | ⊢ (𝜑 → (abs‘((𝐹‘𝐼) − (𝐺‘𝐼))) ≤ (𝐹𝐷𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrnprjdstle.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ) | |
2 | rrnprjdstle.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑋) | |
3 | 1, 2 | ffvelrnd 6829 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐼) ∈ ℝ) |
4 | rrnprjdstle.g | . . . . 5 ⊢ (𝜑 → 𝐺:𝑋⟶ℝ) | |
5 | 4, 2 | ffvelrnd 6829 | . . . 4 ⊢ (𝜑 → (𝐺‘𝐼) ∈ ℝ) |
6 | eqid 2798 | . . . . 5 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
7 | 6 | remetdval 23394 | . . . 4 ⊢ (((𝐹‘𝐼) ∈ ℝ ∧ (𝐺‘𝐼) ∈ ℝ) → ((𝐹‘𝐼)((abs ∘ − ) ↾ (ℝ × ℝ))(𝐺‘𝐼)) = (abs‘((𝐹‘𝐼) − (𝐺‘𝐼)))) |
8 | 3, 5, 7 | syl2anc 587 | . . 3 ⊢ (𝜑 → ((𝐹‘𝐼)((abs ∘ − ) ↾ (ℝ × ℝ))(𝐺‘𝐼)) = (abs‘((𝐹‘𝐼) − (𝐺‘𝐼)))) |
9 | 8 | eqcomd 2804 | . 2 ⊢ (𝜑 → (abs‘((𝐹‘𝐼) − (𝐺‘𝐼))) = ((𝐹‘𝐼)((abs ∘ − ) ↾ (ℝ × ℝ))(𝐺‘𝐼))) |
10 | rrnprjdstle.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
11 | reex 10617 | . . . . . . 7 ⊢ ℝ ∈ V | |
12 | 11 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ∈ V) |
13 | 12, 10 | elmapd 8403 | . . . . 5 ⊢ (𝜑 → (𝐹 ∈ (ℝ ↑m 𝑋) ↔ 𝐹:𝑋⟶ℝ)) |
14 | 1, 13 | mpbird 260 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (ℝ ↑m 𝑋)) |
15 | eqid 2798 | . . . . . 6 ⊢ (ℝ^‘𝑋) = (ℝ^‘𝑋) | |
16 | eqid 2798 | . . . . . 6 ⊢ (Base‘(ℝ^‘𝑋)) = (Base‘(ℝ^‘𝑋)) | |
17 | 10, 15, 16 | rrxbasefi 24014 | . . . . 5 ⊢ (𝜑 → (Base‘(ℝ^‘𝑋)) = (ℝ ↑m 𝑋)) |
18 | 15, 16 | rrxbase 23992 | . . . . . 6 ⊢ (𝑋 ∈ Fin → (Base‘(ℝ^‘𝑋)) = {ℎ ∈ (ℝ ↑m 𝑋) ∣ ℎ finSupp 0}) |
19 | 10, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → (Base‘(ℝ^‘𝑋)) = {ℎ ∈ (ℝ ↑m 𝑋) ∣ ℎ finSupp 0}) |
20 | 17, 19 | eqtr3d 2835 | . . . 4 ⊢ (𝜑 → (ℝ ↑m 𝑋) = {ℎ ∈ (ℝ ↑m 𝑋) ∣ ℎ finSupp 0}) |
21 | 14, 20 | eleqtrd 2892 | . . 3 ⊢ (𝜑 → 𝐹 ∈ {ℎ ∈ (ℝ ↑m 𝑋) ∣ ℎ finSupp 0}) |
22 | 12, 10 | elmapd 8403 | . . . . 5 ⊢ (𝜑 → (𝐺 ∈ (ℝ ↑m 𝑋) ↔ 𝐺:𝑋⟶ℝ)) |
23 | 4, 22 | mpbird 260 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (ℝ ↑m 𝑋)) |
24 | 23, 20 | eleqtrd 2892 | . . 3 ⊢ (𝜑 → 𝐺 ∈ {ℎ ∈ (ℝ ↑m 𝑋) ∣ ℎ finSupp 0}) |
25 | eqid 2798 | . . . 4 ⊢ {ℎ ∈ (ℝ ↑m 𝑋) ∣ ℎ finSupp 0} = {ℎ ∈ (ℝ ↑m 𝑋) ∣ ℎ finSupp 0} | |
26 | rrnprjdstle.d | . . . 4 ⊢ 𝐷 = (dist‘(ℝ^‘𝑋)) | |
27 | 25, 26, 6 | rrxdstprj1 24013 | . . 3 ⊢ (((𝑋 ∈ Fin ∧ 𝐼 ∈ 𝑋) ∧ (𝐹 ∈ {ℎ ∈ (ℝ ↑m 𝑋) ∣ ℎ finSupp 0} ∧ 𝐺 ∈ {ℎ ∈ (ℝ ↑m 𝑋) ∣ ℎ finSupp 0})) → ((𝐹‘𝐼)((abs ∘ − ) ↾ (ℝ × ℝ))(𝐺‘𝐼)) ≤ (𝐹𝐷𝐺)) |
28 | 10, 2, 21, 24, 27 | syl22anc 837 | . 2 ⊢ (𝜑 → ((𝐹‘𝐼)((abs ∘ − ) ↾ (ℝ × ℝ))(𝐺‘𝐼)) ≤ (𝐹𝐷𝐺)) |
29 | 9, 28 | eqbrtrd 5052 | 1 ⊢ (𝜑 → (abs‘((𝐹‘𝐼) − (𝐺‘𝐼))) ≤ (𝐹𝐷𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 class class class wbr 5030 × cxp 5517 ↾ cres 5521 ∘ ccom 5523 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 Fincfn 8492 finSupp cfsupp 8817 ℝcr 10525 0cc0 10526 ≤ cle 10665 − cmin 10859 abscabs 14585 Basecbs 16475 distcds 16566 ℝ^crrx 23987 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ico 12732 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-0g 16707 df-gsum 16708 df-prds 16713 df-pws 16715 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-ghm 18348 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-rnghom 19463 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-xmet 20084 df-met 20085 df-cnfld 20092 df-refld 20294 df-dsmm 20421 df-frlm 20436 df-nm 23189 df-tng 23191 df-tcph 23774 df-rrx 23989 |
This theorem is referenced by: ioorrnopnlem 42946 |
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