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| Mirrors > Home > MPE Home > Th. List > nmrtri | Structured version Visualization version GIF version | ||
| Description: Reverse triangle inequality for the norm of a subtraction. Problem 3 of [Kreyszig] p. 64. (Contributed by NM, 4-Dec-2006.) (Revised by Mario Carneiro, 4-Oct-2015.) |
| Ref | Expression |
|---|---|
| nmf.x | β’ π = (BaseβπΊ) |
| nmf.n | β’ π = (normβπΊ) |
| nmmtri.m | β’ β = (-gβπΊ) |
| Ref | Expression |
|---|---|
| nmrtri | β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (absβ((πβπ΄) β (πβπ΅))) β€ (πβ(π΄ β π΅))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ngpms 24331 | . . . 4 β’ (πΊ β NrmGrp β πΊ β MetSp) | |
| 2 | 1 | 3ad2ant1 1131 | . . 3 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β πΊ β MetSp) |
| 3 | simp2 1135 | . . 3 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β π΄ β π) | |
| 4 | simp3 1136 | . . 3 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β π΅ β π) | |
| 5 | ngpgrp 24330 | . . . . 5 β’ (πΊ β NrmGrp β πΊ β Grp) | |
| 6 | 5 | 3ad2ant1 1131 | . . . 4 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β πΊ β Grp) |
| 7 | nmf.x | . . . . 5 β’ π = (BaseβπΊ) | |
| 8 | eqid 2730 | . . . . 5 β’ (0gβπΊ) = (0gβπΊ) | |
| 9 | 7, 8 | grpidcl 18888 | . . . 4 β’ (πΊ β Grp β (0gβπΊ) β π) |
| 10 | 6, 9 | syl 17 | . . 3 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (0gβπΊ) β π) |
| 11 | eqid 2730 | . . . 4 β’ (distβπΊ) = (distβπΊ) | |
| 12 | 7, 11 | msrtri 24200 | . . 3 β’ ((πΊ β MetSp β§ (π΄ β π β§ π΅ β π β§ (0gβπΊ) β π)) β (absβ((π΄(distβπΊ)(0gβπΊ)) β (π΅(distβπΊ)(0gβπΊ)))) β€ (π΄(distβπΊ)π΅)) |
| 13 | 2, 3, 4, 10, 12 | syl13anc 1370 | . 2 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (absβ((π΄(distβπΊ)(0gβπΊ)) β (π΅(distβπΊ)(0gβπΊ)))) β€ (π΄(distβπΊ)π΅)) |
| 14 | nmf.n | . . . . . 6 β’ π = (normβπΊ) | |
| 15 | 14, 7, 8, 11 | nmval 24320 | . . . . 5 β’ (π΄ β π β (πβπ΄) = (π΄(distβπΊ)(0gβπΊ))) |
| 16 | 15 | 3ad2ant2 1132 | . . . 4 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (πβπ΄) = (π΄(distβπΊ)(0gβπΊ))) |
| 17 | 14, 7, 8, 11 | nmval 24320 | . . . . 5 β’ (π΅ β π β (πβπ΅) = (π΅(distβπΊ)(0gβπΊ))) |
| 18 | 17 | 3ad2ant3 1133 | . . . 4 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (πβπ΅) = (π΅(distβπΊ)(0gβπΊ))) |
| 19 | 16, 18 | oveq12d 7431 | . . 3 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β ((πβπ΄) β (πβπ΅)) = ((π΄(distβπΊ)(0gβπΊ)) β (π΅(distβπΊ)(0gβπΊ)))) |
| 20 | 19 | fveq2d 6896 | . 2 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (absβ((πβπ΄) β (πβπ΅))) = (absβ((π΄(distβπΊ)(0gβπΊ)) β (π΅(distβπΊ)(0gβπΊ))))) |
| 21 | nmmtri.m | . . . 4 β’ β = (-gβπΊ) | |
| 22 | 14, 7, 21, 11 | ngpds 24335 | . . 3 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (π΄(distβπΊ)π΅) = (πβ(π΄ β π΅))) |
| 23 | 22 | eqcomd 2736 | . 2 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (πβ(π΄ β π΅)) = (π΄(distβπΊ)π΅)) |
| 24 | 13, 20, 23 | 3brtr4d 5181 | 1 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (absβ((πβπ΄) β (πβπ΅))) β€ (πβ(π΄ β π΅))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1085 = wceq 1539 β wcel 2104 class class class wbr 5149 βcfv 6544 (class class class)co 7413 β€ cle 11255 β cmin 11450 abscabs 15187 Basecbs 17150 distcds 17212 0gc0g 17391 Grpcgrp 18857 -gcsg 18859 MetSpcms 24046 normcnm 24307 NrmGrpcngp 24308 |
| 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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9441 df-inf 9442 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-div 11878 df-nn 12219 df-2 12281 df-3 12282 df-4 12283 df-5 12284 df-6 12285 df-7 12286 df-8 12287 df-9 12288 df-n0 12479 df-z 12565 df-dec 12684 df-uz 12829 df-q 12939 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-fz 13491 df-seq 13973 df-exp 14034 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17151 df-plusg 17216 df-mulr 17217 df-tset 17222 df-ple 17223 df-ds 17225 df-0g 17393 df-topgen 17395 df-xrs 17454 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-grp 18860 df-minusg 18861 df-sbg 18862 df-psmet 21138 df-xmet 21139 df-met 21140 df-bl 21141 df-mopn 21142 df-top 22618 df-topon 22635 df-topsp 22657 df-bases 22671 df-xms 24048 df-ms 24049 df-nm 24313 df-ngp 24314 |
| This theorem is referenced by: nm2dif 24356 |
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