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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh6eN | Structured version Visualization version GIF version |
Description: Lemmma for mapdh6N 40256. Part (6) in [Baer] p. 47 line 38. (Contributed by NM, 1-May-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mapdh.q | β’ π = (0gβπΆ) |
mapdh.i | β’ πΌ = (π₯ β V β¦ if((2nd βπ₯) = 0 , π, (β©β β π· ((πβ(πβ{(2nd βπ₯)})) = (π½β{β}) β§ (πβ(πβ{((1st β(1st βπ₯)) β (2nd βπ₯))})) = (π½β{((2nd β(1st βπ₯))π β)}))))) |
mapdh.h | β’ π» = (LHypβπΎ) |
mapdh.m | β’ π = ((mapdβπΎ)βπ) |
mapdh.u | β’ π = ((DVecHβπΎ)βπ) |
mapdh.v | β’ π = (Baseβπ) |
mapdh.s | β’ β = (-gβπ) |
mapdhc.o | β’ 0 = (0gβπ) |
mapdh.n | β’ π = (LSpanβπ) |
mapdh.c | β’ πΆ = ((LCDualβπΎ)βπ) |
mapdh.d | β’ π· = (BaseβπΆ) |
mapdh.r | β’ π = (-gβπΆ) |
mapdh.j | β’ π½ = (LSpanβπΆ) |
mapdh.k | β’ (π β (πΎ β HL β§ π β π»)) |
mapdhc.f | β’ (π β πΉ β π·) |
mapdh.mn | β’ (π β (πβ(πβ{π})) = (π½β{πΉ})) |
mapdhcl.x | β’ (π β π β (π β { 0 })) |
mapdh.p | β’ + = (+gβπ) |
mapdh.a | β’ β = (+gβπΆ) |
mapdh6d.xn | β’ (π β Β¬ π β (πβ{π, π})) |
mapdh6d.yz | β’ (π β (πβ{π}) = (πβ{π})) |
mapdh6d.y | β’ (π β π β (π β { 0 })) |
mapdh6d.z | β’ (π β π β (π β { 0 })) |
mapdh6d.w | β’ (π β π€ β (π β { 0 })) |
mapdh6d.wn | β’ (π β Β¬ π€ β (πβ{π, π})) |
Ref | Expression |
---|---|
mapdh6eN | β’ (π β (πΌββ¨π, πΉ, ((π€ + π) + π)β©) = ((πΌββ¨π, πΉ, (π€ + π)β©) β (πΌββ¨π, πΉ, πβ©))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdh.q | . 2 β’ π = (0gβπΆ) | |
2 | mapdh.i | . 2 β’ πΌ = (π₯ β V β¦ if((2nd βπ₯) = 0 , π, (β©β β π· ((πβ(πβ{(2nd βπ₯)})) = (π½β{β}) β§ (πβ(πβ{((1st β(1st βπ₯)) β (2nd βπ₯))})) = (π½β{((2nd β(1st βπ₯))π β)}))))) | |
3 | mapdh.h | . 2 β’ π» = (LHypβπΎ) | |
4 | mapdh.m | . 2 β’ π = ((mapdβπΎ)βπ) | |
5 | mapdh.u | . 2 β’ π = ((DVecHβπΎ)βπ) | |
6 | mapdh.v | . 2 β’ π = (Baseβπ) | |
7 | mapdh.s | . 2 β’ β = (-gβπ) | |
8 | mapdhc.o | . 2 β’ 0 = (0gβπ) | |
9 | mapdh.n | . 2 β’ π = (LSpanβπ) | |
10 | mapdh.c | . 2 β’ πΆ = ((LCDualβπΎ)βπ) | |
11 | mapdh.d | . 2 β’ π· = (BaseβπΆ) | |
12 | mapdh.r | . 2 β’ π = (-gβπΆ) | |
13 | mapdh.j | . 2 β’ π½ = (LSpanβπΆ) | |
14 | mapdh.k | . 2 β’ (π β (πΎ β HL β§ π β π»)) | |
15 | mapdhc.f | . 2 β’ (π β πΉ β π·) | |
16 | mapdh.mn | . 2 β’ (π β (πβ(πβ{π})) = (π½β{πΉ})) | |
17 | mapdhcl.x | . 2 β’ (π β π β (π β { 0 })) | |
