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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem8 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 40880. Baer p. 45, line 4: "...so that (F(x-y))* <= (Fy)*. This would imply that F(x-y) <= F(y)..." (Contributed by NM, 20-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) |
mapdpglem2.j | β’ π½ = (LSpanβπΆ) |
mapdpglem3.f | β’ πΉ = (BaseβπΆ) |
mapdpglem3.te | β’ (π β π‘ β ((πβ(πβ{π})) β (πβ(πβ{π})))) |
mapdpglem3.a | β’ π΄ = (Scalarβπ) |
mapdpglem3.b | β’ π΅ = (Baseβπ΄) |
mapdpglem3.t | β’ Β· = ( Β·π βπΆ) |
mapdpglem3.r | β’ π = (-gβπΆ) |
mapdpglem3.g | β’ (π β πΊ β πΉ) |
mapdpglem3.e | β’ (π β (πβ(πβ{π})) = (π½β{πΊ})) |
mapdpglem4.q | β’ π = (0gβπ) |
mapdpglem.ne | β’ (π β (πβ{π}) β (πβ{π})) |
mapdpglem4.jt | β’ (π β (πβ(πβ{(π β π)})) = (π½β{π‘})) |
mapdpglem4.z | β’ 0 = (0gβπ΄) |
mapdpglem4.g4 | β’ (π β π β π΅) |
mapdpglem4.z4 | β’ (π β π§ β (πβ(πβ{π}))) |
mapdpglem4.t4 | β’ (π β π‘ = ((π Β· πΊ)π π§)) |
mapdpglem4.xn | β’ (π β π β π) |
mapdpglem4.g0 | β’ (π β π = 0 ) |
Ref | Expression |
---|---|
mapdpglem8 | β’ (π β (πβ{(π β π)}) β (πβ{π})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpglem4.jt | . . 3 β’ (π β (πβ(πβ{(π β π)})) = (π½β{π‘})) | |
2 | eqid 2730 | . . . 4 β’ (LSubSpβπΆ) = (LSubSpβπΆ) | |
3 | mapdpglem2.j | . . . 4 β’ π½ = (LSpanβπΆ) | |
4 | mapdpglem.h | . . . . 5 β’ π» = (LHypβπΎ) | |
5 | mapdpglem.c | . . . . 5 β’ πΆ = ((LCDualβπΎ)βπ) | |
6 | mapdpglem.k | . . . . 5 β’ (π β (πΎ β HL β§ π β π»)) | |
7 | 4, 5, 6 | lcdlmod 40766 | . . . 4 β’ (π β πΆ β LMod) |
8 | mapdpglem.m | . . . . 5 β’ π = ((mapdβπΎ)βπ) | |
9 | mapdpglem.u | . . . . 5 β’ π = ((DVecHβπΎ)βπ) | |
10 | eqid 2730 | . . . . 5 β’ (LSubSpβπ) = (LSubSpβπ) | |
11 | 4, 9, 6 | dvhlmod 40284 | . . . . . 6 β’ (π β π β LMod) |
12 | mapdpglem.y | . . . . . 6 β’ (π β π β π) | |
13 | mapdpglem.v | . . . . . . 7 β’ π = (Baseβπ) | |
14 | mapdpglem.n | . . . . . . 7 β’ π = (LSpanβπ) | |
15 | 13, 10, 14 | lspsncl 20732 | . . . . . 6 β’ ((π β LMod β§ π β π) β (πβ{π}) β (LSubSpβπ)) |
16 | 11, 12, 15 | syl2anc 582 | . . . . 5 β’ (π β (πβ{π}) β (LSubSpβπ)) |
17 | 4, 8, 9, 10, 5, 2, 6, 16 | mapdcl2 40830 | . . . 4 β’ (π β (πβ(πβ{π})) β (LSubSpβπΆ)) |
18 | mapdpglem.s | . . . . 5 β’ β = (-gβπ) | |
19 | mapdpglem.x | . . . . 5 β’ (π β π β π) | |
20 | mapdpglem1.p | . . . . 5 β’ β = (LSSumβπΆ) | |
21 | mapdpglem3.f | . . . . 5 β’ πΉ = (BaseβπΆ) | |
22 | mapdpglem3.te | . . . . 5 β’ (π β π‘ β ((πβ(πβ{π})) β (πβ(πβ{π})))) | |
23 | mapdpglem3.a | . . . . 5 β’ π΄ = (Scalarβπ) | |
24 | mapdpglem3.b | . . . . 5 β’ π΅ = (Baseβπ΄) | |
25 | mapdpglem3.t | . . . . 5 β’ Β· = ( Β·π βπΆ) | |
26 | mapdpglem3.r | . . . . 5 β’ π = (-gβπΆ) | |
27 | mapdpglem3.g | . . . . 5 β’ (π β πΊ β πΉ) | |
28 | mapdpglem3.e | . . . . 5 β’ (π β (πβ(πβ{π})) = (π½β{πΊ})) | |
29 | mapdpglem4.q | . . . . 5 β’ π = (0gβπ) | |
30 | mapdpglem.ne | . . . . 5 β’ (π β (πβ{π}) β (πβ{π})) | |
31 | mapdpglem4.z | . . . . 5 β’ 0 = (0gβπ΄) | |
32 | mapdpglem4.g4 | . . . . 5 β’ (π β π β π΅) | |
33 | mapdpglem4.z4 | . . . . 5 β’ (π β π§ β (πβ(πβ{π}))) | |
