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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmapglem5 | Structured version Visualization version GIF version |
Description: Part 1.2 in [Baer] p. 110 line 34, f(u,v) alpha = f(v,u). (Contributed by NM, 12-Jun-2015.) |
Ref | Expression |
---|---|
hdmapglem5.h | β’ π» = (LHypβπΎ) |
hdmapglem5.e | β’ πΈ = β¨( I βΎ (BaseβπΎ)), ( I βΎ ((LTrnβπΎ)βπ))β© |
hdmapglem5.o | β’ π = ((ocHβπΎ)βπ) |
hdmapglem5.u | β’ π = ((DVecHβπΎ)βπ) |
hdmapglem5.v | β’ π = (Baseβπ) |
hdmapglem5.p | β’ + = (+gβπ) |
hdmapglem5.m | β’ β = (-gβπ) |
hdmapglem5.q | β’ Β· = ( Β·π βπ) |
hdmapglem5.r | β’ π = (Scalarβπ) |
hdmapglem5.b | β’ π΅ = (Baseβπ ) |
hdmapglem5.t | β’ Γ = (.rβπ ) |
hdmapglem5.z | β’ 0 = (0gβπ ) |
hdmapglem5.s | β’ π = ((HDMapβπΎ)βπ) |
hdmapglem5.g | β’ πΊ = ((HGMapβπΎ)βπ) |
hdmapglem5.k | β’ (π β (πΎ β HL β§ π β π»)) |
hdmapglem5.c | β’ (π β πΆ β (πβ{πΈ})) |
hdmapglem5.d | β’ (π β π· β (πβ{πΈ})) |
hdmapglem5.i | β’ (π β πΌ β π΅) |
hdmapglem5.j | β’ (π β π½ β π΅) |
Ref | Expression |
---|---|
hdmapglem5 | β’ (π β (πΊβ((πβπ·)βπΆ)) = ((πβπΆ)βπ·)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmapglem5.h | . . . . 5 β’ π» = (LHypβπΎ) | |
2 | hdmapglem5.u | . . . . 5 β’ π = ((DVecHβπΎ)βπ) | |
3 | hdmapglem5.k | . . . . 5 β’ (π β (πΎ β HL β§ π β π»)) | |
4 | 1, 2, 3 | dvhlmod 39576 | . . . 4 β’ (π β π β LMod) |
5 | hdmapglem5.r | . . . . 5 β’ π = (Scalarβπ) | |
6 | 5 | lmodring 20333 | . . . 4 β’ (π β LMod β π β Ring) |
7 | 4, 6 | syl 17 | . . 3 β’ (π β π β Ring) |
8 | hdmapglem5.b | . . . 4 β’ π΅ = (Baseβπ ) | |
9 | hdmapglem5.g | . . . 4 β’ πΊ = ((HGMapβπΎ)βπ) | |
10 | hdmapglem5.v | . . . . 5 β’ π = (Baseβπ) | |
11 | hdmapglem5.s | . . . . 5 β’ π = ((HDMapβπΎ)βπ) | |
12 | eqid 2737 | . . . . . . . . . 10 β’ (BaseβπΎ) = (BaseβπΎ) | |
13 | eqid 2737 | . . . . . . . . . 10 β’ ((LTrnβπΎ)βπ) = ((LTrnβπΎ)βπ) | |
14 | eqid 2737 | . . . . . . . . . 10 β’ (0gβπ) = (0gβπ) | |
15 | hdmapglem5.e | . . . . . . . . . 10 β’ πΈ = β¨( I βΎ (BaseβπΎ)), ( I βΎ ((LTrnβπΎ)βπ))β© | |
16 | 1, 12, 13, 2, 10, 14, 15, 3 | dvheveccl 39578 | . . . . . . . . 9 β’ (π β πΈ β (π β {(0gβπ)})) |
17 | 16 | eldifad 3923 | . . . . . . . 8 β’ (π β πΈ β π) |
18 | 17 | snssd 4770 | . . . . . . 7 β’ (π β {πΈ} β π) |
19 | hdmapglem5.o | . . . . . . . 8 β’ π = ((ocHβπΎ)βπ) | |
20 | 1, 2, 10, 19 | dochssv 39821 | . . . . . . 7 β’ (((πΎ β HL β§ π β π») β§ {πΈ} β π) β (πβ{πΈ}) β π) |