18 | mapdh.p | . 2 β’ + = (+gβπ) | |
19 | mapdh.a | . 2 β’ β = (+gβπΆ) | |
20 | 3, 5, 14 | dvhlmod 39619 | . . . 4 β’ (π β π β LMod) |
21 | mapdh6d.w | . . . . 5 β’ (π β π€ β (π β { 0 })) | |
22 | 21 | eldifad 3923 | . . . 4 β’ (π β π€ β π) |
23 | mapdh6d.y | . . . . 5 β’ (π β π β (π β { 0 })) | |
24 | 23 | eldifad 3923 | . . . 4 β’ (π β π β π) |
25 | 6, 18 | lmodvacl 20351 | . . . 4 β’ ((π β LMod β§ π€ β π β§ π β π) β (π€ + π) β π) |
26 | 20, 22, 24, 25 | syl3anc 1372 | . . 3 β’ (π β (π€ + π) β π) |
27 | 3, 5, 14 | dvhlvec 39618 | . . . . . 6 β’ (π β π β LVec) |
28 | 17 | eldifad 3923 | . . . . . 6 β’ (π β π β π) |
29 | mapdh6d.wn | . . . . . 6 β’ (π β Β¬ π€ β (πβ{π, π})) | |
30 | 6, 9, 27, 22, 28, 24, 29 | lspindpi 20609 | . . . . 5 β’ (π β ((πβ{π€}) β (πβ{π}) β§ (πβ{π€}) β (πβ{π}))) |
31 | 30 | simprd 497 | . . . 4 β’ (π β (πβ{π€}) β (πβ{π})) |
32 | 6, 18, 8, 9, 20, 22, 24, 31 | lmodindp1 20490 | . . 3 β’ (π β (π€ + π) β 0 ) |
33 | eldifsn 4748 | . . 3 β’ ((π€ + π) β (π β { 0 }) β ((π€ + π) β π β§ (π€ + π) β 0 )) | |
34 | 26, 32, 33 | sylanbrc 584 | . 2 β’ (π β (π€ + π) β (π β { 0 })) |
35 | mapdh6d.z | . 2 β’ (π β π β (π β { 0 })) | |
36 | 35 | eldifad 3923 | . . . . 5 β’ (π β π β π) |
37 | mapdh6d.yz | . . . . . 6 β’ (π β (πβ{π}) = (πβ{π})) | |
38 | mapdh6d.xn | . . . . . . . 8 β’ (π β Β¬ π β (πβ{π, π})) | |
39 | 6, 9, 27, 28, 24, 36, 38 | lspindpi 20609 | . . . . . . 7 β’ (π β ((πβ{π}) β (πβ{π}) β§ (πβ{π}) β (πβ{π}))) |
40 | 39 | simpld 496 | . . . . . 6 β’ (π β (πβ{π}) β (πβ{π})) |
41 | 6, 18, 8, 9, 27, 17, 23, 35, 21, 37, 40, 29 | mapdindp3 40231 | . . . . 5 β’ (π β (πβ{π}) β (πβ{(π€ + π)})) |
42 | 6, 18, 8, 9, 27, 17, 23, 35, 21, 37, 40, 29 | mapdindp4 40232 | . . . . 5 β’ (π β Β¬ π β (πβ{π, (π€ + π)})) |
43 | 6, 8, 9, 27, 17, 26, 36, 41, 42 | lspindp1 20610 | . . . 4 β’ (π β ((πβ{π}) β (πβ{(π€ + π)}) β§ Β¬ π β (πβ{π, (π€ + π)}))) |
44 | 43 | simprd 497 | . . 3 β’ (π β Β¬ π β (πβ{π, (π€ + π)})) |
45 | prcom 4694 | . . . . 5 β’ {(π€ + π), π} = {π, (π€ + π)} | |
46 | 45 | fveq2i 6846 | . . . 4 β’ (πβ{(π€ + π), π}) = (πβ{π, (π€ + π)}) |
47 | 46 | eleq2i 2826 | . . 3 β’ (π β (πβ{(π€ + π), π}) β π β (πβ{π, (π€ + π)})) |
48 | 44, 47 | sylnibr 329 | . 2 β’ (π β Β¬ π β (πβ{(π€ + π), π})) |
49 | 6, 9, 27, 36, 28, 26, 42 | lspindpi 20609 | . . . 4 β’ (π β ((πβ{π}) β (πβ{π}) β§ (πβ{π}) β (πβ{(π€ + π)}))) |
