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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem21 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 39002. (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 7151 | . 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 2798 | . . 3 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶) | |
6 | eqid 2798 | . . 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 38888 | . . 3 ⊢ (𝜑 → 𝐶 ∈ LMod) |
12 | mapdpglem.u | . . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
13 | 8, 12, 10 | dvhlvec 38405 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec) |
14 | mapdpglem3.a | . . . . . . 7 ⊢ 𝐴 = (Scalar‘𝑈) | |
15 | 14 | lvecdrng 19870 | . . . . . 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 38978 | . . . . 5 ⊢ (𝜑 → 𝑔 ≠ 0 ) |
37 | eqid 2798 | . . . . . 6 ⊢ (invr‘𝐴) = (invr‘𝐴) | |
38 | 27, 33, 37 | drnginvrcl 19512 | . . . . 5 ⊢ ((𝐴 ∈ DivRing ∧ 𝑔 ∈ 𝐵 ∧ 𝑔 ≠ 0 ) → ((invr‘𝐴)‘𝑔) ∈ 𝐵) |
39 | 16, 17, 36, 38 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → ((invr‘𝐴)‘𝑔) ∈ 𝐵) |
40 | 8, 12, 14, 27, 9, 5, 6, 10 | lcdsbase 38896 | . . . 4 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) = 𝐵) |
41 | 39, 40 | eleqtrrd 2893 | . . 3 ⊢ (𝜑 → ((invr‘𝐴)‘𝑔) ∈ (Base‘(Scalar‘𝐶))) |
42 | 8, 12, 14, 27, 9, 3, 4, 10, 17, 28 | lcdvscl 38901 | . . 3 ⊢ (𝜑 → (𝑔 · 𝐺) ∈ 𝐹) |
43 | eqid 2798 | . . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
44 | eqid 2798 | . . . . . 6 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
45 | 8, 12, 10 | dvhlmod 38406 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod) |
46 | 19, 43, 21 | lspsncl 19742 | . . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
47 | 45, 23, 46 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
48 | 8, 18, 12, 43, 9, 44, 10, 47 | mapdcl2 38952 | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶)) |
49 | 3, 44 | lssss 19701 | . . . . 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 3916 | . . 3 ⊢ (𝜑 → 𝑧 ∈ 𝐹) |
52 | 3, 4, 5, 6, 7, 11, 41, 42, 51 | lmodsubdi 19684 | . 2 ⊢ (𝜑 → (((invr‘𝐴)‘𝑔) · ((𝑔 · 𝐺)𝑅𝑧)) = ((((invr‘𝐴)‘𝑔) · (𝑔 · 𝐺))𝑅(((invr‘𝐴)‘𝑔) · 𝑧))) |
53 | eqid 2798 | . . . . . . . . 9 ⊢ (.r‘𝐴) = (.r‘𝐴) | |
54 | eqid 2798 | . . . . . . . . 9 ⊢ (1r‘𝐴) = (1r‘𝐴) | |
55 | 27, 33, 53, 54, 37 | drnginvrr 19515 | . . . . . . . 8 ⊢ ((𝐴 ∈ DivRing ∧ 𝑔 ∈ 𝐵 ∧ 𝑔 ≠ 0 ) → (𝑔(.r‘𝐴)((invr‘𝐴)‘𝑔)) = (1r‘𝐴)) |
56 | 16, 17, 36, 55 | syl3anc 1368 | . . . . . . 7 ⊢ (𝜑 → (𝑔(.r‘𝐴)((invr‘𝐴)‘𝑔)) = (1r‘𝐴)) |
57 | eqid 2798 | . . . . . . . 8 ⊢ (1r‘(Scalar‘𝐶)) = (1r‘(Scalar‘𝐶)) | |
58 | 8, 12, 14, 54, 9, 5, 57, 10 | lcd1 38905 | . . . . . . 7 ⊢ (𝜑 → (1r‘(Scalar‘𝐶)) = (1r‘𝐴)) |
59 | 56, 58 | eqtr4d 2836 | . . . . . 6 ⊢ (𝜑 → (𝑔(.r‘𝐴)((invr‘𝐴)‘𝑔)) = (1r‘(Scalar‘𝐶))) |
60 | 59 | oveq1d 7150 | . . . . 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 38903 | . . . . 5 ⊢ (𝜑 → ((𝑔(.r‘𝐴)((invr‘𝐴)‘𝑔)) · 𝐺) = (((invr‘𝐴)‘𝑔) · (𝑔 · 𝐺))) |
62 | 3, 5, 4, 57 | lmodvs1 19655 | . . . . . 6 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((1r‘(Scalar‘𝐶)) · 𝐺) = 𝐺) |
63 | 11, 28, 62 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → ((1r‘(Scalar‘𝐶)) · 𝐺) = 𝐺) |
64 | 60, 61, 63 | 3eqtr3d 2841 | . . . 4 ⊢ (𝜑 → (((invr‘𝐴)‘𝑔) · (𝑔 · 𝐺)) = 𝐺) |
65 | 64 | oveq1d 7150 | . . 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 7146 | . . 3 ⊢ (𝐺𝑅𝐸) = (𝐺𝑅(((invr‘𝐴)‘𝑔) · 𝑧)) |
68 | 65, 67 | eqtr4di 2851 | . 2 ⊢ (𝜑 → ((((invr‘𝐴)‘𝑔) · (𝑔 · 𝐺))𝑅(((invr‘𝐴)‘𝑔) · 𝑧)) = (𝐺𝑅𝐸)) |
69 | 2, 52, 68 | 3eqtrd 2837 | 1 ⊢ (𝜑 → (((invr‘𝐴)‘𝑔) · 𝑡) = (𝐺𝑅𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ⊆ wss 3881 {csn 4525 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 .rcmulr 16558 Scalarcsca 16560 ·𝑠 cvsca 16561 0gc0g 16705 -gcsg 18097 LSSumclsm 18751 1rcur 19244 invrcinvr 19417 DivRingcdr 19495 LModclmod 19627 LSubSpclss 19696 LSpanclspn 19736 LVecclvec 19867 HLchlt 36646 LHypclh 37280 DVecHcdvh 38374 LCDualclcd 38882 mapdcmpd 38920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-riotaBAD 36249 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-undef 7922 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-0g 16707 df-mre 16849 df-mrc 16850 df-acs 16852 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-p1 17642 df-lat 17648 df-clat 17710 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cntz 18439 df-oppg 18466 df-lsm 18753 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-drng 19497 df-lmod 19629 df-lss 19697 df-lsp 19737 df-lvec 19868 df-lsatoms 36272 df-lshyp 36273 df-lcv 36315 df-lfl 36354 df-lkr 36382 df-ldual 36420 df-oposet 36472 df-ol 36474 df-oml 36475 df-covers 36562 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 df-llines 36794 df-lplanes 36795 df-lvols 36796 df-lines 36797 df-psubsp 36799 df-pmap 36800 df-padd 37092 df-lhyp 37284 df-laut 37285 df-ldil 37400 df-ltrn 37401 df-trl 37455 df-tgrp 38039 df-tendo 38051 df-edring 38053 df-dveca 38299 df-disoa 38325 df-dvech 38375 df-dib 38435 df-dic 38469 df-dih 38525 df-doch 38644 df-djh 38691 df-lcdual 38883 df-mapd 38921 |
This theorem is referenced by: mapdpglem22 38989 |
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