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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 | ffvelcdmd 7087 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐼) ∈ ℝ) |
4 | rrnprjdstle.g | . . . . 5 ⊢ (𝜑 → 𝐺:𝑋⟶ℝ) | |
5 | 4, 2 | ffvelcdmd 7087 | . . . 4 ⊢ (𝜑 → (𝐺‘𝐼) ∈ ℝ) |
6 | eqid 2731 | . . . . 5 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
7 | 6 | remetdval 24625 | . . . 4 ⊢ (((𝐹‘𝐼) ∈ ℝ ∧ (𝐺‘𝐼) ∈ ℝ) → ((𝐹‘𝐼)((abs ∘ − ) ↾ (ℝ × ℝ))(𝐺‘𝐼)) = (abs‘((𝐹‘𝐼) − (𝐺‘𝐼)))) |
8 | 3, 5, 7 | syl2anc 583 | . . 3 ⊢ (𝜑 → ((𝐹‘𝐼)((abs ∘ − ) ↾ (ℝ × ℝ))(𝐺‘𝐼)) = (abs‘((𝐹‘𝐼) − (𝐺‘𝐼)))) |
9 | 8 | eqcomd 2737 | . 2 ⊢ (𝜑 → (abs‘((𝐹‘𝐼) − (𝐺‘𝐼))) = ((𝐹‘𝐼)((abs ∘ − ) ↾ (ℝ × ℝ))(𝐺‘𝐼))) |
10 | rrnprjdstle.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
11 | reex 11207 | . . . . . . 7 ⊢ ℝ ∈ V | |
12 | 11 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ∈ V) |
13 | 12, 10 | elmapd 8840 | . . . . 5 ⊢ (𝜑 → (𝐹 ∈ (ℝ ↑m 𝑋) ↔ 𝐹:𝑋⟶ℝ)) |
14 | 1, 13 | mpbird 257 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (ℝ ↑m 𝑋)) |
15 | eqid 2731 | . . . . . 6 ⊢ (ℝ^‘𝑋) = (ℝ^‘𝑋) | |
16 | eqid 2731 | . . . . . 6 ⊢ (Base‘(ℝ^‘𝑋)) = (Base‘(ℝ^‘𝑋)) | |
17 | 10, 15, 16 | rrxbasefi 25258 | . . . . 5 ⊢ (𝜑 → (Base‘(ℝ^‘𝑋)) = (ℝ ↑m 𝑋)) |
18 | 15, 16 | rrxbase 25236 | . . . . . 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 2773 | . . . 4 ⊢ (𝜑 → (ℝ ↑m 𝑋) = {ℎ ∈ (ℝ ↑m 𝑋) ∣ ℎ finSupp 0}) |
21 | 14, 20 | eleqtrd 2834 | . . 3 ⊢ (𝜑 → 𝐹 ∈ {ℎ ∈ (ℝ ↑m 𝑋) ∣ ℎ finSupp 0}) |
22 | 12, 10 | elmapd 8840 | . . . . 5 ⊢ (𝜑 → (𝐺 ∈ (ℝ ↑m 𝑋) ↔ 𝐺:𝑋⟶ℝ)) |
23 | 4, 22 | mpbird 257 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (ℝ ↑m 𝑋)) |
24 | 23, 20 | eleqtrd 2834 | . . 3 ⊢ (𝜑 → 𝐺 ∈ {ℎ ∈ (ℝ ↑m 𝑋) ∣ ℎ finSupp 0}) |
25 | eqid 2731 | . . . 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 25257 | . . 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 5170 | 1 ⊢ (𝜑 → (abs‘((𝐹‘𝐼) − (𝐺‘𝐼))) ≤ (𝐹𝐷𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 {crab 3431 Vcvv 3473 class class class wbr 5148 × cxp 5674 ↾ cres 5678 ∘ ccom 5680 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ↑m cmap 8826 Fincfn 8945 finSupp cfsupp 9367 ℝcr 11115 0cc0 11116 ≤ cle 11256 − cmin 11451 abscabs 15188 Basecbs 17151 distcds 17213 ℝ^crrx 25231 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 ax-mulf 11196 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-sup 9443 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-ico 13337 df-fz 13492 df-fzo 13635 df-seq 13974 df-exp 14035 df-hash 14298 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15439 df-sum 15640 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-0g 17394 df-gsum 17395 df-prds 17400 df-pws 17402 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19046 df-ghm 19135 df-cntz 19229 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-cring 20137 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-rhm 20370 df-subrng 20442 df-subrg 20467 df-drng 20585 df-field 20586 df-staf 20684 df-srng 20685 df-lmod 20704 df-lss 20775 df-sra 21019 df-rgmod 21020 df-xmet 21226 df-met 21227 df-cnfld 21234 df-refld 21468 df-dsmm 21597 df-frlm 21612 df-nm 24411 df-tng 24413 df-tcph 25017 df-rrx 25233 |
This theorem is referenced by: ioorrnopnlem 45479 |
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