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| Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem1 | Structured version Visualization version GIF version | ||
| Description: Lemma for mapdpg 41088. Baer p. 44, last line: "(F(x-y))* <= (Fx)*+(Fy)*." (Contributed by NM, 15-Mar-2015.) |
| Ref | Expression |
|---|---|
| mapdpglem.h | β’ π» = (LHypβπΎ) |
| mapdpglem.m | β’ π = ((mapdβπΎ)βπ) |
| mapdpglem.u | β’ π = ((DVecHβπΎ)βπ) |
| mapdpglem.v | β’ π = (Baseβπ) |
| mapdpglem.s | β’ β = (-gβπ) |
| mapdpglem.n | β’ π = (LSpanβπ) |
| mapdpglem.c | β’ πΆ = ((LCDualβπΎ)βπ) |
| mapdpglem.k | β’ (π β (πΎ β HL β§ π β π»)) |
| mapdpglem.x | β’ (π β π β π) |
| mapdpglem.y | β’ (π β π β π) |
| mapdpglem1.p | β’ β = (LSSumβπΆ) |
| Ref | Expression |
|---|---|
| mapdpglem1 | β’ (π β (πβ(πβ{(π β π)})) β ((πβ(πβ{π})) β (πβ(πβ{π})))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mapdpglem.h | . . . . 5 β’ π» = (LHypβπΎ) | |
| 2 | mapdpglem.u | . . . . 5 β’ π = ((DVecHβπΎ)βπ) | |
| 3 | mapdpglem.k | . . . . 5 β’ (π β (πΎ β HL β§ π β π»)) | |
| 4 | 1, 2, 3 | dvhlmod 40492 | . . . 4 β’ (π β π β LMod) |
| 5 | mapdpglem.x | . . . 4 β’ (π β π β π) | |
| 6 | mapdpglem.y | . . . 4 β’ (π β π β π) | |
| 7 | mapdpglem.v | . . . . 5 β’ π = (Baseβπ) | |
| 8 | mapdpglem.s | . . . . 5 β’ β = (-gβπ) | |
| 9 | eqid 2726 | . . . . 5 β’ (LSSumβπ) = (LSSumβπ) | |
| 10 | mapdpglem.n | . . . . 5 β’ π = (LSpanβπ) | |
| 11 | 7, 8, 9, 10 | lspsntrim 20944 | . . . 4 β’ ((π β LMod β§ π β π β§ π β π) β (πβ{(π β π)}) β ((πβ{π})(LSSumβπ)(πβ{π}))) |
| 12 | 4, 5, 6, 11 | syl3anc 1368 | . . 3 β’ (π β (πβ{(π β π)}) β ((πβ{π})(LSSumβπ)(πβ{π}))) |
| 13 | eqid 2726 | . . . 4 β’ (LSubSpβπ) = (LSubSpβπ) | |
| 14 | mapdpglem.m | . . . 4 β’ π = ((mapdβπΎ)βπ) | |
| 15 | 7, 8 | lmodvsubcl 20751 | . . . . . 6 β’ ((π β LMod β§ π β π β§ π β π) β (π β π) β π) |
| 16 | 4, 5, 6, 15 | syl3anc 1368 | . . . . 5 β’ (π β (π β π) β π) |
| 17 | 7, 13, 10 | lspsncl 20822 | . . . . 5 β’ ((π β LMod β§ (π β π) β π) β (πβ{(π β π)}) β (LSubSpβπ)) |
| 18 | 4, 16, 17 | syl2anc 583 | . . . 4 β’ (π β (πβ{(π β π)}) β (LSubSpβπ)) |
| 19 | 7, 13, 10 | lspsncl 20822 | . . . . . 6 β’ ((π β LMod β§ π β π) β (πβ{π}) β (LSubSpβπ)) |
| 20 | 4, 5, 19 | syl2anc 583 | . . . . 5 β’ (π β (πβ{π}) β (LSubSpβπ)) |
| 21 | 7, 13, 10 | lspsncl 20822 | . . . . . 6 β’ ((π β LMod β§ π β π) β (πβ{π}) β (LSubSpβπ)) |
| 22 | 4, 6, 21 | syl2anc 583 | . . . . 5 β’ (π β (πβ{π}) β (LSubSpβπ)) |
| 23 | 13, 9 | lsmcl 20929 | . . . . 5 β’ ((π β LMod β§ (πβ{π}) β (LSubSpβπ) β§ (πβ{π}) β (LSubSpβπ)) β ((πβ{π})(LSSumβπ)(πβ{π})) β (LSubSpβπ)) |
