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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem18 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 37780. Baer p. 45, line 7: "Then y =/= 0..." (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 |
---|---|
mapdpglem18 | ⊢ (𝜑 → 𝐸 ≠ (0g‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpglem.h | . . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | mapdpglem.u | . . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | mapdpglem.k | . . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | 1, 2, 3 | dvhlvec 37183 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec) |
5 | mapdpglem3.a | . . . . . . 7 ⊢ 𝐴 = (Scalar‘𝑈) | |
6 | 5 | lvecdrng 19471 | . . . . . 6 ⊢ (𝑈 ∈ LVec → 𝐴 ∈ DivRing) |
7 | 4, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ DivRing) |
8 | mapdpglem4.g4 | . . . . 5 ⊢ (𝜑 → 𝑔 ∈ 𝐵) | |
9 | mapdpglem.m | . . . . . 6 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
10 | mapdpglem.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑈) | |
11 | mapdpglem.s | . . . . . 6 ⊢ − = (-g‘𝑈) | |
12 | mapdpglem.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈) | |
13 | mapdpglem.c | . . . . . 6 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
14 | mapdpglem.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
15 | mapdpglem.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
16 | mapdpglem1.p | . . . . . 6 ⊢ ⊕ = (LSSum‘𝐶) | |
17 | mapdpglem2.j | . . . . . 6 ⊢ 𝐽 = (LSpan‘𝐶) | |
18 | mapdpglem3.f | . . . . . 6 ⊢ 𝐹 = (Base‘𝐶) | |
19 | mapdpglem3.te | . . . . . 6 ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) | |
20 | mapdpglem3.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐴) | |
21 | mapdpglem3.t | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝐶) | |
22 | mapdpglem3.r | . . . . . 6 ⊢ 𝑅 = (-g‘𝐶) | |
23 | mapdpglem3.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
24 | mapdpglem3.e | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) | |
25 | mapdpglem4.q | . . . . . 6 ⊢ 𝑄 = (0g‘𝑈) | |
26 | mapdpglem.ne | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
27 | mapdpglem4.jt | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) | |
28 | mapdpglem4.z | . . . . . 6 ⊢ 0 = (0g‘𝐴) | |
29 | mapdpglem4.z4 | . . . . . 6 ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) | |
30 | mapdpglem4.t4 | . . . . . 6 ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) | |
31 | mapdpglem4.xn | . . . . . 6 ⊢ (𝜑 → 𝑋 ≠ 𝑄) | |
32 | 1, 9, 2, 10, 11, 12, 13, 3, 14, 15, 16, 17, 18, 19, 5, 20, 21, 22, 23, 24, 25, 26, 27, 28, 8, 29, 30, 31 | mapdpglem11 37756 | . . . . 5 ⊢ (𝜑 → 𝑔 ≠ 0 ) |
33 | eqid 2825 | . . . . . 6 ⊢ (invr‘𝐴) = (invr‘𝐴) | |
34 | 20, 28, 33 | drnginvrn0 19128 | . . . . 5 ⊢ ((𝐴 ∈ DivRing ∧ 𝑔 ∈ 𝐵 ∧ 𝑔 ≠ 0 ) → ((invr‘𝐴)‘𝑔) ≠ 0 ) |
35 | 7, 8, 32, 34 | syl3anc 1494 | . . . 4 ⊢ (𝜑 → ((invr‘𝐴)‘𝑔) ≠ 0 ) |
36 | eqid 2825 | . . . . 5 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶) | |
37 | eqid 2825 | . . . . 5 ⊢ (0g‘(Scalar‘𝐶)) = (0g‘(Scalar‘𝐶)) | |
38 | 1, 2, 5, 28, 13, 36, 37, 3 | lcd0 37682 | . . . 4 ⊢ (𝜑 → (0g‘(Scalar‘𝐶)) = 0 ) |
39 | 35, 38 | neeqtrrd 3073 | . . 3 ⊢ (𝜑 → ((invr‘𝐴)‘𝑔) ≠ (0g‘(Scalar‘𝐶))) |
40 | mapdpglem12.yn | . . . 4 ⊢ (𝜑 → 𝑌 ≠ 𝑄) | |
41 | 1, 9, 2, 10, 11, 12, 13, 3, 14, 15, 16, 17, 18, 19, 5, 20, 21, 22, 23, 24, 25, 26, 27, 28, 8, 29, 30, 31, 40 | mapdpglem16 37761 | . . 3 ⊢ (𝜑 → 𝑧 ≠ (0g‘𝐶)) |
42 | eqid 2825 | . . . 4 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶)) | |
43 | eqid 2825 | . . . 4 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
