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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap14lem13 | Structured version Visualization version GIF version |
Description: Lemma for proof of part 14 in [Baer] p. 50. (Contributed by NM, 6-Jun-2015.) |
Ref | Expression |
---|---|
hdmap14lem12.h | β’ π» = (LHypβπΎ) |
hdmap14lem12.u | β’ π = ((DVecHβπΎ)βπ) |
hdmap14lem12.v | β’ π = (Baseβπ) |
hdmap14lem12.t | β’ Β· = ( Β·π βπ) |
hdmap14lem12.r | β’ π = (Scalarβπ) |
hdmap14lem12.b | β’ π΅ = (Baseβπ ) |
hdmap14lem12.c | β’ πΆ = ((LCDualβπΎ)βπ) |
hdmap14lem12.e | β’ β = ( Β·π βπΆ) |
hdmap14lem12.s | β’ π = ((HDMapβπΎ)βπ) |
hdmap14lem12.k | β’ (π β (πΎ β HL β§ π β π»)) |
hdmap14lem12.f | β’ (π β πΉ β π΅) |
hdmap14lem12.p | β’ π = (ScalarβπΆ) |
hdmap14lem12.a | β’ π΄ = (Baseβπ) |
hdmap14lem12.o | β’ 0 = (0gβπ) |
hdmap14lem12.x | β’ (π β π β (π β { 0 })) |
hdmap14lem12.g | β’ (π β πΊ β π΄) |
Ref | Expression |
---|---|
hdmap14lem13 | β’ (π β ((πβ(πΉ Β· π)) = (πΊ β (πβπ)) β βπ¦ β π (πβ(πΉ Β· π¦)) = (πΊ β (πβπ¦)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap14lem12.h | . . 3 β’ π» = (LHypβπΎ) | |
2 | hdmap14lem12.u | . . 3 β’ π = ((DVecHβπΎ)βπ) | |
3 | hdmap14lem12.v | . . 3 β’ π = (Baseβπ) | |
4 | hdmap14lem12.t | . . 3 β’ Β· = ( Β·π βπ) | |
5 | hdmap14lem12.r | . . 3 β’ π = (Scalarβπ) | |
6 | hdmap14lem12.b | . . 3 β’ π΅ = (Baseβπ ) | |
7 | hdmap14lem12.c | . . 3 β’ πΆ = ((LCDualβπΎ)βπ) | |
8 | hdmap14lem12.e | . . 3 β’ β = ( Β·π βπΆ) | |
9 | hdmap14lem12.s | . . 3 β’ π = ((HDMapβπΎ)βπ) | |
10 | hdmap14lem12.k | . . 3 β’ (π β (πΎ β HL β§ π β π»)) | |
11 | hdmap14lem12.f | . . 3 β’ (π β πΉ β π΅) | |
12 | hdmap14lem12.p | . . 3 β’ π = (ScalarβπΆ) | |
13 | hdmap14lem12.a | . . 3 β’ π΄ = (Baseβπ) | |
14 | hdmap14lem12.o | . . 3 β’ 0 = (0gβπ) | |
15 | hdmap14lem12.x | . . 3 β’ (π β π β (π β { 0 })) | |
16 | hdmap14lem12.g | . . 3 β’ (π β πΊ β π΄) | |
17 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 | hdmap14lem12 40345 | . 2 β’ (π β ((πβ(πΉ Β· π)) = (πΊ β (πβπ)) β βπ¦ β (π β { 0 })(πβ(πΉ Β· π¦)) = (πΊ β (πβπ¦)))) |
18 | velsn 4603 | . . . . . 6 β’ (π¦ β { 0 } β π¦ = 0 ) | |
19 | 1, 7, 10 | lcdlmod 40058 | . . . . . . . . 9 β’ (π β πΆ β LMod) |
20 | eqid 2737 | . . . . . . . . . 10 β’ (0gβπΆ) = (0gβπΆ) | |
