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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh6cN | Structured version Visualization version GIF version |
Description: Lemmma for mapdh6N 41121. (Contributed by NM, 24-Apr-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) |
mapdh6c.y | β’ (π β π β π) |
mapdh6c.z | β’ (π β π = 0 ) |
mapdh6c.ne | β’ (π β Β¬ π β (πβ{π, π})) |
Ref | Expression |
---|---|
mapdh6cN | β’ (π β (πΌββ¨π, πΉ, (π + π)β©) = ((πΌββ¨π, πΉ, πβ©) β (πΌββ¨π, πΉ, πβ©))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdh.h | . . . . 5 β’ π» = (LHypβπΎ) | |
2 | mapdh.c | . . . . 5 β’ πΆ = ((LCDualβπΎ)βπ) | |
3 | mapdh.k | . . . . 5 β’ (π β (πΎ β HL β§ π β π»)) | |
4 | 1, 2, 3 | lcdlmod 40966 | . . . 4 β’ (π β πΆ β LMod) |
5 | lmodgrp 20709 | . . . 4 β’ (πΆ β LMod β πΆ β Grp) | |
6 | 4, 5 | syl 17 | . . 3 β’ (π β πΆ β Grp) |
7 | mapdh.q | . . . 4 β’ π = (0gβπΆ) | |
8 | mapdh.i | . . . 4 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)}))))) | |
9 | mapdh.m | . . . 4 β’ π = ((mapdβπΎ)βπ) | |
10 | mapdh.u | . . . 4 β’ π = ((DVecHβπΎ)βπ) | |
11 | mapdh.v | . . . 4 β’ π = (Baseβπ) | |
12 | mapdh.s | . . . 4 β’ β = (-gβπ) | |
13 | mapdhc.o | . . . 4 β’ 0 = (0gβπ) | |
14 | mapdh.n | . . . 4 β’ π = (LSpanβπ) | |
15 | mapdh.d | . . . 4 β’ π· = (BaseβπΆ) | |
16 | mapdh.r | . . . 4 β’ π = (-gβπΆ) | |
17 | mapdh.j | . . . 4 β’ π½ = (LSpanβπΆ) | |
18 | mapdhc.f | . . . 4 β’ (π β πΉ β π·) | |
19 | mapdh.mn | . . . 4 β’ (π β (πβ(πβ{π})) = (π½β{πΉ})) | |
20 | mapdhcl.x | . . . 4 β’ (π β π β (π β { 0 })) | |
21 | mapdh6c.y | . . . 4 β’ (π β π β π) | |
22 | 1, 10, 3 | dvhlvec 40483 | . . . . . 6 β’ (π β π β LVec) |
23 | 20 | eldifad 3953 | . . . . . 6 β’ (π β π β π) |
24 | mapdh6c.z | . . . . . . 7 β’ (π β π = 0 ) | |
25 | 1, 10, 3 | dvhlmod 40484 | . . . . . . . 8 β’ (π β π β LMod) |
26 | 11, 13 | lmod0vcl 20733 | . . . . . . . 8 β’ (π β LMod β 0 β π) |
27 | 25, 26 | syl 17 | . . . . . . 7 β’ (π β 0 β π) |
28 | 24, 27 | eqeltrd 2825 | . . . . . 6 β’ (π β π β π) |
29 | mapdh6c.ne | . . . . . 6 β’ (π β Β¬ π β (πβ{π, π})) | |
30 | 11, 14, 22, 23, 21, 28, 29 | lspindpi 20979 | . . . . 5 β’ (π β ((πβ{π}) β (πβ{π}) β§ (πβ{π}) β (πβ{π}))) |
31 | 30 | simpld 494 | . . . 4 β’ (π β (πβ{π}) β (πβ{π})) |
32 | 7, 8, 1, 9, 10, 11, 12, 13, 14, 2, 15, 16, 17, 3, 18, 19, 20, 21, 31 | mapdhcl 41101 | . . 3 β’ (π β (πΌββ¨π, πΉ, πβ©) β π·) |
33 | mapdh.a | . . . 4 β’ β = (+gβπΆ) | |
34 | 15, 33, 7 | grprid 18894 | . . 3 β’ ((πΆ β Grp β§ (πΌββ¨π, πΉ, πβ©) β π·) β ((πΌββ¨π, πΉ, πβ©) β π) = (πΌββ¨π, πΉ, πβ©)) |
35 | 6, 32, 34 | syl2anc 583 | . 2 β’ (π β ((πΌββ¨π, πΉ, πβ©) β π) = (πΌββ¨π, πΉ, πβ©)) |
36 | 24 | oteq3d 4880 | . . . . 5 β’ (π β β¨π, πΉ, πβ© = β¨π, πΉ, 0 β©) |
37 | 36 | fveq2d 6886 | . . . 4 β’ (π β (πΌββ¨π, πΉ, πβ©) = (πΌββ¨π, πΉ, 0 β©)) |