44 | 1, 13, 3 | lcdlvec 37665 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LVec) |
45 | 20, 28, 33 | drnginvrcl 19127 | . . . . . 6 ⊢ ((𝐴 ∈ DivRing ∧ 𝑔 ∈ 𝐵 ∧ 𝑔 ≠ 0 ) → ((invr‘𝐴)‘𝑔) ∈ 𝐵) |
46 | 7, 8, 32, 45 | syl3anc 1494 | . . . . 5 ⊢ (𝜑 → ((invr‘𝐴)‘𝑔) ∈ 𝐵) |
47 | 1, 2, 5, 20, 13, 36, 42, 3 | lcdsbase 37674 | . . . . 5 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) = 𝐵) |
48 | 46, 47 | eleqtrrd 2909 | . . . 4 ⊢ (𝜑 → ((invr‘𝐴)‘𝑔) ∈ (Base‘(Scalar‘𝐶))) |
49 | eqid 2825 | . . . . . . 7 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
50 | eqid 2825 | . . . . . . 7 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
51 | 1, 2, 3 | dvhlmod 37184 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod) |
52 | 10, 49, 12 | lspsncl 19343 | . . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
53 | 51, 15, 52 | syl2anc 579 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
54 | 1, 9, 2, 49, 13, 50, 3, 53 | mapdcl2 37730 | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶)) |
55 | 18, 50 | lssss 19300 | . . . . . 6 ⊢ ((𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶) → (𝑀‘(𝑁‘{𝑌})) ⊆ 𝐹) |
56 | 54, 55 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ⊆ 𝐹) |
57 | 56, 29 | sseldd 3828 | . . . 4 ⊢ (𝜑 → 𝑧 ∈ 𝐹) |
58 | 18, 21, 36, 42, 37, 43, 44, 48, 57 | lvecvsn0 19475 | . . 3 ⊢ (𝜑 → ((((invr‘𝐴)‘𝑔) · 𝑧) ≠ (0g‘𝐶) ↔ (((invr‘𝐴)‘𝑔) ≠ (0g‘(Scalar‘𝐶)) ∧ 𝑧 ≠ (0g‘𝐶)))) |
59 | 39, 41, 58 | mpbir2and 704 | . 2 ⊢ (𝜑 → (((invr‘𝐴)‘𝑔) · 𝑧) ≠ (0g‘𝐶)) |
60 | mapdpglem17.ep | . 2 ⊢ 𝐸 = (((invr‘𝐴)‘𝑔) · 𝑧) | |
61 | 59, 60, 43 | 3netr4g 3078 | 1 ⊢ (𝜑 → 𝐸 ≠ (0g‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 ⊆ wss 3798 {csn 4399 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 Scalarcsca 16315 ·𝑠 cvsca 16316 0gc0g 16460 -gcsg 17785 LSSumclsm 18407 invrcinvr 19032 DivRingcdr 19110 LModclmod 19226 LSubSpclss 19295 LSpanclspn 19337 LVecclvec 19468 HLchlt 35424 LHypclh 36058 DVecHcdvh 37152 LCDualclcd 37660 mapdcmpd 37698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-riotaBAD 35027 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-om 7332 df-1st 7433 df-2nd 7434 df-tpos 7622 df-undef 7669 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-n0 11626 df-z 11712 df-uz 11976 df-fz 12627 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-sca 16328 df-vsca 16329 df-0g 16462 df-mre 16606 df-mrc 16607 df-acs 16609 df-proset 17288 df-poset 17306 df-plt 17318 df-lub 17334 df-glb 17335 df-join 17336 df-meet 17337 df-p0 17399 df-p1 17400 df-lat 17406 df-clat 17468 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-submnd 17696 df-grp 17786 df-minusg 17787 df-sbg 17788 df-subg 17949 df-cntz 18107 df-oppg 18133 df-lsm 18409 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-ring 18910 df-oppr 18984 df-dvdsr 19002 df-unit 19003 df-invr 19033 df-dvr 19044 df-drng 19112 df-lmod 19228 df-lss 19296 df-lsp 19338 df-lvec 19469 df-lsatoms 35050 df-lshyp 35051 df-lcv 35093 df-lfl 35132 df-lkr 35160 df-ldual 35198 df-oposet 35250 df-ol 35252 df-oml 35253 df-covers 35340 df-ats 35341 df-atl 35372 df-cvlat 35396 df-hlat 35425 df-llines 35572 df-lplanes 35573 df-lvols 35574 df-lines 35575 df-psubsp 35577 df-pmap 35578 df-padd 35870 df-lhyp 36062 df-laut 36063 df-ldil 36178 df-ltrn 36179 df-trl 36233 df-tgrp 36817 df-tendo 36829 df-edring 36831 df-dveca 37077 df-disoa 37103 df-dvech 37153 df-dib 37213 df-dic 37247 df-dih 37303 df-doch 37422 df-djh 37469 df-lcdual 37661 df-mapd 37699 |
This theorem is referenced by: mapdpglem20 37765 |
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