21 | 12, 8, 13, 20 | lmodvs0 20359 | . . . . . . . . 9 β’ ((πΆ β LMod β§ πΊ β π΄) β (πΊ β (0gβπΆ)) = (0gβπΆ)) |
22 | 19, 16, 21 | syl2anc 585 | . . . . . . . 8 β’ (π β (πΊ β (0gβπΆ)) = (0gβπΆ)) |
23 | 1, 2, 14, 7, 20, 9, 10 | hdmapval0 40299 | . . . . . . . . 9 β’ (π β (πβ 0 ) = (0gβπΆ)) |
24 | 23 | oveq2d 7374 | . . . . . . . 8 β’ (π β (πΊ β (πβ 0 )) = (πΊ β (0gβπΆ))) |
25 | 1, 2, 10 | dvhlmod 39576 | . . . . . . . . . . 11 β’ (π β π β LMod) |
26 | 5, 4, 6, 14 | lmodvs0 20359 | . . . . . . . . . . 11 β’ ((π β LMod β§ πΉ β π΅) β (πΉ Β· 0 ) = 0 ) |
27 | 25, 11, 26 | syl2anc 585 | . . . . . . . . . 10 β’ (π β (πΉ Β· 0 ) = 0 ) |
28 | 27 | fveq2d 6847 | . . . . . . . . 9 β’ (π β (πβ(πΉ Β· 0 )) = (πβ 0 )) |
29 | 28, 23 | eqtrd 2777 | . . . . . . . 8 β’ (π β (πβ(πΉ Β· 0 )) = (0gβπΆ)) |
30 | 22, 24, 29 | 3eqtr4rd 2788 | . . . . . . 7 β’ (π β (πβ(πΉ Β· 0 )) = (πΊ β (πβ 0 ))) |
31 | oveq2 7366 | . . . . . . . . 9 β’ (π¦ = 0 β (πΉ Β· π¦) = (πΉ Β· 0 )) | |
32 | 31 | fveq2d 6847 | . . . . . . . 8 β’ (π¦ = 0 β (πβ(πΉ Β· π¦)) = (πβ(πΉ Β· 0 ))) |
33 | fveq2 6843 | . . . . . . . . 9 β’ (π¦ = 0 β (πβπ¦) = (πβ 0 )) | |
34 | 33 | oveq2d 7374 | . . . . . . . 8 β’ (π¦ = 0 β (πΊ β (πβπ¦)) = (πΊ β (πβ 0 ))) |
35 | 32, 34 | eqeq12d 2753 | . . . . . . 7 β’ (π¦ = 0 β ((πβ(πΉ Β· π¦)) = (πΊ β (πβπ¦)) β (πβ(πΉ Β· 0 )) = (πΊ β (πβ 0 )))) |
36 | 30, 35 | syl5ibrcom 247 | . . . . . 6 β’ (π β (π¦ = 0 β (πβ(πΉ Β· π¦)) = (πΊ β (πβπ¦)))) |
37 | 18, 36 | biimtrid 241 | . . . . 5 β’ (π β (π¦ β { 0 } β (πβ(πΉ Β· π¦)) = (πΊ β (πβπ¦)))) |
38 | 37 | ralrimiv 3143 | . . . 4 β’ (π β βπ¦ β { 0 } (πβ(πΉ Β· π¦)) = (πΊ β (πβπ¦))) |
39 | 38 | biantrud 533 | . . 3 β’ (π β (βπ¦ β (π β { 0 })(πβ(πΉ Β· π¦)) = (πΊ β (πβπ¦)) β (βπ¦ β (π β { 0 })(πβ(πΉ Β· π¦)) = (πΊ β (πβπ¦)) β§ βπ¦ β { 0 } (πβ(πΉ Β· π¦)) = (πΊ β (πβπ¦))))) |
40 | ralunb 4152 | . . 3 β’ (βπ¦ β ((π β { 0 }) βͺ { 0 })(πβ(πΉ Β· π¦)) = (πΊ β (πβπ¦)) β (βπ¦ β (π β { 0 })(πβ(πΉ Β· π¦)) = (πΊ β (πβπ¦)) β§ βπ¦ β { 0 } (πβ(πΉ Β· π¦)) = (πΊ β (πβπ¦)))) | |
41 | 39, 40 | bitr4di 289 | . 2 β’ (π β (βπ¦ β (π β { 0 })(πβ(πΉ Β· π¦)) = (πΊ β (πβπ¦)) β βπ¦ β ((π β { 0 }) βͺ { 0 })(πβ(πΉ Β· π¦)) = (πΊ β (πβπ¦)))) |
42 | 3, 14 | lmod0vcl 20354 | . . . 4 β’ (π β LMod β 0 β π) |
43 | difsnid 4771 | . . . 4 β’ ( 0 β π β ((π β { 0 }) βͺ { 0 }) = π) | |
