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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 23210 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) | |
2 | nmtri2.x | . . . . . 6 ⊢ 𝑋 = (Base‘𝐺) | |
3 | eqid 2823 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | nmtri2.m | . . . . . 6 ⊢ − = (-g‘𝐺) | |
5 | 2, 3, 4 | grpnpncan 18196 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴 − 𝐵)(+g‘𝐺)(𝐵 − 𝐶)) = (𝐴 − 𝐶)) |
6 | 5 | eqcomd 2829 | . . . 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 6676 | . 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 18181 | . . . 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 18181 | . . . 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 23237 | . . 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 5090 | 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 1537 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 + caddc 10542 ≤ cle 10678 Basecbs 16485 +gcplusg 16567 Grpcgrp 18105 -gcsg 18107 normcnm 23188 NrmGrpcngp 23189 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 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 16717 df-topgen 16719 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-sbg 18110 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-xms 22932 df-ms 22933 df-nm 23194 df-ngp 23195 |
This theorem is referenced by: (None) |
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