![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem2 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 40663. Baer p. 45, lines 1 and 2: "we have (F(x-y))* = Gt where t necessarily belongs to (Fx)*+(Fy)*." (We scope $d π‘π locally to avoid clashes with later substitutions into π.) (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) |
mapdpglem2.j | β’ π½ = (LSpanβπΆ) |
Ref | Expression |
---|---|
mapdpglem2 | β’ (π β βπ‘ β ((πβ(πβ{π})) β (πβ(πβ{π})))(πβ(πβ{(π β π)})) = (π½β{π‘})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpglem.h | . . . 4 β’ π» = (LHypβπΎ) | |
2 | mapdpglem.m | . . . 4 β’ π = ((mapdβπΎ)βπ) | |
3 | mapdpglem.u | . . . 4 β’ π = ((DVecHβπΎ)βπ) | |
4 | mapdpglem.v | . . . 4 β’ π = (Baseβπ) | |
5 | mapdpglem.n | . . . 4 β’ π = (LSpanβπ) | |
6 | mapdpglem.c | . . . 4 β’ πΆ = ((LCDualβπΎ)βπ) | |
7 | eqid 2732 | . . . 4 β’ (BaseβπΆ) = (BaseβπΆ) | |
8 | mapdpglem2.j | . . . 4 β’ π½ = (LSpanβπΆ) | |
9 | mapdpglem.k | . . . 4 β’ (π β (πΎ β HL β§ π β π»)) | |
10 | 1, 3, 9 | dvhlmod 40067 | . . . . 5 β’ (π β π β LMod) |
11 | mapdpglem.x | . . . . 5 β’ (π β π β π) | |
12 | mapdpglem.y | . . . . 5 β’ (π β π β π) | |
13 | mapdpglem.s | . . . . . 6 β’ β = (-gβπ) | |
14 | 4, 13 | lmodvsubcl 20522 | . . . . 5 β’ ((π β LMod β§ π β π β§ π β π) β (π β π) β π) |
15 | 10, 11, 12, 14 | syl3anc 1371 | . . . 4 β’ (π β (π β π) β π) |
16 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 15 | mapdspex 40625 | . . 3 β’ (π β βπ‘ β (BaseβπΆ)(πβ(πβ{(π β π)})) = (π½β{π‘})) |
17 | 1, 6, 9 | lcdlmod 40549 | . . . . . 6 β’ (π β πΆ β LMod) |
18 | 7, 8 | lspsnid 20609 | . . . . . 6 β’ ((πΆ β LMod β§ π‘ β (BaseβπΆ)) β π‘ β (π½β{π‘})) |
19 | 17, 18 | sylan 580 | . . . . 5 β’ ((π β§ π‘ β (BaseβπΆ)) β π‘ β (π½β{π‘})) |
20 | 19 | adantrr 715 | . . . 4 β’ ((π β§ (π‘ β (BaseβπΆ) β§ (πβ(πβ{(π β π)})) = (π½β{π‘}))) β π‘ β (π½β{π‘})) |
21 | simprr 771 | . . . 4 β’ ((π β§ (π‘ β (BaseβπΆ) β§ (πβ(πβ{(π β π)})) = (π½β{π‘}))) β (πβ(πβ{(π β π)})) = (π½β{π‘})) | |
22 | 20, 21 | eleqtrrd 2836 | . . 3 β’ ((π β§ (π‘ β (BaseβπΆ) β§ (πβ(πβ{(π β π)})) = (π½β{π‘}))) β π‘ β (πβ(πβ{(π β π)}))) |
23 | 16, 22, 21 | reximssdv 3172 | . 2 β’ (π β βπ‘ β (πβ(πβ{(π β π)}))(πβ(πβ{(π β π)})) = (π½β{π‘})) |
24 | mapdpglem1.p | . . . . . 6 β’ β = (LSSumβπΆ) | |
25 | 1, 2, 3, 4, 13, 5, 6, 9, 11, 12, 24 | mapdpglem1 40629 | . . . . 5 β’ (π β (πβ(πβ{(π β π)})) β ((πβ(πβ{π})) β (πβ(πβ{π})))) |
26 | 25 | sseld 3981 | . . . 4 β’ (π β (π‘ β (πβ(πβ{(π β π)})) β π‘ β ((πβ(πβ{π})) β (πβ(πβ{π}))))) |
27 | 26 | anim1d 611 | . . 3 β’ (π β ((π‘ β (πβ(πβ{(π β π)})) β§ (πβ(πβ{(π β π)})) = (π½β{π‘})) β (π‘ β ((πβ(πβ{π})) β (πβ(πβ{π}))) β§ (πβ(πβ{(π β π)})) = (π½β{π‘})))) |
28 | 27 | reximdv2 3164 | . 2 β’ (π β (βπ‘ β (πβ(πβ{(π β π)}))(πβ(πβ{(π β π)})) = (π½β{π‘}) β βπ‘ β ((πβ(πβ{π})) β (πβ(πβ{π})))(πβ(πβ{(π β π)})) = (π½β{π‘}))) |
29 | 23, 28 | mpd 15 | 1 β’ (π β βπ‘ β ((πβ(πβ{π})) β (πβ(πβ{π})))(πβ(πβ{(π β π)})) = (π½β{π‘})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βwrex 3070 {csn 4628 βcfv 6543 (class class class)co 7411 Basecbs 17146 -gcsg 18823 LSSumclsm 19504 LModclmod 20475 LSpanclspn 20587 HLchlt 38306 LHypclh 38941 DVecHcdvh 40035 LCDualclcd 40543 mapdcmpd 40581 |
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 ax-riotaBAD 37909 |
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-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 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-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 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-n0 12475 df-z 12561 df-uz 12825 df-fz 13487 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-sca 17215 df-vsca 17216 df-0g 17389 df-mre 17532 df-mrc 17533 df-acs 17535 df-proset 18250 df-poset 18268 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-p1 18381 df-lat 18387 df-clat 18454 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-submnd 18674 df-grp 18824 df-minusg 18825 df-sbg 18826 df-subg 19005 df-cntz 19183 df-oppg 19212 df-lsm 19506 df-cmn 19652 df-abl 19653 df-mgp 19990 df-ur 20007 df-ring 20060 df-oppr 20154 df-dvdsr 20175 df-unit 20176 df-invr 20206 df-dvr 20219 df-drng 20363 df-lmod 20477 df-lss 20548 df-lsp 20588 df-lvec 20719 df-lsatoms 37932 df-lshyp 37933 df-lcv 37975 df-lfl 38014 df-lkr 38042 df-ldual 38080 df-oposet 38132 df-ol 38134 df-oml 38135 df-covers 38222 df-ats 38223 df-atl 38254 df-cvlat 38278 df-hlat 38307 df-llines 38455 df-lplanes 38456 df-lvols 38457 df-lines 38458 df-psubsp 38460 df-pmap 38461 df-padd 38753 df-lhyp 38945 df-laut 38946 df-ldil 39061 df-ltrn 39062 df-trl 39116 df-tgrp 39700 df-tendo 39712 df-edring 39714 df-dveca 39960 df-disoa 39986 df-dvech 40036 df-dib 40096 df-dic 40130 df-dih 40186 df-doch 40305 df-djh 40352 df-lcdual 40544 df-mapd 40582 |
This theorem is referenced by: mapdpglem24 40661 |
Copyright terms: Public domain | W3C validator |