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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem21 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 36995. (Contributed by NM, 20-Mar-2015.) |
Ref | Expression |
---|---|
mapdpglem.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdpglem.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdpglem.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdpglem.v | ⊢ 𝑉 = (Base‘𝑈) |
mapdpglem.s | ⊢ − = (-g‘𝑈) |
mapdpglem.n | ⊢ 𝑁 = (LSpan‘𝑈) |
mapdpglem.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
mapdpglem.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdpglem.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
mapdpglem.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
mapdpglem1.p | ⊢ ⊕ = (LSSum‘𝐶) |
mapdpglem2.j | ⊢ 𝐽 = (LSpan‘𝐶) |
mapdpglem3.f | ⊢ 𝐹 = (Base‘𝐶) |
mapdpglem3.te | ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) |
mapdpglem3.a | ⊢ 𝐴 = (Scalar‘𝑈) |
mapdpglem3.b | ⊢ 𝐵 = (Base‘𝐴) |
mapdpglem3.t | ⊢ · = ( ·𝑠 ‘𝐶) |
mapdpglem3.r | ⊢ 𝑅 = (-g‘𝐶) |
mapdpglem3.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
mapdpglem3.e | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) |
mapdpglem4.q | ⊢ 𝑄 = (0g‘𝑈) |
mapdpglem.ne | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
mapdpglem4.jt | ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) |
mapdpglem4.z | ⊢ 0 = (0g‘𝐴) |
mapdpglem4.g4 | ⊢ (𝜑 → 𝑔 ∈ 𝐵) |
mapdpglem4.z4 | ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) |
mapdpglem4.t4 | ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) |
mapdpglem4.xn | ⊢ (𝜑 → 𝑋 ≠ 𝑄) |
mapdpglem12.yn | ⊢ (𝜑 → 𝑌 ≠ 𝑄) |
mapdpglem17.ep | ⊢ 𝐸 = (((invr‘𝐴)‘𝑔) · 𝑧) |
Ref | Expression |
---|---|
mapdpglem21 | ⊢ (𝜑 → (((invr‘𝐴)‘𝑔) · 𝑡) = (𝐺𝑅𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpglem4.t4 | . . 3 ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) | |
2 | 1 | oveq2d 6666 | . 2 ⊢ (𝜑 → (((invr‘𝐴)‘𝑔) · 𝑡) = (((invr‘𝐴)‘𝑔) · ((𝑔 · 𝐺)𝑅𝑧))) |
3 | mapdpglem3.f | . . 3 ⊢ 𝐹 = (Base‘𝐶) | |
4 | mapdpglem3.t | . . 3 ⊢ · = ( ·𝑠 ‘𝐶) | |
5 | eqid 2622 | . . 3 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶) | |
6 | eqid 2622 | . . 3 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶)) | |
7 | mapdpglem3.r | . . 3 ⊢ 𝑅 = (-g‘𝐶) | |
8 | mapdpglem.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
9 | mapdpglem.c | . . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
10 | mapdpglem.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
11 | 8, 9, 10 | lcdlmod 36881 | . . 3 ⊢ (𝜑 → 𝐶 ∈ LMod) |
12 | mapdpglem.u | . . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
13 | 8, 12, 10 | dvhlvec 36398 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec) |
14 | mapdpglem3.a | . . . . . . 7 ⊢ 𝐴 = (Scalar‘𝑈) | |
15 | 14 | lvecdrng 19105 | . . . . . 6 ⊢ (𝑈 ∈ LVec → 𝐴 ∈ DivRing) |
