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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap14lem8 | Structured version Visualization version GIF version |
Description: Part of proof of part 14 in [Baer] p. 49 lines 33-35. (Contributed by NM, 1-Jun-2015.) |
Ref | Expression |
---|---|
hdmap14lem8.h | β’ π» = (LHypβπΎ) |
hdmap14lem8.u | β’ π = ((DVecHβπΎ)βπ) |
hdmap14lem8.v | β’ π = (Baseβπ) |
hdmap14lem8.q | β’ + = (+gβπ) |
hdmap14lem8.t | β’ Β· = ( Β·π βπ) |
hdmap14lem8.o | β’ 0 = (0gβπ) |
hdmap14lem8.n | β’ π = (LSpanβπ) |
hdmap14lem8.r | β’ π = (Scalarβπ) |
hdmap14lem8.b | β’ π΅ = (Baseβπ ) |
hdmap14lem8.c | β’ πΆ = ((LCDualβπΎ)βπ) |
hdmap14lem8.d | β’ β = (+gβπΆ) |
hdmap14lem8.e | β’ β = ( Β·π βπΆ) |
hdmap14lem8.p | β’ π = (ScalarβπΆ) |
hdmap14lem8.a | β’ π΄ = (Baseβπ) |
hdmap14lem8.s | β’ π = ((HDMapβπΎ)βπ) |
hdmap14lem8.k | β’ (π β (πΎ β HL β§ π β π»)) |
hdmap14lem8.x | β’ (π β π β (π β { 0 })) |
hdmap14lem8.y | β’ (π β π β (π β { 0 })) |
hdmap14lem8.f | β’ (π β πΉ β π΅) |
hdmap14lem8.g | β’ (π β πΊ β π΄) |
hdmap14lem8.i | β’ (π β πΌ β π΄) |
hdmap14lem8.xx | β’ (π β (πβ(πΉ Β· π)) = (πΊ β (πβπ))) |
hdmap14lem8.yy | β’ (π β (πβ(πΉ Β· π)) = (πΌ β (πβπ))) |
hdmap14lem8.ne | β’ (π β (πβ{π}) β (πβ{π})) |
hdmap14lem8.j | β’ (π β π½ β π΄) |
hdmap14lem8.xy | β’ (π β (πβ(πΉ Β· (π + π))) = (π½ β (πβ(π + π)))) |
Ref | Expression |
---|---|
hdmap14lem8 | β’ (π β ((π½ β (πβπ)) β (π½ β (πβπ))) = ((πΊ β (πβπ)) β (πΌ β (πβπ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap14lem8.h | . . . 4 β’ π» = (LHypβπΎ) | |
2 | hdmap14lem8.c | . . . 4 β’ πΆ = ((LCDualβπΎ)βπ) | |
3 | hdmap14lem8.k | . . . 4 β’ (π β (πΎ β HL β§ π β π»)) | |
4 | 1, 2, 3 | lcdlmod 41120 | . . 3 β’ (π β πΆ β LMod) |
5 | hdmap14lem8.j | . . 3 β’ (π β π½ β π΄) | |
6 | hdmap14lem8.u | . . . 4 β’ π = ((DVecHβπΎ)βπ) | |
7 | hdmap14lem8.v | . . . 4 β’ π = (Baseβπ) | |
8 | eqid 2725 | . . . 4 β’ (BaseβπΆ) = (BaseβπΆ) | |
9 | hdmap14lem8.s | . . . 4 β’ π = ((HDMapβπΎ)βπ) | |
10 | hdmap14lem8.x | . . . . 5 β’ (π β π β (π β { 0 })) | |
11 | 10 | eldifad 3952 | . . . 4 β’ (π β π β π) |
12 | 1, 6, 7, 2, 8, 9, 3, 11 | hdmapcl 41358 | . . 3 β’ (π β (πβπ) β (BaseβπΆ)) |
13 | hdmap14lem8.y | . . . . 5 β’ (π β π β (π β { 0 })) | |
14 | 13 | eldifad 3952 | . . . 4 β’ (π β π β π) |
