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 6959 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐼) ∈ ℝ) |
4 | rrnprjdstle.g | . . . . 5 ⊢ (𝜑 → 𝐺:𝑋⟶ℝ) | |
5 | 4, 2 | ffvelrnd 6959 | . . . 4 ⊢ (𝜑 → (𝐺‘𝐼) ∈ ℝ) |
6 | eqid 2740 | . . . . 5 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
7 | 6 | remetdval 23950 | . . . 4 ⊢ (((𝐹‘𝐼) ∈ ℝ ∧ (𝐺‘𝐼) ∈ ℝ) → ((𝐹‘𝐼)((abs ∘ − ) ↾ (ℝ × ℝ))(𝐺‘𝐼)) = (abs‘((𝐹‘𝐼) − (𝐺‘𝐼)))) |
8 | 3, 5, 7 | syl2anc 584 | . . 3 ⊢ (𝜑 → ((𝐹‘𝐼)((abs ∘ − ) ↾ (ℝ × ℝ))(𝐺‘𝐼)) = (abs‘((𝐹‘𝐼) − (𝐺‘𝐼)))) |
9 | 8 | eqcomd 2746 | . 2 ⊢ (𝜑 → (abs‘((𝐹‘𝐼) − (𝐺‘𝐼))) = ((𝐹‘𝐼)((abs ∘ − ) ↾ (ℝ × ℝ))(𝐺‘𝐼))) |
10 | rrnprjdstle.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
11 | reex 10963 | . . . . . . 7 ⊢ ℝ ∈ V | |
12 | 11 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ∈ V) |
13 | 12, 10 | elmapd 8612 | . . . . 5 ⊢ (𝜑 → (𝐹 ∈ (ℝ ↑m 𝑋) ↔ 𝐹:𝑋⟶ℝ)) |
14 | 1, 13 | mpbird 256 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (ℝ ↑m 𝑋)) |
15 | eqid 2740 | . . . . . 6 ⊢ (ℝ^‘𝑋) = (ℝ^‘𝑋) | |
16 | eqid 2740 | . . . . . 6 ⊢ (Base‘(ℝ^‘𝑋)) = (Base‘(ℝ^‘𝑋)) | |
17 | 10, 15, 16 | rrxbasefi 24572 | . . . . 5 ⊢ (𝜑 → (Base‘(ℝ^‘𝑋)) = (ℝ ↑m 𝑋)) |
18 | 15, 16 | rrxbase 24550 | . . . . . 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 2782 | . . . 4 ⊢ (𝜑 → (ℝ ↑m 𝑋) = {ℎ ∈ (ℝ ↑m 𝑋) ∣ ℎ finSupp 0}) |
21 | 14, 20 | eleqtrd 2843 | . . 3 ⊢ (𝜑 → 𝐹 ∈ {ℎ ∈ (ℝ ↑m 𝑋) ∣ ℎ finSupp 0}) |
22 | 12, 10 | elmapd 8612 | . . . . 5 ⊢ (𝜑 → (𝐺 ∈ (ℝ ↑m 𝑋) ↔ 𝐺:𝑋⟶ℝ)) |
23 | 4, 22 | mpbird 256 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (ℝ ↑m 𝑋)) |
24 | 23, 20 | eleqtrd 2843 | . . 3 ⊢ (𝜑 → 𝐺 ∈ {ℎ ∈ (ℝ ↑m 𝑋) ∣ ℎ finSupp 0}) |
25 | eqid 2740 | . . . 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 24571 | . . 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 5101 | 1 ⊢ (𝜑 → (abs‘((𝐹‘𝐼) − (𝐺‘𝐼))) ≤ (𝐹𝐷𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 {crab 3070 Vcvv 3431 class class class wbr 5079 × cxp 5588 ↾ cres 5592 ∘ ccom 5594 ⟶wf 6428 ‘cfv 6432 (class class class)co 7271 ↑m cmap 8598 Fincfn 8716 finSupp cfsupp 9106 ℝcr 10871 0cc0 10872 ≤ cle 11011 − cmin 11205 abscabs 14943 Basecbs 16910 distcds 16969 ℝ^crrx 24545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-inf2 9377 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 ax-addf 10951 ax-mulf 10952 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-om 7707 df-1st 7824 df-2nd 7825 df-supp 7969 df-tpos 8033 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-ixp 8669 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fsupp 9107 df-sup 9179 df-oi 9247 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-rp 12730 df-xneg 12847 df-xadd 12848 df-xmul 12849 df-ico 13084 df-fz 13239 df-fzo 13382 df-seq 13720 df-exp 13781 df-hash 14043 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-clim 15195 df-sum 15396 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-hom 16984 df-cco 16985 df-0g 17150 df-gsum 17151 df-prds 17156 df-pws 17158 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-mhm 18428 df-grp 18578 df-minusg 18579 df-sbg 18580 df-subg 18750 df-ghm 18830 df-cntz 18921 df-cmn 19386 df-abl 19387 df-mgp 19719 df-ur 19736 df-ring 19783 df-cring 19784 df-oppr 19860 df-dvdsr 19881 df-unit 19882 df-invr 19912 df-dvr 19923 df-rnghom 19957 df-drng 19991 df-field 19992 df-subrg 20020 df-staf 20103 df-srng 20104 df-lmod 20123 df-lss 20192 df-sra 20432 df-rgmod 20433 df-xmet 20588 df-met 20589 df-cnfld 20596 df-refld 20808 df-dsmm 20937 df-frlm 20952 df-nm 23736 df-tng 23738 df-tcph 24331 df-rrx 24547 |
This theorem is referenced by: ioorrnopnlem 43816 |
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