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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem21 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 39332. (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 7180 | . 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 2738 | . . 3 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶) | |
6 | eqid 2738 | . . 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 39218 | . . 3 ⊢ (𝜑 → 𝐶 ∈ LMod) |
12 | mapdpglem.u | . . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
13 | 8, 12, 10 | dvhlvec 38735 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec) |
14 | mapdpglem3.a | . . . . . . 7 ⊢ 𝐴 = (Scalar‘𝑈) | |
15 | 14 | lvecdrng 19989 | . . . . . 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 39308 | . . . . 5 ⊢ (𝜑 → 𝑔 ≠ 0 ) |
37 | eqid 2738 | . . . . . 6 ⊢ (invr‘𝐴) = (invr‘𝐴) | |
38 | 27, 33, 37 | drnginvrcl 19631 | . . . . 5 ⊢ ((𝐴 ∈ DivRing ∧ 𝑔 ∈ 𝐵 ∧ 𝑔 ≠ 0 ) → ((invr‘𝐴)‘𝑔) ∈ 𝐵) |
39 | 16, 17, 36, 38 | syl3anc 1372 | . . . 4 ⊢ (𝜑 → ((invr‘𝐴)‘𝑔) ∈ 𝐵) |
40 | 8, 12, 14, 27, 9, 5, 6, 10 | lcdsbase 39226 | . . . 4 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) = 𝐵) |
41 | 39, 40 | eleqtrrd 2836 | . . 3 ⊢ (𝜑 → ((invr‘𝐴)‘𝑔) ∈ (Base‘(Scalar‘𝐶))) |
42 | 8, 12, 14, 27, 9, 3, 4, 10, 17, 28 | lcdvscl 39231 | . . 3 ⊢ (𝜑 → (𝑔 · 𝐺) ∈ 𝐹) |
43 | eqid 2738 | . . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
44 | eqid 2738 | . . . . . 6 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
45 | 8, 12, 10 | dvhlmod 38736 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod) |
46 | 19, 43, 21 | lspsncl 19861 | . . . . . . 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 39282 | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶)) |
49 | 3, 44 | lssss 19820 | . . . . 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 3876 | . . 3 ⊢ (𝜑 → 𝑧 ∈ 𝐹) |
52 | 3, 4, 5, 6, 7, 11, 41, 42, 51 | lmodsubdi 19803 | . 2 ⊢ (𝜑 → (((invr‘𝐴)‘𝑔) · ((𝑔 · 𝐺)𝑅𝑧)) = ((((invr‘𝐴)‘𝑔) · (𝑔 · 𝐺))𝑅(((invr‘𝐴)‘𝑔) · 𝑧))) |
53 | eqid 2738 | . . . . . . . . 9 ⊢ (.r‘𝐴) = (.r‘𝐴) | |
54 | eqid 2738 | . . . . . . . . 9 ⊢ (1r‘𝐴) = (1r‘𝐴) | |
55 | 27, 33, 53, 54, 37 | drnginvrr 19634 | . . . . . . . 8 ⊢ ((𝐴 ∈ DivRing ∧ 𝑔 ∈ 𝐵 ∧ 𝑔 ≠ 0 ) → (𝑔(.r‘𝐴)((invr‘𝐴)‘𝑔)) = (1r‘𝐴)) |
56 | 16, 17, 36, 55 | syl3anc 1372 | . . . . . . 7 ⊢ (𝜑 → (𝑔(.r‘𝐴)((invr‘𝐴)‘𝑔)) = (1r‘𝐴)) |
57 | eqid 2738 | . . . . . . . 8 ⊢ (1r‘(Scalar‘𝐶)) = (1r‘(Scalar‘𝐶)) | |
58 | 8, 12, 14, 54, 9, 5, 57, 10 | lcd1 39235 | . . . . . . 7 ⊢ (𝜑 → (1r‘(Scalar‘𝐶)) = (1r‘𝐴)) |
59 | 56, 58 | eqtr4d 2776 | . . . . . 6 ⊢ (𝜑 → (𝑔(.r‘𝐴)((invr‘𝐴)‘𝑔)) = (1r‘(Scalar‘𝐶))) |
60 | 59 | oveq1d 7179 | . . . . 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 39233 | . . . . 5 ⊢ (𝜑 → ((𝑔(.r‘𝐴)((invr‘𝐴)‘𝑔)) · 𝐺) = (((invr‘𝐴)‘𝑔) · (𝑔 · 𝐺))) |