| 24 | 4, 20, 22, 23 | syl3anc 1368 | . . . 4 β’ (π β ((πβ{π})(LSSumβπ)(πβ{π})) β (LSubSpβπ)) |
| 25 | 1, 2, 13, 14, 3, 18, 24 | mapdord 41020 | . . 3 β’ (π β ((πβ(πβ{(π β π)})) β (πβ((πβ{π})(LSSumβπ)(πβ{π}))) β (πβ{(π β π)}) β ((πβ{π})(LSSumβπ)(πβ{π})))) |
| 26 | 12, 25 | mpbird 257 | . 2 β’ (π β (πβ(πβ{(π β π)})) β (πβ((πβ{π})(LSSumβπ)(πβ{π})))) |
| 27 | mapdpglem.c | . . 3 β’ πΆ = ((LCDualβπΎ)βπ) | |
| 28 | mapdpglem1.p | . . 3 β’ β = (LSSumβπΆ) | |
| 29 | 1, 14, 2, 13, 9, 27, 28, 3, 20, 22 | mapdlsm 41046 | . 2 β’ (π β (πβ((πβ{π})(LSSumβπ)(πβ{π}))) = ((πβ(πβ{π})) β (πβ(πβ{π})))) |
| 30 | 26, 29 | sseqtrd 4017 | 1 β’ (π β (πβ(πβ{(π β π)})) β ((πβ(πβ{π})) β (πβ(πβ{π})))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wss 3943 {csn 4623 βcfv 6536 (class class class)co 7404 Basecbs 17151 -gcsg 18863 LSSumclsm 19552 LModclmod 20704 LSubSpclss 20776 LSpanclspn 20816 HLchlt 38731 LHypclh 39366 DVecHcdvh 40460 LCDualclcd 40968 mapdcmpd 41006 |
| 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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-riotaBAD 38334 |
| 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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8209 df-undef 8256 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-0g 17394 df-mre 17537 df-mrc 17538 df-acs 17540 df-proset 18258 df-poset 18276 df-plt 18293 df-lub 18309 df-glb 18310 df-join 18311 df-meet 18312 df-p0 18388 df-p1 18389 df-lat 18395 df-clat 18462 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19048 df-cntz 19231 df-oppg 19260 df-lsm 19554 df-cmn 19700 df-abl 19701 df-mgp 20038 df-rng 20056 df-ur 20085 df-ring 20138 df-oppr 20234 df-dvdsr 20257 df-unit 20258 df-invr 20288 df-dvr 20301 df-drng 20587 df-lmod 20706 df-lss 20777 df-lsp 20817 df-lvec 20949 df-lsatoms 38357 df-lshyp 38358 df-lcv 38400 df-lfl 38439 df-lkr 38467 df-ldual 38505 df-oposet 38557 df-ol 38559 df-oml 38560 df-covers 38647 df-ats 38648 df-atl 38679 df-cvlat 38703 df-hlat 38732 df-llines 38880 df-lplanes 38881 df-lvols 38882 df-lines 38883 df-psubsp 38885 df-pmap 38886 df-padd 39178 df-lhyp 39370 df-laut 39371 df-ldil 39486 df-ltrn 39487 df-trl 39541 df-tgrp 40125 df-tendo 40137 df-edring 40139 df-dveca 40385 df-disoa 40411 df-dvech 40461 df-dib 40521 df-dic 40555 df-dih 40611 df-doch 40730 df-djh 40777 df-lcdual 40969 df-mapd 41007 |
| This theorem is referenced by: mapdpglem2 41055 |
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