38 | 7, 8, 13, 20, 18 | mapdhval0 41099 | . . . 4 β’ (π β (πΌββ¨π, πΉ, 0 β©) = π) |
39 | 37, 38 | eqtrd 2764 | . . 3 β’ (π β (πΌββ¨π, πΉ, πβ©) = π) |
40 | 39 | oveq2d 7418 | . 2 β’ (π β ((πΌββ¨π, πΉ, πβ©) β (πΌββ¨π, πΉ, πβ©)) = ((πΌββ¨π, πΉ, πβ©) β π)) |
41 | 24 | oveq2d 7418 | . . . . 5 β’ (π β (π + π) = (π + 0 )) |
42 | lmodgrp 20709 | . . . . . . 7 β’ (π β LMod β π β Grp) | |
43 | 25, 42 | syl 17 | . . . . . 6 β’ (π β π β Grp) |
44 | mapdh.p | . . . . . . 7 β’ + = (+gβπ) | |
45 | 11, 44, 13 | grprid 18894 | . . . . . 6 β’ ((π β Grp β§ π β π) β (π + 0 ) = π) |
46 | 43, 21, 45 | syl2anc 583 | . . . . 5 β’ (π β (π + 0 ) = π) |
47 | 41, 46 | eqtrd 2764 | . . . 4 β’ (π β (π + π) = π) |
48 | 47 | oteq3d 4880 | . . 3 β’ (π β β¨π, πΉ, (π + π)β© = β¨π, πΉ, πβ©) |
49 | 48 | fveq2d 6886 | . 2 β’ (π β (πΌββ¨π, πΉ, (π + π)β©) = (πΌββ¨π, πΉ, πβ©)) |
50 | 35, 40, 49 | 3eqtr4rd 2775 | 1 β’ (π β (πΌββ¨π, πΉ, (π + π)β©) = ((πΌββ¨π, πΉ, πβ©) β (πΌββ¨π, πΉ, πβ©))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wne 2932 Vcvv 3466 β cdif 3938 ifcif 4521 {csn 4621 {cpr 4623 β¨cotp 4629 β¦ cmpt 5222 βcfv 6534 β©crio 7357 (class class class)co 7402 1st c1st 7967 2nd c2nd 7968 Basecbs 17149 +gcplusg 17202 0gc0g 17390 Grpcgrp 18859 -gcsg 18861 LModclmod 20702 LSpanclspn 20814 HLchlt 38723 LHypclh 39358 DVecHcdvh 40452 LCDualclcd 40960 mapdcmpd 40998 |
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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-riotaBAD 38326 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-ot 4630 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8207 df-undef 8254 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13486 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-sca 17218 df-vsca 17219 df-0g 17392 df-mre 17535 df-mrc 17536 df-acs 17538 df-proset 18256 df-poset 18274 df-plt 18291 df-lub 18307 df-glb 18308 df-join 18309 df-meet 18310 df-p0 18386 df-p1 18387 df-lat 18393 df-clat 18460 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18710 df-grp 18862 df-minusg 18863 df-sbg 18864 df-subg 19046 df-cntz 19229 df-oppg 19258 df-lsm 19552 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-drng 20585 df-lmod 20704 df-lss 20775 df-lsp 20815 df-lvec 20947 df-lsatoms 38349 df-lshyp 38350 df-lcv 38392 df-lfl 38431 df-lkr 38459 df-ldual 38497 df-oposet 38549 df-ol 38551 df-oml 38552 df-covers 38639 df-ats 38640 df-atl 38671 df-cvlat 38695 df-hlat 38724 df-llines 38872 df-lplanes 38873 df-lvols 38874 df-lines 38875 df-psubsp 38877 df-pmap 38878 df-padd 39170 df-lhyp 39362 df-laut 39363 df-ldil 39478 df-ltrn 39479 df-trl 39533 df-tgrp 40117 df-tendo 40129 df-edring 40131 df-dveca 40377 df-disoa 40403 df-dvech 40453 df-dib 40513 df-dic 40547 df-dih 40603 df-doch 40722 df-djh 40769 df-lcdual 40961 df-mapd 40999 |
This theorem is referenced by: mapdh6kN 41120 |
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