44 | 25, 42, 43 | 3syl 18 | . . 3 β’ (π β ((π β { 0 }) βͺ { 0 }) = π) |
45 | 44 | raleqdv 3314 | . 2 β’ (π β (βπ¦ β ((π β { 0 }) βͺ { 0 })(πβ(πΉ Β· π¦)) = (πΊ β (πβπ¦)) β βπ¦ β π (πβ(πΉ Β· π¦)) = (πΊ β (πβπ¦)))) |
46 | 17, 41, 45 | 3bitrd 305 | 1 β’ (π β ((πβ(πΉ Β· π)) = (πΊ β (πβπ)) β βπ¦ β π (πβ(πΉ Β· π¦)) = (πΊ β (πβπ¦)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3065 β cdif 3908 βͺ cun 3909 {csn 4587 βcfv 6497 (class class class)co 7358 Basecbs 17084 Scalarcsca 17137 Β·π cvsca 17138 0gc0g 17322 LModclmod 20325 HLchlt 37815 LHypclh 38450 DVecHcdvh 39544 LCDualclcd 40052 HDMapchdma 40258 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-riotaBAD 37418 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 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 8649 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-n0 12415 df-z 12501 df-uz 12765 df-fz 13426 df-struct 17020 df-sets 17037 df-slot 17055 df-ndx 17067 df-base 17085 df-ress 17114 df-plusg 17147 df-mulr 17148 df-sca 17150 df-vsca 17151 df-0g 17324 df-mre 17467 df-mrc 17468 df-acs 17470 df-proset 18185 df-poset 18203 df-plt 18220 df-lub 18236 df-glb 18237 df-join 18238 df-meet 18239 df-p0 18315 df-p1 18316 df-lat 18322 df-clat 18389 df-mgm 18498 df-sgrp 18547 df-mnd 18558 df-submnd 18603 df-grp 18752 df-minusg 18753 df-sbg 18754 df-subg 18926 df-cntz 19098 df-oppg 19125 df-lsm 19419 df-cmn 19565 df-abl 19566 df-mgp 19898 df-ur 19915 df-ring 19967 df-oppr 20050 df-dvdsr 20071 df-unit 20072 df-invr 20102 df-dvr 20113 df-drng 20188 df-lmod 20327 df-lss 20396 df-lsp 20436 df-lvec 20567 df-lsatoms 37441 df-lshyp 37442 df-lcv 37484 df-lfl 37523 df-lkr 37551 df-ldual 37589 df-oposet 37641 df-ol 37643 df-oml 37644 df-covers 37731 df-ats 37732 df-atl 37763 df-cvlat 37787 df-hlat 37816 df-llines 37964 df-lplanes 37965 df-lvols 37966 df-lines 37967 df-psubsp 37969 df-pmap 37970 df-padd 38262 df-lhyp 38454 df-laut 38455 df-ldil 38570 df-ltrn 38571 df-trl 38625 df-tgrp 39209 df-tendo 39221 df-edring 39223 df-dveca 39469 df-disoa 39495 df-dvech 39545 df-dib 39605 df-dic 39639 df-dih 39695 df-doch 39814 df-djh 39861 df-lcdual 40053 df-mapd 40091 df-hvmap 40223 df-hdmap1 40259 df-hdmap 40260 |
This theorem is referenced by: hdmap14lem14 40347 |
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