16 | 13, 15 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ DivRing) |
17 | mapdpglem4.g4 | . . . . 5 ⊢ (𝜑 → 𝑔 ∈ 𝐵) | |
18 | mapdpglem.m | . . . . . 6 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
19 | mapdpglem.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑈) | |
20 | mapdpglem.s | . . . . . 6 ⊢ − = (-g‘𝑈) | |
21 | mapdpglem.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈) | |
22 | mapdpglem.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
23 | mapdpglem.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
24 | mapdpglem1.p | . . . . . 6 ⊢ ⊕ = (LSSum‘𝐶) | |
25 | mapdpglem2.j | . . . . . 6 ⊢ 𝐽 = (LSpan‘𝐶) | |
26 | mapdpglem3.te | . . . . . 6 ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) | |
27 | mapdpglem3.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐴) | |
28 | mapdpglem3.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
29 | mapdpglem3.e | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) | |
30 | mapdpglem4.q | . . . . . 6 ⊢ 𝑄 = (0g‘𝑈) | |
31 | mapdpglem.ne | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
32 | mapdpglem4.jt | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) | |
33 | mapdpglem4.z | . . . . . 6 ⊢ 0 = (0g‘𝐴) | |
34 | mapdpglem4.z4 | . . . . . 6 ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) | |
35 | mapdpglem4.xn | . . . . . 6 ⊢ (𝜑 → 𝑋 ≠ 𝑄) | |
36 | 8, 18, 12, 19, 20, 21, 9, 10, 22, 23, 24, 25, 3, 26, 14, 27, 4, 7, 28, 29, 30, 31, 32, 33, 17, 34, 1, 35 | mapdpglem11 36971 | . . . . 5 ⊢ (𝜑 → 𝑔 ≠ 0 ) |
37 | eqid 2622 | . . . . . 6 ⊢ (invr‘𝐴) = (invr‘𝐴) | |
38 | 27, 33, 37 | drnginvrcl 18764 | . . . . 5 ⊢ ((𝐴 ∈ DivRing ∧ 𝑔 ∈ 𝐵 ∧ 𝑔 ≠ 0 ) → ((invr‘𝐴)‘𝑔) ∈ 𝐵) |
39 | 16, 17, 36, 38 | syl3anc 1326 | . . . 4 ⊢ (𝜑 → ((invr‘𝐴)‘𝑔) ∈ 𝐵) |
40 | 8, 12, 14, 27, 9, 5, 6, 10 | lcdsbase 36889 | . . . 4 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) = 𝐵) |
41 | 39, 40 | eleqtrrd 2704 | . . 3 ⊢ (𝜑 → ((invr‘𝐴)‘𝑔) ∈ (Base‘(Scalar‘𝐶))) |
42 | 8, 12, 14, 27, 9, 3, 4, 10, 17, 28 | lcdvscl 36894 | . . 3 ⊢ (𝜑 → (𝑔 · 𝐺) ∈ 𝐹) |
43 | eqid 2622 | . . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
44 | eqid 2622 | . . . . . 6 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
45 | 8, 12, 10 | dvhlmod 36399 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod) |
46 | 19, 43, 21 | lspsncl 18977 | . . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
47 | 45, 23, 46 | syl2anc 693 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
48 | 8, 18, 12, 43, 9, 44, 10, 47 | mapdcl2 36945 | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶)) |
49 | 3, 44 | lssss 18937 | . . . . 5 ⊢ ((𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶) → (𝑀‘(𝑁‘{𝑌})) ⊆ 𝐹) |
50 | 48, 49 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ⊆ 𝐹) |
51 | 50, 34 | sseldd 3604 | . . 3 ⊢ (𝜑 → 𝑧 ∈ 𝐹) |
52 | 3, 4, 5, 6, 7, 11, 41, 42, 51 | lmodsubdi 18920 | . 2 ⊢ (𝜑 → (((invr‘𝐴)‘𝑔) · ((𝑔 · 𝐺)𝑅𝑧)) = ((((invr‘𝐴)‘𝑔) · (𝑔 · 𝐺))𝑅(((invr‘𝐴)‘𝑔) · 𝑧))) |