34 | mapdpglem4.t4 | . . . . 5 β’ (π β π‘ = ((π Β· πΊ)π π§)) | |
35 | mapdpglem4.xn | . . . . 5 β’ (π β π β π) | |
36 | mapdpglem4.g0 | . . . . 5 β’ (π β π = 0 ) | |
37 | 4, 8, 9, 13, 18, 14, 5, 6, 19, 12, 20, 3, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 1, 31, 32, 33, 34, 35, 36 | mapdpglem6 40852 | . . . 4 β’ (π β π‘ β (πβ(πβ{π}))) |
38 | 2, 3, 7, 17, 37 | lspsnel5a 20751 | . . 3 β’ (π β (π½β{π‘}) β (πβ(πβ{π}))) |
39 | 1, 38 | eqsstrd 4019 | . 2 β’ (π β (πβ(πβ{(π β π)})) β (πβ(πβ{π}))) |
40 | 13, 18 | lmodvsubcl 20661 | . . . . 5 β’ ((π β LMod β§ π β π β§ π β π) β (π β π) β π) |
41 | 11, 19, 12, 40 | syl3anc 1369 | . . . 4 β’ (π β (π β π) β π) |
42 | 13, 10, 14 | lspsncl 20732 | . . . 4 β’ ((π β LMod β§ (π β π) β π) β (πβ{(π β π)}) β (LSubSpβπ)) |
43 | 11, 41, 42 | syl2anc 582 | . . 3 β’ (π β (πβ{(π β π)}) β (LSubSpβπ)) |
44 | 4, 9, 10, 8, 6, 43, 16 | mapdord 40812 | . 2 β’ (π β ((πβ(πβ{(π β π)})) β (πβ(πβ{π})) β (πβ{(π β π)}) β (πβ{π}))) |
45 | 39, 44 | mpbid 231 | 1 β’ (π β (πβ{(π β π)}) β (πβ{π})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1539 β wcel 2104 β wne 2938 β wss 3947 {csn 4627 βcfv 6542 (class class class)co 7411 Basecbs 17148 Scalarcsca 17204 Β·π cvsca 17205 0gc0g 17389 -gcsg 18857 LSSumclsm 19543 LModclmod 20614 LSubSpclss 20686 LSpanclspn 20726 HLchlt 38523 LHypclh 39158 DVecHcdvh 40252 LCDualclcd 40760 mapdcmpd 40798 |
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-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 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 ax-riotaBAD 38126 |
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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-tpos 8213 df-undef 8260 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 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-3 12280 df-4 12281 df-5 12282 df-6 12283 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-sca 17217 df-vsca 17218 df-0g 17391 df-mre 17534 df-mrc 17535 df-acs 17537 df-proset 18252 df-poset 18270 df-plt 18287 df-lub 18303 df-glb 18304 df-join 18305 df-meet 18306 df-p0 18382 df-p1 18383 df-lat 18389 df-clat 18456 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-oppg 19251 df-lsm 19545 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-oppr 20225 df-dvdsr 20248 df-unit 20249 df-invr 20279 df-dvr 20292 df-drng 20502 df-lmod 20616 df-lss 20687 df-lsp 20727 df-lvec 20858 df-lsatoms 38149 df-lshyp 38150 df-lcv 38192 df-lfl 38231 df-lkr 38259 df-ldual 38297 df-oposet 38349 df-ol 38351 df-oml 38352 df-covers 38439 df-ats 38440 df-atl 38471 df-cvlat 38495 df-hlat 38524 df-llines 38672 df-lplanes 38673 df-lvols 38674 df-lines 38675 df-psubsp 38677 df-pmap 38678 df-padd 38970 df-lhyp 39162 df-laut 39163 df-ldil 39278 df-ltrn 39279 df-trl 39333 df-tgrp 39917 df-tendo 39929 df-edring 39931 df-dveca 40177 df-disoa 40203 df-dvech 40253 df-dib 40313 df-dic 40347 df-dih 40403 df-doch 40522 df-djh 40569 df-lcdual 40761 df-mapd 40799 |
This theorem is referenced by: mapdpglem9 40854 |
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