![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
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 40989 | . . 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 2727 | . . . 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 3956 | . . . 4 β’ (π β π β π) |
12 | 1, 6, 7, 2, 8, 9, 3, 11 | hdmapcl 41227 | . . 3 β’ (π β (πβπ) β (BaseβπΆ)) |
13 | hdmap14lem8.y | . . . . 5 β’ (π β π β (π β { 0 })) | |
14 | 13 | eldifad 3956 | . . . 4 β’ (π β π β π) |
15 | 1, 6, 7, 2, 8, 9, 3, 14 | hdmapcl 41227 | . . 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 20750 | . . 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 1370 | . 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 41240 | . . . 4 β’ (π β (πβ(π + π)) = ((πβπ) β (πβπ))) |
24 | 23 | oveq2d 7430 | . . 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 40507 | . . . . . . 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 20750 | . . . . . . 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 1370 | . . . . . 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 20743 | . . . . . . 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 1369 | . . . . . 6 β’ (π β (πΉ Β· π) β π) |
36 | 7, 28, 29, 30 | lmodvscl 20743 | . . . . . . 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 1369 | . . . . . 6 β’ (π β (πΉ Β· π) β π) |
38 | 1, 6, 7, 22, 2, 16, 9, 3, 35, 37 | hdmapadd 41240 | . . . . 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 7432 | . . . . 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 2771 | . . . 4 β’ (π β (πβ(πΉ Β· (π + π))) = ((πΊ β (πβπ)) β (πΌ β (πβπ)))) |
43 | 25, 42 | eqtr3d 2769 | . . 3 β’ (π β (π½ β (πβ(π + π))) = ((πΊ β (πβπ)) β (πΌ β (πβπ)))) |
44 | 24, 43 | eqtr3d 2769 | . 2 β’ (π β (π½ β ((πβπ) β (πβπ))) = ((πΊ β (πβπ)) β (πΌ β (πβπ)))) |
45 | 21, 44 | eqtr3d 2769 | 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 395 = wceq 1534 β wcel 2099 β wne 2935 β cdif 3941 {csn 4624 βcfv 6542 (class class class)co 7414 Basecbs 17165 +gcplusg 17218 Scalarcsca 17221 Β·π cvsca 17222 0gc0g 17406 LModclmod 20725 LSpanclspn 20837 HLchlt 38746 LHypclh 39381 DVecHcdvh 40475 LCDualclcd 40983 HDMapchdma 41189 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-riotaBAD 38349 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 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 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 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 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7677 df-om 7863 df-1st 7985 df-2nd 7986 df-tpos 8223 df-undef 8270 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-n0 12489 df-z 12575 df-uz 12839 df-fz 13503 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-0g 17408 df-mre 17551 df-mrc 17552 df-acs 17554 df-proset 18272 df-poset 18290 df-plt 18307 df-lub 18323 df-glb 18324 df-join 18325 df-meet 18326 df-p0 18402 df-p1 18403 df-lat 18409 df-clat 18476 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-submnd 18726 df-grp 18878 df-minusg 18879 df-sbg 18880 df-subg 19062 df-cntz 19252 df-oppg 19281 df-lsm 19575 df-cmn 19721 df-abl 19722 df-mgp 20059 df-rng 20077 df-ur 20106 df-ring 20159 df-oppr 20255 df-dvdsr 20278 df-unit 20279 df-invr 20309 df-dvr 20322 df-drng 20608 df-lmod 20727 df-lss 20798 df-lsp 20838 df-lvec 20970 df-lsatoms 38372 df-lshyp 38373 df-lcv 38415 df-lfl 38454 df-lkr 38482 df-ldual 38520 df-oposet 38572 df-ol 38574 df-oml 38575 df-covers 38662 df-ats 38663 df-atl 38694 df-cvlat 38718 df-hlat 38747 df-llines 38895 df-lplanes 38896 df-lvols 38897 df-lines 38898 df-psubsp 38900 df-pmap 38901 df-padd 39193 df-lhyp 39385 df-laut 39386 df-ldil 39501 df-ltrn 39502 df-trl 39556 df-tgrp 40140 df-tendo 40152 df-edring 40154 df-dveca 40400 df-disoa 40426 df-dvech 40476 df-dib 40536 df-dic 40570 df-dih 40626 df-doch 40745 df-djh 40792 df-lcdual 40984 df-mapd 41022 df-hvmap 41154 df-hdmap1 41190 df-hdmap 41191 |
This theorem is referenced by: hdmap14lem9 41273 |
Copyright terms: Public domain | W3C validator |