53 | eqid 2622 | . . . . . . . . 9 ⊢ (.r‘𝐴) = (.r‘𝐴) | |
54 | eqid 2622 | . . . . . . . . 9 ⊢ (1r‘𝐴) = (1r‘𝐴) | |
55 | 27, 33, 53, 54, 37 | drnginvrr 18767 | . . . . . . . 8 ⊢ ((𝐴 ∈ DivRing ∧ 𝑔 ∈ 𝐵 ∧ 𝑔 ≠ 0 ) → (𝑔(.r‘𝐴)((invr‘𝐴)‘𝑔)) = (1r‘𝐴)) |
56 | 16, 17, 36, 55 | syl3anc 1326 | . . . . . . 7 ⊢ (𝜑 → (𝑔(.r‘𝐴)((invr‘𝐴)‘𝑔)) = (1r‘𝐴)) |
57 | eqid 2622 | . . . . . . . 8 ⊢ (1r‘(Scalar‘𝐶)) = (1r‘(Scalar‘𝐶)) | |
58 | 8, 12, 14, 54, 9, 5, 57, 10 | lcd1 36898 | . . . . . . 7 ⊢ (𝜑 → (1r‘(Scalar‘𝐶)) = (1r‘𝐴)) |
59 | 56, 58 | eqtr4d 2659 | . . . . . 6 ⊢ (𝜑 → (𝑔(.r‘𝐴)((invr‘𝐴)‘𝑔)) = (1r‘(Scalar‘𝐶))) |
60 | 59 | oveq1d 6665 | . . . . 5 ⊢ (𝜑 → ((𝑔(.r‘𝐴)((invr‘𝐴)‘𝑔)) · 𝐺) = ((1r‘(Scalar‘𝐶)) · 𝐺)) |
61 | 8, 12, 14, 27, 53, 9, 3, 4, 10, 39, 17, 28 | lcdvsass 36896 | . . . . 5 ⊢ (𝜑 → ((𝑔(.r‘𝐴)((invr‘𝐴)‘𝑔)) · 𝐺) = (((invr‘𝐴)‘𝑔) · (𝑔 · 𝐺))) |
62 | 3, 5, 4, 57 | lmodvs1 18891 | . . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((1r‘(Scalar‘𝐶)) · 𝐺) = 𝐺) |
63 | 11, 28, 62 | syl2anc 693 | . . . . 5 ⊢ (𝜑 → ((1r‘(Scalar‘𝐶)) · 𝐺) = 𝐺) |
64 | 60, 61, 63 | 3eqtr3d 2664 | . . . 4 ⊢ (𝜑 → (((invr‘𝐴)‘𝑔) · (𝑔 · 𝐺)) = 𝐺) |
65 | 64 | oveq1d 6665 | . . 3 ⊢ (𝜑 → ((((invr‘𝐴)‘𝑔) · (𝑔 · 𝐺))𝑅(((invr‘𝐴)‘𝑔) · 𝑧)) = (𝐺𝑅(((invr‘𝐴)‘𝑔) · 𝑧))) |
66 | mapdpglem17.ep | . . . 4 ⊢ 𝐸 = (((invr‘𝐴)‘𝑔) · 𝑧) | |
67 | 66 | oveq2i 6661 | . . 3 ⊢ (𝐺𝑅𝐸) = (𝐺𝑅(((invr‘𝐴)‘𝑔) · 𝑧)) |
68 | 65, 67 | syl6eqr 2674 | . 2 ⊢ (𝜑 → ((((invr‘𝐴)‘𝑔) · (𝑔 · 𝐺))𝑅(((invr‘𝐴)‘𝑔) · 𝑧)) = (𝐺𝑅𝐸)) |
69 | 2, 52, 68 | 3eqtrd 2660 | 1 ⊢ (𝜑 → (((invr‘𝐴)‘𝑔) · 𝑡) = (𝐺𝑅𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ⊆ wss 3574 {csn 4177 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 .rcmulr 15942 Scalarcsca 15944 ·𝑠 cvsca 15945 0gc0g 16100 -gcsg 17424 LSSumclsm 18049 1rcur 18501 invrcinvr 18671 DivRingcdr 18747 LModclmod 18863 LSubSpclss 18932 LSpanclspn 18971 LVecclvec 19102 HLchlt 34637 LHypclh 35270 DVecHcdvh 36367 LCDualclcd 36875 mapdcmpd 36913 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-riotaBAD 34239 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-undef 7399 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-0g 16102 df-mre 16246 df-mrc 16247 df-acs 16249 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cntz 17750 df-oppg 17776 df-lsm 18051 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103 df-lsatoms 34263 df-lshyp 34264 df-lcv 34306 df-lfl 34345 df-lkr 34373 df-ldual 34411 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 df-lvols 34786 df-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446 df-tgrp 36031 df-tendo 36043 df-edring 36045 df-dveca 36291 df-disoa 36318 df-dvech 36368 df-dib 36428 df-dic 36462 df-dih 36518 df-doch 36637 df-djh 36684 df-lcdual 36876 df-mapd 36914 |
This theorem is referenced by: mapdpglem22 36982 |
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