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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 41131. 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 40494 | . . . 4 β’ (π β π β LMod) |
21 | mapdh6d.w | . . . . 5 β’ (π β π€ β (π β { 0 })) | |
22 | 21 | eldifad 3955 | . . . 4 β’ (π β π€ β π) |
23 | mapdh6d.y | . . . . 5 β’ (π β π β (π β { 0 })) | |
24 | 23 | eldifad 3955 | . . . 4 β’ (π β π β π) |
25 | 6, 18 | lmodvacl 20721 | . . . 4 β’ ((π β LMod β§ π€ β π β§ π β π) β (π€ + π) β π) |
26 | 20, 22, 24, 25 | syl3anc 1368 | . . 3 β’ (π β (π€ + π) β π) |
27 | 3, 5, 14 | dvhlvec 40493 | . . . . . 6 β’ (π β π β LVec) |
28 | 17 | eldifad 3955 | . . . . . 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 20983 | . . . . 5 β’ (π β ((πβ{π€}) β (πβ{π}) β§ (πβ{π€}) β (πβ{π}))) |
31 | 30 | simprd 495 | . . . 4 β’ (π β (πβ{π€}) β (πβ{π})) |
32 | 6, 18, 8, 9, 20, 22, 24, 31 | lmodindp1 20861 | . . 3 β’ (π β (π€ + π) β 0 ) |
33 | eldifsn 4785 | . . 3 β’ ((π€ + π) β (π β { 0 }) β ((π€ + π) β π β§ (π€ + π) β 0 )) | |
34 | 26, 32, 33 | sylanbrc 582 | . 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 3955 | . . . . 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 20983 | . . . . . . 7 β’ (π β ((πβ{π}) β (πβ{π}) β§ (πβ{π}) β (πβ{π}))) |
40 | 39 | simpld 494 | . . . . . 6 β’ (π β (πβ{π}) β (πβ{π})) |
41 | 6, 18, 8, 9, 27, 17, 23, 35, 21, 37, 40, 29 | mapdindp3 41106 | . . . . 5 β’ (π β (πβ{π}) β (πβ{(π€ + π)})) |
42 | 6, 18, 8, 9, 27, 17, 23, 35, 21, 37, 40, 29 | mapdindp4 41107 | . . . . 5 β’ (π β Β¬ π β (πβ{π, (π€ + π)})) |
43 | 6, 8, 9, 27, 17, 26, 36, 41, 42 | lspindp1 20984 | . . . 4 β’ (π β ((πβ{π}) β (πβ{(π€ + π)}) β§ Β¬ π β (πβ{π, (π€ + π)}))) |
44 | 43 | simprd 495 | . . 3 β’ (π β Β¬ π β (πβ{π, (π€ + π)})) |
45 | prcom 4731 | . . . . 5 β’ {(π€ + π), π} = {π, (π€ + π)} | |
46 | 45 | fveq2i 6888 | . . . 4 β’ (πβ{(π€ + π), π}) = (πβ{π, (π€ + π)}) |
47 | 46 | eleq2i 2819 | . . 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 20983 | . . . 4 β’ (π β ((πβ{π}) β (πβ{π}) β§ (πβ{π}) β (πβ{(π€ + π)}))) |
50 | 49 | simprd 495 | . . 3 β’ (π β (πβ{π}) β (πβ{(π€ + π)})) |
51 | 50 | necomd 2990 | . 2 β’ (π β (πβ{(π€ + π)}) β (πβ{π})) |
52 | eqidd 2727 | . 2 β’ (π β (πΌββ¨π, πΉ, (π€ + π)β©) = (πΌββ¨π, πΉ, (π€ + π)β©)) | |
53 | eqidd 2727 | . 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 41119 | 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 395 = wceq 1533 β wcel 2098 β wne 2934 Vcvv 3468 β cdif 3940 ifcif 4523 {csn 4623 {cpr 4625 β¨cotp 4631 β¦ cmpt 5224 βcfv 6537 β©crio 7360 (class class class)co 7405 1st c1st 7972 2nd c2nd 7973 Basecbs 17153 +gcplusg 17206 0gc0g 17394 -gcsg 18865 LModclmod 20706 LSpanclspn 20818 HLchlt 38733 LHypclh 39368 DVecHcdvh 40462 LCDualclcd 40970 mapdcmpd 41008 |
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 7722 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 38336 |
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-ot 4632 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 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-tpos 8212 df-undef 8259 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 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 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-0g 17396 df-mre 17539 df-mrc 17540 df-acs 17542 df-proset 18260 df-poset 18278 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-p1 18391 df-lat 18397 df-clat 18464 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19050 df-cntz 19233 df-oppg 19262 df-lsm 19556 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-dvr 20303 df-drng 20589 df-lmod 20708 df-lss 20779 df-lsp 20819 df-lvec 20951 df-lsatoms 38359 df-lshyp 38360 df-lcv 38402 df-lfl 38441 df-lkr 38469 df-ldual 38507 df-oposet 38559 df-ol 38561 df-oml 38562 df-covers 38649 df-ats 38650 df-atl 38681 df-cvlat 38705 df-hlat 38734 df-llines 38882 df-lplanes 38883 df-lvols 38884 df-lines 38885 df-psubsp 38887 df-pmap 38888 df-padd 39180 df-lhyp 39372 df-laut 39373 df-ldil 39488 df-ltrn 39489 df-trl 39543 df-tgrp 40127 df-tendo 40139 df-edring 40141 df-dveca 40387 df-disoa 40413 df-dvech 40463 df-dib 40523 df-dic 40557 df-dih 40613 df-doch 40732 df-djh 40779 df-lcdual 40971 df-mapd 41009 |
This theorem is referenced by: mapdh6gN 41126 |
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