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| Mirrors > Home > MPE Home > Th. List > lspsntri | Structured version Visualization version GIF version | ||
| Description: Triangle-type inequality for span of a singleton. (Contributed by NM, 24-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.) |
| Ref | Expression |
|---|---|
| lspsntri.v | β’ π = (Baseβπ) |
| lspsntri.a | β’ + = (+gβπ) |
| lspsntri.n | β’ π = (LSpanβπ) |
| lspsntri.p | β’ β = (LSSumβπ) |
| Ref | Expression |
|---|---|
| lspsntri | β’ ((π β LMod β§ π β π β§ π β π) β (πβ{(π + π)}) β ((πβ{π}) β (πβ{π}))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lspsntri.v | . . . 4 β’ π = (Baseβπ) | |
| 2 | lspsntri.a | . . . 4 β’ + = (+gβπ) | |
| 3 | lspsntri.n | . . . 4 β’ π = (LSpanβπ) | |
| 4 | 1, 2, 3 | lspvadd 20851 | . . 3 β’ ((π β LMod β§ π β π β§ π β π) β (πβ{(π + π)}) β (πβ{π, π})) |
| 5 | df-pr 4631 | . . . 4 β’ {π, π} = ({π} βͺ {π}) | |
| 6 | 5 | fveq2i 6894 | . . 3 β’ (πβ{π, π}) = (πβ({π} βͺ {π})) |
| 7 | 4, 6 | sseqtrdi 4032 | . 2 β’ ((π β LMod β§ π β π β§ π β π) β (πβ{(π + π)}) β (πβ({π} βͺ {π}))) |
| 8 | simp1 1136 | . . 3 β’ ((π β LMod β§ π β π β§ π β π) β π β LMod) | |
| 9 | simp2 1137 | . . . 4 β’ ((π β LMod β§ π β π β§ π β π) β π β π) | |
| 10 | 9 | snssd 4812 | . . 3 β’ ((π β LMod β§ π β π β§ π β π) β {π} β π) |
| 11 | simp3 1138 | . . . 4 β’ ((π β LMod β§ π β π β§ π β π) β π β π) | |
| 12 | 11 | snssd 4812 | . . 3 β’ ((π β LMod β§ π β π β§ π β π) β {π} β π) |
| 13 | lspsntri.p | . . . 4 β’ β = (LSSumβπ) | |
| 14 | 1, 3, 13 | lsmsp2 20842 | . . 3 β’ ((π β LMod β§ {π} β π β§ {π} β π) β ((πβ{π}) β (πβ{π})) = (πβ({π} βͺ {π}))) |
| 15 | 8, 10, 12, 14 | syl3anc 1371 | . 2 β’ ((π β LMod β§ π β π β§ π β π) β ((πβ{π}) β (πβ{π})) = (πβ({π} βͺ {π}))) |
| 16 | 7, 15 | sseqtrrd 4023 | 1 β’ ((π β LMod β§ π β π β§ π β π) β (πβ{(π + π)}) β ((πβ{π}) β (πβ{π}))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1087 = wceq 1541 β wcel 2106 βͺ cun 3946 β wss 3948 {csn 4628 {cpr 4630 βcfv 6543 (class class class)co 7411 Basecbs 17148 +gcplusg 17201 LSSumclsm 19543 LModclmod 20614 LSpanclspn 20726 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-0g 17391 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-subg 19039 df-cntz 19222 df-lsm 19545 df-cmn 19691 df-abl 19692 df-mgp 20029 df-ur 20076 df-ring 20129 df-lmod 20616 df-lss 20687 df-lsp 20727 |
| This theorem is referenced by: lspsntrim 20853 cdlemn4a 40373 lcfrlem23 40739 baerlem5blem2 40886 |
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