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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem13 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 40101. (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) |
mapdpglem12.g0 | ⊢ (𝜑 → 𝑧 = (0g‘𝐶)) |
Ref | Expression |
---|---|
mapdpglem13 | ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ⊆ (𝑁‘{𝑋})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpglem4.jt | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) | |
2 | eqid 2738 | . . . 4 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
3 | mapdpglem2.j | . . . 4 ⊢ 𝐽 = (LSpan‘𝐶) | |
4 | mapdpglem.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | mapdpglem.c | . . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
6 | mapdpglem.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | 4, 5, 6 | lcdlmod 39987 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod) |
8 | mapdpglem.m | . . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
9 | mapdpglem.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
10 | eqid 2738 | . . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
11 | 4, 9, 6 | dvhlmod 39505 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod) |
12 | mapdpglem.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
13 | mapdpglem.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑈) | |
14 | mapdpglem.n | . . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑈) | |
15 | 13, 10, 14 | lspsncl 20391 | . . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
16 | 11, 12, 15 | syl2anc 585 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
17 | 4, 8, 9, 10, 5, 2, 6, 16 | mapdcl2 40051 | . . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) ∈ (LSubSp‘𝐶)) |
18 | mapdpglem.s | . . . . 5 ⊢ − = (-g‘𝑈) | |
19 | mapdpglem.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
20 | mapdpglem1.p | . . . . 5 ⊢ ⊕ = (LSSum‘𝐶) | |
21 | mapdpglem3.f | . . . . 5 ⊢ 𝐹 = (Base‘𝐶) | |
22 | mapdpglem3.te | . . . . 5 ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) | |
23 | mapdpglem3.a | . . . . 5 ⊢ 𝐴 = (Scalar‘𝑈) | |
24 | mapdpglem3.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐴) | |
25 | mapdpglem3.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝐶) | |
26 | mapdpglem3.r | . . . . 5 ⊢ 𝑅 = (-g‘𝐶) | |
27 | mapdpglem3.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
28 | mapdpglem3.e | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) | |
29 | mapdpglem4.q | . . . . 5 ⊢ 𝑄 = (0g‘𝑈) | |
30 | mapdpglem.ne | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
31 | mapdpglem4.z | . . . . 5 ⊢ 0 = (0g‘𝐴) | |
32 | mapdpglem4.g4 | . . . . 5 ⊢ (𝜑 → 𝑔 ∈ 𝐵) | |
33 | mapdpglem4.z4 | . . . . 5 ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) | |
34 | mapdpglem4.t4 | . . . . 5 ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) | |
35 | mapdpglem4.xn | . . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝑄) | |
36 | mapdpglem12.yn | . . . . 5 ⊢ (𝜑 → 𝑌 ≠ 𝑄) | |
37 | mapdpglem12.g0 | . . . . 5 ⊢ (𝜑 → 𝑧 = (0g‘𝐶)) | |
38 | 4, 8, 9, 13, 18, 14, 5, 6, 12, 19, 20, 3, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 1, 31, 32, 33, 34, 35, 36, 37 | mapdpglem12 40078 | . . . 4 ⊢ (𝜑 → 𝑡 ∈ (𝑀‘(𝑁‘{𝑋}))) |
39 | 2, 3, 7, 17, 38 | lspsnel5a 20410 | . . 3 ⊢ (𝜑 → (𝐽‘{𝑡}) ⊆ (𝑀‘(𝑁‘{𝑋}))) |
40 | 1, 39 | eqsstrd 3981 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) ⊆ (𝑀‘(𝑁‘{𝑋}))) |
41 | 13, 18 | lmodvsubcl 20320 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉) |
42 | 11, 12, 19, 41 | syl3anc 1372 | . . . 4 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝑉) |
43 | 13, 10, 14 | lspsncl 20391 | . . . 4 ⊢ ((𝑈 ∈ LMod ∧ (𝑋 − 𝑌) ∈ 𝑉) → (𝑁‘{(𝑋 − 𝑌)}) ∈ (LSubSp‘𝑈)) |
44 | 11, 42, 43 | syl2anc 585 | . . 3 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ∈ (LSubSp‘𝑈)) |
45 | 4, 9, 10, 8, 6, 44, 16 | mapdord 40033 | . 2 ⊢ (𝜑 → ((𝑀‘(𝑁‘{(𝑋 − 𝑌)})) ⊆ (𝑀‘(𝑁‘{𝑋})) ↔ (𝑁‘{(𝑋 − 𝑌)}) ⊆ (𝑁‘{𝑋}))) |
46 | 40, 45 | mpbid 231 | 1 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ⊆ (𝑁‘{𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2942 ⊆ wss 3909 {csn 4585 ‘cfv 6494 (class class class)co 7352 Basecbs 17043 Scalarcsca 17096 ·𝑠 cvsca 17097 0gc0g 17281 -gcsg 18710 LSSumclsm 19375 LModclmod 20275 LSubSpclss 20345 LSpanclspn 20385 HLchlt 37744 LHypclh 38379 DVecHcdvh 39473 LCDualclcd 39981 mapdcmpd 40019 |
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 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 ax-riotaBAD 37347 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7610 df-om 7796 df-1st 7914 df-2nd 7915 df-tpos 8150 df-undef 8197 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-1o 8405 df-er 8607 df-map 8726 df-en 8843 df-dom 8844 df-sdom 8845 df-fin 8846 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-nn 12113 df-2 12175 df-3 12176 df-4 12177 df-5 12178 df-6 12179 df-n0 12373 df-z 12459 df-uz 12723 df-fz 13380 df-struct 16979 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-ress 17073 df-plusg 17106 df-mulr 17107 df-sca 17109 df-vsca 17110 df-0g 17283 df-mre 17426 df-mrc 17427 df-acs 17429 df-proset 18144 df-poset 18162 df-plt 18179 df-lub 18195 df-glb 18196 df-join 18197 df-meet 18198 df-p0 18274 df-p1 18275 df-lat 18281 df-clat 18348 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-submnd 18562 df-grp 18711 df-minusg 18712 df-sbg 18713 df-subg 18884 df-cntz 19056 df-oppg 19083 df-lsm 19377 df-cmn 19523 df-abl 19524 df-mgp 19856 df-ur 19873 df-ring 19920 df-oppr 20002 df-dvdsr 20023 df-unit 20024 df-invr 20054 df-dvr 20065 df-drng 20140 df-lmod 20277 df-lss 20346 df-lsp 20386 df-lvec 20517 df-lsatoms 37370 df-lshyp 37371 df-lcv 37413 df-lfl 37452 df-lkr 37480 df-ldual 37518 df-oposet 37570 df-ol 37572 df-oml 37573 df-covers 37660 df-ats 37661 df-atl 37692 df-cvlat 37716 df-hlat 37745 df-llines 37893 df-lplanes 37894 df-lvols 37895 df-lines 37896 df-psubsp 37898 df-pmap 37899 df-padd 38191 df-lhyp 38383 df-laut 38384 df-ldil 38499 df-ltrn 38500 df-trl 38554 df-tgrp 39138 df-tendo 39150 df-edring 39152 df-dveca 39398 df-disoa 39424 df-dvech 39474 df-dib 39534 df-dic 39568 df-dih 39624 df-doch 39743 df-djh 39790 df-lcdual 39982 df-mapd 40020 |
This theorem is referenced by: mapdpglem14 40080 |
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