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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem20 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 37312. Baer p. 45, line 8: "...so that (Fy)*=Gy'." (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 |
---|---|
mapdpglem20 | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐸})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2651 | . 2 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
2 | mapdpglem2.j | . 2 ⊢ 𝐽 = (LSpan‘𝐶) | |
3 | eqid 2651 | . 2 ⊢ (LSAtoms‘𝐶) = (LSAtoms‘𝐶) | |
4 | mapdpglem.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | mapdpglem.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
6 | mapdpglem.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | 4, 5, 6 | lcdlvec 37197 | . 2 ⊢ (𝜑 → 𝐶 ∈ LVec) |
8 | mapdpglem.m | . . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
9 | mapdpglem.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
10 | eqid 2651 | . . 3 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈) | |
11 | mapdpglem.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
12 | mapdpglem.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑈) | |
13 | mapdpglem4.q | . . . 4 ⊢ 𝑄 = (0g‘𝑈) | |
14 | 4, 9, 6 | dvhlmod 36716 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
15 | mapdpglem.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
16 | mapdpglem12.yn | . . . . 5 ⊢ (𝜑 → 𝑌 ≠ 𝑄) | |
17 | eldifsn 4350 | . . . . 5 ⊢ (𝑌 ∈ (𝑉 ∖ {𝑄}) ↔ (𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 𝑄)) | |
18 | 15, 16, 17 | sylanbrc 699 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ {𝑄})) |
19 | 11, 12, 13, 10, 14, 18 | lsatlspsn 34598 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSAtoms‘𝑈)) |
20 | 4, 8, 9, 10, 5, 3, 6, 19 | mapdat 37273 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (LSAtoms‘𝐶)) |
21 | mapdpglem.s | . . 3 ⊢ − = (-g‘𝑈) | |
22 | mapdpglem.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
23 | mapdpglem1.p | . . 3 ⊢ ⊕ = (LSSum‘𝐶) | |
24 | mapdpglem3.f | . . 3 ⊢ 𝐹 = (Base‘𝐶) | |
25 | mapdpglem3.te | . . 3 ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) | |
26 | mapdpglem3.a | . . 3 ⊢ 𝐴 = (Scalar‘𝑈) | |
27 | mapdpglem3.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
28 | mapdpglem3.t | . . 3 ⊢ · = ( ·𝑠 ‘𝐶) | |
29 | mapdpglem3.r | . . 3 ⊢ 𝑅 = (-g‘𝐶) | |
30 | mapdpglem3.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
31 | mapdpglem3.e | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) | |
32 | mapdpglem.ne | . . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
33 | mapdpglem4.jt | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) | |
34 | mapdpglem4.z | . . 3 ⊢ 0 = (0g‘𝐴) | |
35 | mapdpglem4.g4 | . . 3 ⊢ (𝜑 → 𝑔 ∈ 𝐵) | |
36 | mapdpglem4.z4 | . . 3 ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) | |
37 | mapdpglem4.t4 | . . 3 ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) | |
38 | mapdpglem4.xn | . . 3 ⊢ (𝜑 → 𝑋 ≠ 𝑄) | |
39 | mapdpglem17.ep | . . 3 ⊢ 𝐸 = (((invr‘𝐴)‘𝑔) · 𝑧) | |
40 | 4, 8, 9, 11, 21, 12, 5, 6, 22, 15, 23, 2, 24, 25, 26, 27, 28, 29, 30, 31, 13, 32, 33, 34, 35, 36, 37, 38, 16, 39 | mapdpglem19 37296 | . 2 ⊢ (𝜑 → 𝐸 ∈ (𝑀‘(𝑁‘{𝑌}))) |
41 | 4, 8, 9, 11, 21, 12, 5, 6, 22, 15, 23, 2, 24, 25, 26, 27, 28, 29, 30, 31, 13, 32, 33, 34, 35, 36, 37, 38, 16, 39 | mapdpglem18 37295 | . 2 ⊢ (𝜑 → 𝐸 ≠ (0g‘𝐶)) |
42 | 1, 2, 3, 7, 20, 40, 41 | lsatel 34610 | 1 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐸})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ∖ cdif 3604 {csn 4210 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 Scalarcsca 15991 ·𝑠 cvsca 15992 0gc0g 16147 -gcsg 17471 LSSumclsm 18095 invrcinvr 18717 LSpanclspn 19019 LSAtomsclsa 34579 HLchlt 34955 LHypclh 35588 DVecHcdvh 36684 LCDualclcd 37192 mapdcmpd 37230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-riotaBAD 34557 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-tpos 7397 df-undef 7444 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-sca 16004 df-vsca 16005 df-0g 16149 df-mre 16293 df-mrc 16294 df-acs 16296 df-preset 16975 df-poset 16993 df-plt 17005 df-lub 17021 df-glb 17022 df-join 17023 df-meet 17024 df-p0 17086 df-p1 17087 df-lat 17093 df-clat 17155 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-grp 17472 df-minusg 17473 df-sbg 17474 df-subg 17638 df-cntz 17796 df-oppg 17822 df-lsm 18097 df-cmn 18241 df-abl 18242 df-mgp 18536 df-ur 18548 df-ring 18595 df-oppr 18669 df-dvdsr 18687 df-unit 18688 df-invr 18718 df-dvr 18729 df-drng 18797 df-lmod 18913 df-lss 18981 df-lsp 19020 df-lvec 19151 df-lsatoms 34581 df-lshyp 34582 df-lcv 34624 df-lfl 34663 df-lkr 34691 df-ldual 34729 df-oposet 34781 df-ol 34783 df-oml 34784 df-covers 34871 df-ats 34872 df-atl 34903 df-cvlat 34927 df-hlat 34956 df-llines 35102 df-lplanes 35103 df-lvols 35104 df-lines 35105 df-psubsp 35107 df-pmap 35108 df-padd 35400 df-lhyp 35592 df-laut 35593 df-ldil 35708 df-ltrn 35709 df-trl 35764 df-tgrp 36348 df-tendo 36360 df-edring 36362 df-dveca 36608 df-disoa 36635 df-dvech 36685 df-dib 36745 df-dic 36779 df-dih 36835 df-doch 36954 df-djh 37001 df-lcdual 37193 df-mapd 37231 |
This theorem is referenced by: mapdpglem23 37300 |
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