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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 24518 | . . . 4 β’ ((πΊ β NrmGrp β§ π΄ β π) β (πβπ΄) β β) |
| 4 | 3 | 3adant3 1130 | . . 3 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (πβπ΄) β β) |
| 5 | 1, 2 | nmcl 24518 | . . . 4 β’ ((πΊ β NrmGrp β§ π΅ β π) β (πβπ΅) β β) |
| 6 | 5 | 3adant2 1129 | . . 3 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (πβπ΅) β β) |
| 7 | 4, 6 | resubcld 11666 | . 2 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β ((πβπ΄) β (πβπ΅)) β β) |
| 8 | 7 | recnd 11266 | . . 3 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β ((πβπ΄) β (πβπ΅)) β β) |
| 9 | 8 | abscld 15409 | . 2 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (absβ((πβπ΄) β (πβπ΅))) β β) |
| 10 | simp1 1134 | . . 3 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β πΊ β NrmGrp) | |
| 11 | ngpgrp 24501 | . . . 4 β’ (πΊ β NrmGrp β πΊ β Grp) | |
| 12 | nmmtri.m | . . . . 5 β’ β = (-gβπΊ) | |
| 13 | 1, 12 | grpsubcl 18969 | . . . 4 β’ ((πΊ β Grp β§ π΄ β π β§ π΅ β π) β (π΄ β π΅) β π) |
| 14 | 11, 13 | syl3an1 1161 | . . 3 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (π΄ β π΅) β π) |
| 15 | 1, 2 | nmcl 24518 | . . 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 15387 | . 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 24526 | . 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 11395 | 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 1085 = wceq 1534 β wcel 2099 class class class wbr 5142 βcfv 6542 (class class class)co 7414 βcr 11131 β€ cle 11273 β cmin 11468 abscabs 15207 Basecbs 17173 Grpcgrp 18883 -gcsg 18885 normcnm 24478 NrmGrpcngp 24479 |
| 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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 |
| 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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9459 df-inf 9460 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-q 12957 df-rp 13001 df-xneg 13118 df-xadd 13119 df-xmul 13120 df-fz 13511 df-seq 13993 df-exp 14053 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-struct 17109 df-slot 17144 df-ndx 17156 df-base 17174 df-plusg 17239 df-mulr 17240 df-tset 17245 df-ple 17246 df-ds 17248 df-0g 17416 df-topgen 17418 df-xrs 17477 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-grp 18886 df-minusg 18887 df-sbg 18888 df-psmet 21264 df-xmet 21265 df-met 21266 df-bl 21267 df-mopn 21268 df-top 22789 df-topon 22806 df-topsp 22828 df-bases 22842 df-xms 24219 df-ms 24220 df-nm 24484 df-ngp 24485 |
| This theorem is referenced by: nlmvscnlem2 24595 nrginvrcnlem 24601 ipcnlem2 25165 |
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