15 | 1, 6, 7, 2, 8, 9, 3, 14 | hdmapcl 41358 | . . 3 β’ (π β (πβπ) β (BaseβπΆ)) |
16 | hdmap14lem8.d | . . . 4 β’ β = (+gβπΆ) | |
17 | hdmap14lem8.p | . . . 4 β’ π = (ScalarβπΆ) | |
18 | hdmap14lem8.e | . . . 4 β’ β = ( Β·π βπΆ) | |
19 | hdmap14lem8.a | . . . 4 β’ π΄ = (Baseβπ) | |
20 | 8, 16, 17, 18, 19 | lmodvsdi 20770 | . . 3 β’ ((πΆ β LMod β§ (π½ β π΄ β§ (πβπ) β (BaseβπΆ) β§ (πβπ) β (BaseβπΆ))) β (π½ β ((πβπ) β (πβπ))) = ((π½ β (πβπ)) β (π½ β (πβπ)))) |
21 | 4, 5, 12, 15, 20 | syl13anc 1369 | . 2 β’ (π β (π½ β ((πβπ) β (πβπ))) = ((π½ β (πβπ)) β (π½ β (πβπ)))) |
22 | hdmap14lem8.q | . . . . 5 β’ + = (+gβπ) | |
23 | 1, 6, 7, 22, 2, 16, 9, 3, 11, 14 | hdmapadd 41371 | . . . 4 β’ (π β (πβ(π + π)) = ((πβπ) β (πβπ))) |
24 | 23 | oveq2d 7431 | . . 3 β’ (π β (π½ β (πβ(π + π))) = (π½ β ((πβπ) β (πβπ)))) |
25 | hdmap14lem8.xy | . . . 4 β’ (π β (πβ(πΉ Β· (π + π))) = (π½ β (πβ(π + π)))) | |
26 | 1, 6, 3 | dvhlmod 40638 | . . . . . . 7 β’ (π β π β LMod) |
27 | hdmap14lem8.f | . . . . . . 7 β’ (π β πΉ β π΅) | |
28 | hdmap14lem8.r | . . . . . . . 8 β’ π = (Scalarβπ) | |
29 | hdmap14lem8.t | . . . . . . . 8 β’ Β· = ( Β·π βπ) | |
30 | hdmap14lem8.b | . . . . . . . 8 β’ π΅ = (Baseβπ ) | |
31 | 7, 22, 28, 29, 30 | lmodvsdi 20770 | . . . . . . 7 β’ ((π β LMod β§ (πΉ β π΅ β§ π β π β§ π β π)) β (πΉ Β· (π + π)) = ((πΉ Β· π) + (πΉ Β· π))) |
32 | 26, 27, 11, 14, 31 | syl13anc 1369 | . . . . . 6 β’ (π β (πΉ Β· (π + π)) = ((πΉ Β· π) + (πΉ Β· π))) |
33 | 32 | fveq2d 6895 | . . . . 5 β’ (π β (πβ(πΉ Β· (π + π))) = (πβ((πΉ Β· π) + (πΉ Β· π)))) |
34 | 7, 28, 29, 30 | lmodvscl 20763 | . . . . . . 7 β’ ((π β LMod β§ πΉ β π΅ β§ π β π) β (πΉ Β· π) β π) |
35 | 26, 27, 11, 34 | syl3anc 1368 | . . . . . 6 β’ (π β (πΉ Β· π) β π) |
36 | 7, 28, 29, 30 | lmodvscl 20763 | . . . . . . 7 β’ ((π β LMod β§ πΉ β π΅ β§ π β π) β (πΉ Β· π) β π) |
37 | 26, 27, 14, 36 | syl3anc 1368 | . . . . . 6 β’ (π β (πΉ Β· π) β π) |
38 | 1, 6, 7, 22, 2, 16, 9, 3, 35, 37 | hdmapadd 41371 | . . . . 5 β’ (π β (πβ((πΉ Β· π) + (πΉ Β· π))) = ((πβ(πΉ Β· π)) β (πβ(πΉ Β· π)))) |
39 | hdmap14lem8.xx | . . . . . 6 β’ (π β (πβ(πΉ Β· π)) = (πΊ β (πβπ))) | |
40 | hdmap14lem8.yy | . . . . . 6 β’ (π β (πβ(πΉ Β· π)) = (πΌ β (πβπ))) | |
41 | 39, 40 | oveq12d 7433 | . . . . 5 β’ (π β ((πβ(πΉ Β· π)) β (πβ(πΉ Β· π))) = ((πΊ β (πβπ)) β (πΌ β (πβπ)))) |
