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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 | β’ π = (BaseβπΊ) |
| nmtri2.n | β’ π = (normβπΊ) |
| nmtri2.m | β’ β = (-gβπΊ) |
| Ref | Expression |
|---|---|
| nmtri2 | β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (πβ(π΄ β πΆ)) β€ ((πβ(π΄ β π΅)) + (πβ(π΅ β πΆ)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ngpgrp 24521 | . . . 4 β’ (πΊ β NrmGrp β πΊ β Grp) | |
| 2 | nmtri2.x | . . . . . 6 β’ π = (BaseβπΊ) | |
| 3 | eqid 2728 | . . . . . 6 β’ (+gβπΊ) = (+gβπΊ) | |
| 4 | nmtri2.m | . . . . . 6 β’ β = (-gβπΊ) | |
| 5 | 2, 3, 4 | grpnpncan 18991 | . . . . 5 β’ ((πΊ β Grp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((π΄ β π΅)(+gβπΊ)(π΅ β πΆ)) = (π΄ β πΆ)) |
| 6 | 5 | eqcomd 2734 | . . . 4 β’ ((πΊ β Grp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄ β πΆ) = ((π΄ β π΅)(+gβπΊ)(π΅ β πΆ))) |
| 7 | 1, 6 | sylan 579 | . . 3 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄ β πΆ) = ((π΄ β π΅)(+gβπΊ)(π΅ β πΆ))) |
| 8 | 7 | fveq2d 6901 | . 2 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (πβ(π΄ β πΆ)) = (πβ((π΄ β π΅)(+gβπΊ)(π΅ β πΆ)))) |
| 9 | simpl 482 | . . 3 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β πΊ β NrmGrp) | |
| 10 | 1 | adantr 480 | . . . 4 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β πΊ β Grp) |
| 11 | simpr1 1192 | . . . 4 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π΄ β π) | |
| 12 | simpr2 1193 | . . . 4 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π΅ β π) | |
| 13 | 2, 4 | grpsubcl 18976 | . . . 4 β’ ((πΊ β Grp β§ π΄ β π β§ π΅ β π) β (π΄ β π΅) β π) |
| 14 | 10, 11, 12, 13 | syl3anc 1369 | . . 3 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄ β π΅) β π) |
| 15 | simpr3 1194 | . . . 4 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β πΆ β π) | |
| 16 | 2, 4 | grpsubcl 18976 | . . . 4 β’ ((πΊ β Grp β§ π΅ β π β§ πΆ β π) β (π΅ β πΆ) β π) |
| 17 | 10, 12, 15, 16 | syl3anc 1369 | . . 3 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΅ β πΆ) β π) |
| 18 | nmtri2.n | . . . 4 β’ π = (normβπΊ) | |
| 19 | 2, 18, 3 | nmtri 24548 | . . 3 β’ ((πΊ β NrmGrp β§ (π΄ β π΅) β π β§ (π΅ β πΆ) β π) β (πβ((π΄ β π΅)(+gβπΊ)(π΅ β πΆ))) β€ ((πβ(π΄ β π΅)) + (πβ(π΅ β πΆ)))) |
| 20 | 9, 14, 17, 19 | syl3anc 1369 | . 2 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (πβ((π΄ β π΅)(+gβπΊ)(π΅ β πΆ))) β€ ((πβ(π΄ β π΅)) + (πβ(π΅ β πΆ)))) |
| 21 | 8, 20 | eqbrtrd 5170 | 1 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (πβ(π΄ β πΆ)) β€ ((πβ(π΄ β π΅)) + (πβ(π΅ β πΆ)))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 β§ w3a 1085 = wceq 1534 β wcel 2099 class class class wbr 5148 βcfv 6548 (class class class)co 7420 + caddc 11142 β€ cle 11280 Basecbs 17180 +gcplusg 17233 Grpcgrp 18890 -gcsg 18892 normcnm 24498 NrmGrpcngp 24499 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9466 df-inf 9467 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-n0 12504 df-z 12590 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-0g 17423 df-topgen 17425 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-minusg 18894 df-sbg 18895 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-xms 24239 df-ms 24240 df-nm 24504 df-ngp 24505 |
| This theorem is referenced by: (None) |
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