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 6838 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐼) ∈ ℝ) |
4 | rrnprjdstle.g | . . . . 5 ⊢ (𝜑 → 𝐺:𝑋⟶ℝ) | |
5 | 4, 2 | ffvelrnd 6838 | . . . 4 ⊢ (𝜑 → (𝐺‘𝐼) ∈ ℝ) |
6 | eqid 2821 | . . . . 5 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
7 | 6 | remetdval 23380 | . . . 4 ⊢ (((𝐹‘𝐼) ∈ ℝ ∧ (𝐺‘𝐼) ∈ ℝ) → ((𝐹‘𝐼)((abs ∘ − ) ↾ (ℝ × ℝ))(𝐺‘𝐼)) = (abs‘((𝐹‘𝐼) − (𝐺‘𝐼)))) |
8 | 3, 5, 7 | syl2anc 586 | . . 3 ⊢ (𝜑 → ((𝐹‘𝐼)((abs ∘ − ) ↾ (ℝ × ℝ))(𝐺‘𝐼)) = (abs‘((𝐹‘𝐼) − (𝐺‘𝐼)))) |
9 | 8 | eqcomd 2827 | . 2 ⊢ (𝜑 → (abs‘((𝐹‘𝐼) − (𝐺‘𝐼))) = ((𝐹‘𝐼)((abs ∘ − ) ↾ (ℝ × ℝ))(𝐺‘𝐼))) |
10 | rrnprjdstle.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
11 | reex 10614 | . . . . . . 7 ⊢ ℝ ∈ V | |
12 | 11 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ∈ V) |
13 | 12, 10 | elmapd 8406 | . . . . 5 ⊢ (𝜑 → (𝐹 ∈ (ℝ ↑m 𝑋) ↔ 𝐹:𝑋⟶ℝ)) |
14 | 1, 13 | mpbird 259 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (ℝ ↑m 𝑋)) |
15 | eqid 2821 | . . . . . 6 ⊢ (ℝ^‘𝑋) = (ℝ^‘𝑋) | |
16 | eqid 2821 | . . . . . 6 ⊢ (Base‘(ℝ^‘𝑋)) = (Base‘(ℝ^‘𝑋)) | |
17 | 10, 15, 16 | rrxbasefi 23996 | . . . . 5 ⊢ (𝜑 → (Base‘(ℝ^‘𝑋)) = (ℝ ↑m 𝑋)) |
18 | 15, 16 | rrxbase 23974 | . . . . . 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 2858 | . . . 4 ⊢ (𝜑 → (ℝ ↑m 𝑋) = {ℎ ∈ (ℝ ↑m 𝑋) ∣ ℎ finSupp 0}) |
21 | 14, 20 | eleqtrd 2915 | . . 3 ⊢ (𝜑 → 𝐹 ∈ {ℎ ∈ (ℝ ↑m 𝑋) ∣ ℎ finSupp 0}) |
22 | 12, 10 | elmapd 8406 | . . . . 5 ⊢ (𝜑 → (𝐺 ∈ (ℝ ↑m 𝑋) ↔ 𝐺:𝑋⟶ℝ)) |
23 | 4, 22 | mpbird 259 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (ℝ ↑m 𝑋)) |
24 | 23, 20 | eleqtrd 2915 | . . 3 ⊢ (𝜑 → 𝐺 ∈ {ℎ ∈ (ℝ ↑m 𝑋) ∣ ℎ finSupp 0}) |
25 | eqid 2821 | . . . 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 23995 | . . 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 836 | . 2 ⊢ (𝜑 → ((𝐹‘𝐼)((abs ∘ − ) ↾ (ℝ × ℝ))(𝐺‘𝐼)) ≤ (𝐹𝐷𝐺)) |
29 | 9, 28 | eqbrtrd 5074 | 1 ⊢ (𝜑 → (abs‘((𝐹‘𝐼) − (𝐺‘𝐼))) ≤ (𝐹𝐷𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {crab 3142 Vcvv 3486 class class class wbr 5052 × cxp 5539 ↾ cres 5543 ∘ ccom 5545 ⟶wf 6337 ‘cfv 6341 (class class class)co 7142 ↑m cmap 8392 Fincfn 8495 finSupp cfsupp 8819 ℝcr 10522 0cc0 10523 ≤ cle 10662 − cmin 10856 abscabs 14578 Basecbs 16466 distcds 16557 ℝ^crrx 23969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-inf2 9090 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 ax-pre-sup 10601 ax-addf 10602 ax-mulf 10603 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-se 5501 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-isom 6350 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-of 7395 df-om 7567 df-1st 7675 df-2nd 7676 df-supp 7817 df-tpos 7878 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-oadd 8092 df-er 8275 df-map 8394 df-ixp 8448 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-fsupp 8820 df-sup 8892 df-oi 8960 df-card 9354 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-div 11284 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-7 11692 df-8 11693 df-9 11694 df-n0 11885 df-z 11969 df-dec 12086 df-uz 12231 df-rp 12377 df-xneg 12494 df-xadd 12495 df-xmul 12496 df-ico 12731 df-fz 12883 df-fzo 13024 df-seq 13360 df-exp 13420 df-hash 13681 df-cj 14443 df-re 14444 df-im 14445 df-sqrt 14579 df-abs 14580 df-clim 14830 df-sum 15028 df-struct 16468 df-ndx 16469 df-slot 16470 df-base 16472 df-sets 16473 df-ress 16474 df-plusg 16561 df-mulr 16562 df-starv 16563 df-sca 16564 df-vsca 16565 df-ip 16566 df-tset 16567 df-ple 16568 df-ds 16570 df-unif 16571 df-hom 16572 df-cco 16573 df-0g 16698 df-gsum 16699 df-prds 16704 df-pws 16706 df-mgm 17835 df-sgrp 17884 df-mnd 17895 df-mhm 17939 df-grp 18089 df-minusg 18090 df-sbg 18091 df-subg 18259 df-ghm 18339 df-cntz 18430 df-cmn 18891 df-abl 18892 df-mgp 19223 df-ur 19235 df-ring 19282 df-cring 19283 df-oppr 19356 df-dvdsr 19374 df-unit 19375 df-invr 19405 df-dvr 19416 df-rnghom 19450 df-drng 19487 df-field 19488 df-subrg 19516 df-staf 19599 df-srng 19600 df-lmod 19619 df-lss 19687 df-sra 19927 df-rgmod 19928 df-xmet 20521 df-met 20522 df-cnfld 20529 df-refld 20732 df-dsmm 20859 df-frlm 20874 df-nm 23175 df-tng 23177 df-tcph 23756 df-rrx 23971 |
This theorem is referenced by: ioorrnopnlem 42679 |
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