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Mirrors > Home > MPE Home > Th. List > nmtri2 | Structured version Visualization version GIF version |
Description: Triangle inequality for the norm of two subtractions. (Contributed by NM, 24-Feb-2008.) (Revised by AV, 8-Oct-2021.) |
Ref | Expression |
---|---|
nmtri2.x | ⊢ 𝑋 = (Base‘𝐺) |
nmtri2.n | ⊢ 𝑁 = (norm‘𝐺) |
nmtri2.m | ⊢ − = (-g‘𝐺) |
Ref | Expression |
---|---|
nmtri2 | ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝑁‘(𝐴 − 𝐶)) ≤ ((𝑁‘(𝐴 − 𝐵)) + (𝑁‘(𝐵 − 𝐶)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ngpgrp 23211 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) | |
2 | nmtri2.x | . . . . . 6 ⊢ 𝑋 = (Base‘𝐺) | |
3 | eqid 2824 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | nmtri2.m | . . . . . 6 ⊢ − = (-g‘𝐺) | |
5 | 2, 3, 4 | grpnpncan 18197 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴 − 𝐵)(+g‘𝐺)(𝐵 − 𝐶)) = (𝐴 − 𝐶)) |
6 | 5 | eqcomd 2830 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴 − 𝐶) = ((𝐴 − 𝐵)(+g‘𝐺)(𝐵 − 𝐶))) |
7 | 1, 6 | sylan 582 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴 − 𝐶) = ((𝐴 − 𝐵)(+g‘𝐺)(𝐵 − 𝐶))) |
8 | 7 | fveq2d 6677 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝑁‘(𝐴 − 𝐶)) = (𝑁‘((𝐴 − 𝐵)(+g‘𝐺)(𝐵 − 𝐶)))) |
9 | simpl 485 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐺 ∈ NrmGrp) | |
10 | 1 | adantr 483 | . . . 4 ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐺 ∈ Grp) |
11 | simpr1 1190 | . . . 4 ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐴 ∈ 𝑋) | |
12 | simpr2 1191 | . . . 4 ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐵 ∈ 𝑋) | |
13 | 2, 4 | grpsubcl 18182 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 − 𝐵) ∈ 𝑋) |
14 | 10, 11, 12, 13 | syl3anc 1367 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴 − 𝐵) ∈ 𝑋) |
15 | simpr3 1192 | . . . 4 ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐶 ∈ 𝑋) | |
16 | 2, 4 | grpsubcl 18182 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵 − 𝐶) ∈ 𝑋) |
17 | 10, 12, 15, 16 | syl3anc 1367 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵 − 𝐶) ∈ 𝑋) |
18 | nmtri2.n | . . . 4 ⊢ 𝑁 = (norm‘𝐺) | |
19 | 2, 18, 3 | nmtri 23238 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 − 𝐵) ∈ 𝑋 ∧ (𝐵 − 𝐶) ∈ 𝑋) → (𝑁‘((𝐴 − 𝐵)(+g‘𝐺)(𝐵 − 𝐶))) ≤ ((𝑁‘(𝐴 − 𝐵)) + (𝑁‘(𝐵 − 𝐶)))) |
20 | 9, 14, 17, 19 | syl3anc 1367 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝑁‘((𝐴 − 𝐵)(+g‘𝐺)(𝐵 − 𝐶))) ≤ ((𝑁‘(𝐴 − 𝐵)) + (𝑁‘(𝐵 − 𝐶)))) |
21 | 8, 20 | eqbrtrd 5091 | 1 ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝑁‘(𝐴 − 𝐶)) ≤ ((𝑁‘(𝐴 − 𝐵)) + (𝑁‘(𝐵 − 𝐶)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 + caddc 10543 ≤ cle 10679 Basecbs 16486 +gcplusg 16568 Grpcgrp 18106 -gcsg 18108 normcnm 23189 NrmGrpcngp 23190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-sup 8909 df-inf 8910 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-0g 16718 df-topgen 16720 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 df-minusg 18110 df-sbg 18111 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 df-xms 22933 df-ms 22934 df-nm 23195 df-ngp 23196 |
This theorem is referenced by: (None) |
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