50 | 49 | simprd 497 | . . 3 β’ (π β (πβ{π}) β (πβ{(π€ + π)})) |
51 | 50 | necomd 2996 | . 2 β’ (π β (πβ{(π€ + π)}) β (πβ{π})) |
52 | eqidd 2734 | . 2 β’ (π β (πΌββ¨π, πΉ, (π€ + π)β©) = (πΌββ¨π, πΉ, (π€ + π)β©)) | |
53 | eqidd 2734 | . 2 β’ (π β (πΌββ¨π, πΉ, πβ©) = (πΌββ¨π, πΉ, πβ©)) | |
54 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 34, 35, 48, 51, 52, 53 | mapdh6aN 40244 | 1 β’ (π β (πΌββ¨π, πΉ, ((π€ + π) + π)β©) = ((πΌββ¨π, πΉ, (π€ + π)β©) β (πΌββ¨π, πΉ, πβ©))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2940 Vcvv 3444 β cdif 3908 ifcif 4487 {csn 4587 {cpr 4589 β¨cotp 4595 β¦ cmpt 5189 βcfv 6497 β©crio 7313 (class class class)co 7358 1st c1st 7920 2nd c2nd 7921 Basecbs 17088 +gcplusg 17138 0gc0g 17326 -gcsg 18755 LModclmod 20336 LSpanclspn 20447 HLchlt 37858 LHypclh 38493 DVecHcdvh 39587 LCDualclcd 40095 mapdcmpd 40133 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-riotaBAD 37461 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-ot 4596 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-tpos 8158 df-undef 8205 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-sca 17154 df-vsca 17155 df-0g 17328 df-mre 17471 df-mrc 17472 df-acs 17474 df-proset 18189 df-poset 18207 df-plt 18224 df-lub 18240 df-glb 18241 df-join 18242 df-meet 18243 df-p0 18319 df-p1 18320 df-lat 18326 df-clat 18393 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-submnd 18607 df-grp 18756 df-minusg 18757 df-sbg 18758 df-subg 18930 df-cntz 19102 df-oppg 19129 df-lsm 19423 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-ring 19971 df-oppr 20054 df-dvdsr 20075 df-unit 20076 df-invr 20106 df-dvr 20117 df-drng 20199 df-lmod 20338 df-lss 20408 df-lsp 20448 df-lvec 20579 df-lsatoms 37484 df-lshyp 37485 df-lcv 37527 df-lfl 37566 df-lkr 37594 df-ldual 37632 df-oposet 37684 df-ol 37686 df-oml 37687 df-covers 37774 df-ats 37775 df-atl 37806 df-cvlat 37830 df-hlat 37859 df-llines 38007 df-lplanes 38008 df-lvols 38009 df-lines 38010 df-psubsp 38012 df-pmap 38013 df-padd 38305 df-lhyp 38497 df-laut 38498 df-ldil 38613 df-ltrn 38614 df-trl 38668 df-tgrp 39252 df-tendo 39264 df-edring 39266 df-dveca 39512 df-disoa 39538 df-dvech 39588 df-dib 39648 df-dic 39682 df-dih 39738 df-doch 39857 df-djh 39904 df-lcdual 40096 df-mapd 40134 |
This theorem is referenced by: mapdh6gN 40251 |
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