21 | 3, 18, 20 | syl2anc 585 | . . . . . 6 β’ (π β (πβ{πΈ}) β π) |
22 | hdmapglem5.c | . . . . . 6 β’ (π β πΆ β (πβ{πΈ})) | |
23 | 21, 22 | sseldd 3946 | . . . . 5 β’ (π β πΆ β π) |
24 | hdmapglem5.d | . . . . . 6 β’ (π β π· β (πβ{πΈ})) | |
25 | 21, 24 | sseldd 3946 | . . . . 5 β’ (π β π· β π) |
26 | 1, 2, 10, 5, 8, 11, 3, 23, 25 | hdmapipcl 40371 | . . . 4 β’ (π β ((πβπ·)βπΆ) β π΅) |
27 | 1, 2, 5, 8, 9, 3, 26 | hgmapcl 40355 | . . 3 β’ (π β (πΊβ((πβπ·)βπΆ)) β π΅) |
28 | hdmapglem5.t | . . . 4 β’ Γ = (.rβπ ) | |
29 | eqid 2737 | . . . 4 β’ (1rβπ ) = (1rβπ ) | |
30 | 8, 28, 29 | ringlidm 19993 | . . 3 β’ ((π β Ring β§ (πΊβ((πβπ·)βπΆ)) β π΅) β ((1rβπ ) Γ (πΊβ((πβπ·)βπΆ))) = (πΊβ((πβπ·)βπΆ))) |
31 | 7, 27, 30 | syl2anc 585 | . 2 β’ (π β ((1rβπ ) Γ (πΊβ((πβπ·)βπΆ))) = (πΊβ((πβπ·)βπΆ))) |
32 | hdmapglem5.p | . . 3 β’ + = (+gβπ) | |
33 | hdmapglem5.m | . . 3 β’ β = (-gβπ) | |
34 | hdmapglem5.q | . . 3 β’ Β· = ( Β·π βπ) | |
35 | hdmapglem5.z | . . 3 β’ 0 = (0gβπ ) | |
36 | 8, 29 | ringidcl 19990 | . . . 4 β’ (π β Ring β (1rβπ ) β π΅) |
37 | 7, 36 | syl 17 | . . 3 β’ (π β (1rβπ ) β π΅) |
38 | 1, 2, 5, 29, 9, 3 | hgmapval1 40359 | . . . . 5 β’ (π β (πΊβ(1rβπ )) = (1rβπ )) |
39 | 38 | oveq2d 7374 | . . . 4 β’ (π β (((πβπ·)βπΆ) Γ (πΊβ(1rβπ ))) = (((πβπ·)βπΆ) Γ (1rβπ ))) |
40 | 8, 28, 29 | ringridm 19994 | . . . . 5 β’ ((π β Ring β§ ((πβπ·)βπΆ) β π΅) β (((πβπ·)βπΆ) Γ (1rβπ )) = ((πβπ·)βπΆ)) |
41 | 7, 26, 40 | syl2anc 585 | . . . 4 β’ (π β (((πβπ·)βπΆ) Γ (1rβπ )) = ((πβπ·)βπΆ)) |
42 | 39, 41 | eqtrd 2777 | . . 3 β’ (π β (((πβπ·)βπΆ) Γ (πΊβ(1rβπ ))) = ((πβπ·)βπΆ)) |
43 | 1, 15, 19, 2, 10, 32, 33, 34, 5, 8, 28, 35, 11, 9, 3, 22, 24, 26, 37, 42 | hdmapinvlem4 40387 | . 2 β’ (π β ((1rβπ ) Γ (πΊβ((πβπ·)βπΆ))) = ((πβπΆ)βπ·)) |
44 | 31, 43 | eqtr3d 2779 | 1 β’ (π β (πΊβ((πβπ·)βπΆ)) = ((πβπΆ)βπ·)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wss 3911 {csn 4587 β¨cop 4593 I cid 5531 βΎ cres 5636 βcfv 6497 (class class class)co 7358 Basecbs 17084 +gcplusg 17134 .rcmulr 17135 Scalarcsca 17137 Β·π cvsca 17138 0gc0g 17322 -gcsg 18751 1rcur 19914 Ringcrg 19965 LModclmod 20325 HLchlt 37815 LHypclh 38450 LTrncltrn 38567 DVecHcdvh 39544 ocHcoch 39813 HDMapchdma 40258 HGMapchg 40349 |
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 df-hgmap 40350 |
This theorem is referenced by: hgmapvvlem1 40389 hdmapglem7 40395 |
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