42 | 33, 38, 41 | 3eqtrd 2769 | . . . 4 β’ (π β (πβ(πΉ Β· (π + π))) = ((πΊ β (πβπ)) β (πΌ β (πβπ)))) |
43 | 25, 42 | eqtr3d 2767 | . . 3 β’ (π β (π½ β (πβ(π + π))) = ((πΊ β (πβπ)) β (πΌ β (πβπ)))) |
44 | 24, 43 | eqtr3d 2767 | . 2 β’ (π β (π½ β ((πβπ) β (πβπ))) = ((πΊ β (πβπ)) β (πΌ β (πβπ)))) |
45 | 21, 44 | eqtr3d 2767 | 1 β’ (π β ((π½ β (πβπ)) β (π½ β (πβπ))) = ((πΊ β (πβπ)) β (πΌ β (πβπ)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2930 β cdif 3937 {csn 4624 βcfv 6542 (class class class)co 7415 Basecbs 17177 +gcplusg 17230 Scalarcsca 17233 Β·π cvsca 17234 0gc0g 17418 LModclmod 20745 LSpanclspn 20857 HLchlt 38877 LHypclh 39512 DVecHcdvh 40606 LCDualclcd 41114 HDMapchdma 41320 |
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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-riotaBAD 38480 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-om 7868 df-1st 7989 df-2nd 7990 df-tpos 8228 df-undef 8275 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-n0 12501 df-z 12587 df-uz 12851 df-fz 13515 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-0g 17420 df-mre 17563 df-mrc 17564 df-acs 17566 df-proset 18284 df-poset 18302 df-plt 18319 df-lub 18335 df-glb 18336 df-join 18337 df-meet 18338 df-p0 18414 df-p1 18415 df-lat 18421 df-clat 18488 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18738 df-grp 18895 df-minusg 18896 df-sbg 18897 df-subg 19080 df-cntz 19270 df-oppg 19299 df-lsm 19593 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-oppr 20275 df-dvdsr 20298 df-unit 20299 df-invr 20329 df-dvr 20342 df-drng 20628 df-lmod 20747 df-lss 20818 df-lsp 20858 df-lvec 20990 df-lsatoms 38503 df-lshyp 38504 df-lcv 38546 df-lfl 38585 df-lkr 38613 df-ldual 38651 df-oposet 38703 df-ol 38705 df-oml 38706 df-covers 38793 df-ats 38794 df-atl 38825 df-cvlat 38849 df-hlat 38878 df-llines 39026 df-lplanes 39027 df-lvols 39028 df-lines 39029 df-psubsp 39031 df-pmap 39032 df-padd 39324 df-lhyp 39516 df-laut 39517 df-ldil 39632 df-ltrn 39633 df-trl 39687 df-tgrp 40271 df-tendo 40283 df-edring 40285 df-dveca 40531 df-disoa 40557 df-dvech 40607 df-dib 40667 df-dic 40701 df-dih 40757 df-doch 40876 df-djh 40923 df-lcdual 41115 df-mapd 41153 df-hvmap 41285 df-hdmap1 41321 df-hdmap 41322 |
This theorem is referenced by: hdmap14lem9 41404 |
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