62 | 3, 5, 4, 57 | lmodvs1 19774 | . . . . . 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 2781 | . . . 4 ⊢ (𝜑 → (((invr‘𝐴)‘𝑔) · (𝑔 · 𝐺)) = 𝐺) |
65 | 64 | oveq1d 7179 | . . 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 7175 | . . 3 ⊢ (𝐺𝑅𝐸) = (𝐺𝑅(((invr‘𝐴)‘𝑔) · 𝑧)) |
68 | 65, 67 | eqtr4di 2791 | . 2 ⊢ (𝜑 → ((((invr‘𝐴)‘𝑔) · (𝑔 · 𝐺))𝑅(((invr‘𝐴)‘𝑔) · 𝑧)) = (𝐺𝑅𝐸)) |
69 | 2, 52, 68 | 3eqtrd 2777 | 1 ⊢ (𝜑 → (((invr‘𝐴)‘𝑔) · 𝑡) = (𝐺𝑅𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 ⊆ wss 3841 {csn 4513 ‘cfv 6333 (class class class)co 7164 Basecbs 16579 .rcmulr 16662 Scalarcsca 16664 ·𝑠 cvsca 16665 0gc0g 16809 -gcsg 18214 LSSumclsm 18870 1rcur 19363 invrcinvr 19536 DivRingcdr 19614 LModclmod 19746 LSubSpclss 19815 LSpanclspn 19855 LVecclvec 19986 HLchlt 36976 LHypclh 37610 DVecHcdvh 38704 LCDualclcd 39212 mapdcmpd 39250 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-riotaBAD 36579 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-iin 4881 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 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 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-of 7419 df-om 7594 df-1st 7707 df-2nd 7708 df-tpos 7914 df-undef 7961 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-map 8432 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-n0 11970 df-z 12056 df-uz 12318 df-fz 12975 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-mulr 16675 df-sca 16677 df-vsca 16678 df-0g 16811 df-mre 16953 df-mrc 16954 df-acs 16956 df-proset 17647 df-poset 17665 df-plt 17677 df-lub 17693 df-glb 17694 df-join 17695 df-meet 17696 df-p0 17758 df-p1 17759 df-lat 17765 df-clat 17827 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-submnd 18066 df-grp 18215 df-minusg 18216 df-sbg 18217 df-subg 18387 df-cntz 18558 df-oppg 18585 df-lsm 18872 df-cmn 19019 df-abl 19020 df-mgp 19352 df-ur 19364 df-ring 19411 df-oppr 19488 df-dvdsr 19506 df-unit 19507 df-invr 19537 df-dvr 19548 df-drng 19616 df-lmod 19748 df-lss 19816 df-lsp 19856 df-lvec 19987 df-lsatoms 36602 df-lshyp 36603 df-lcv 36645 df-lfl 36684 df-lkr 36712 df-ldual 36750 df-oposet 36802 df-ol 36804 df-oml 36805 df-covers 36892 df-ats 36893 df-atl 36924 df-cvlat 36948 df-hlat 36977 df-llines 37124 df-lplanes 37125 df-lvols 37126 df-lines 37127 df-psubsp 37129 df-pmap 37130 df-padd 37422 df-lhyp 37614 df-laut 37615 df-ldil 37730 df-ltrn 37731 df-trl 37785 df-tgrp 38369 df-tendo 38381 df-edring 38383 df-dveca 38629 df-disoa 38655 df-dvech 38705 df-dib 38765 df-dic 38799 df-dih 38855 df-doch 38974 df-djh 39021 df-lcdual 39213 df-mapd 39251 |
This theorem is referenced by: mapdpglem22 39319 |
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