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| Mirrors > Home > MPE Home > Th. List > nm2dif | Structured version Visualization version GIF version | ||
| Description: Inequality for the difference of norms. (Contributed by Mario Carneiro, 6-Oct-2015.) |
| Ref | Expression |
|---|---|
| nmf.x | β’ π = (BaseβπΊ) |
| nmf.n | β’ π = (normβπΊ) |
| nmmtri.m | β’ β = (-gβπΊ) |
| Ref | Expression |
|---|---|
| nm2dif | β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β ((πβπ΄) β (πβπ΅)) β€ (πβ(π΄ β π΅))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nmf.x | . . . . 5 β’ π = (BaseβπΊ) | |
| 2 | nmf.n | . . . . 5 β’ π = (normβπΊ) | |
| 3 | 1, 2 | nmcl 24447 | . . . 4 β’ ((πΊ β NrmGrp β§ π΄ β π) β (πβπ΄) β β) |
| 4 | 3 | 3adant3 1129 | . . 3 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (πβπ΄) β β) |
| 5 | 1, 2 | nmcl 24447 | . . . 4 β’ ((πΊ β NrmGrp β§ π΅ β π) β (πβπ΅) β β) |
| 6 | 5 | 3adant2 1128 | . . 3 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (πβπ΅) β β) |
| 7 | 4, 6 | resubcld 11639 | . 2 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β ((πβπ΄) β (πβπ΅)) β β) |
| 8 | 7 | recnd 11239 | . . 3 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β ((πβπ΄) β (πβπ΅)) β β) |
| 9 | 8 | abscld 15380 | . 2 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (absβ((πβπ΄) β (πβπ΅))) β β) |
| 10 | simp1 1133 | . . 3 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β πΊ β NrmGrp) | |
| 11 | ngpgrp 24430 | . . . 4 β’ (πΊ β NrmGrp β πΊ β Grp) | |
| 12 | nmmtri.m | . . . . 5 β’ β = (-gβπΊ) | |
| 13 | 1, 12 | grpsubcl 18938 | . . . 4 β’ ((πΊ β Grp β§ π΄ β π β§ π΅ β π) β (π΄ β π΅) β π) |
| 14 | 11, 13 | syl3an1 1160 | . . 3 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (π΄ β π΅) β π) |
| 15 | 1, 2 | nmcl 24447 | . . 3 β’ ((πΊ β NrmGrp β§ (π΄ β π΅) β π) β (πβ(π΄ β π΅)) β β) |
| 16 | 10, 14, 15 | syl2anc 583 | . 2 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (πβ(π΄ β π΅)) β β) |
| 17 | 7 | leabsd 15358 | . 2 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β ((πβπ΄) β (πβπ΅)) β€ (absβ((πβπ΄) β (πβπ΅)))) |
| 18 | 1, 2, 12 | nmrtri 24455 | . 2 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (absβ((πβπ΄) β (πβπ΅))) β€ (πβ(π΄ β π΅))) |
| 19 | 7, 9, 16, 17, 18 | letrd 11368 | 1 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β ((πβπ΄) β (πβπ΅)) β€ (πβ(π΄ β π΅))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5138 βcfv 6533 (class class class)co 7401 βcr 11105 β€ cle 11246 β cmin 11441 abscabs 15178 Basecbs 17143 Grpcgrp 18853 -gcsg 18855 normcnm 24407 NrmGrpcngp 24408 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-fz 13482 df-seq 13964 df-exp 14025 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-struct 17079 df-slot 17114 df-ndx 17126 df-base 17144 df-plusg 17209 df-mulr 17210 df-tset 17215 df-ple 17216 df-ds 17218 df-0g 17386 df-topgen 17388 df-xrs 17447 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-grp 18856 df-minusg 18857 df-sbg 18858 df-psmet 21220 df-xmet 21221 df-met 21222 df-bl 21223 df-mopn 21224 df-top 22718 df-topon 22735 df-topsp 22757 df-bases 22771 df-xms 24148 df-ms 24149 df-nm 24413 df-ngp 24414 |
| This theorem is referenced by: nlmvscnlem2 24524 nrginvrcnlem 24530 ipcnlem2 25094 |
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