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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem20 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 38722. 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 2818 | . 2 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
2 | mapdpglem2.j | . 2 ⊢ 𝐽 = (LSpan‘𝐶) | |
3 | eqid 2818 | . 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 38607 | . 2 ⊢ (𝜑 → 𝐶 ∈ LVec) |
8 | mapdpglem.m | . . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
9 | mapdpglem.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
10 | eqid 2818 | . . 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 38126 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
15 | mapdpglem.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
16 | mapdpglem12.yn | . . . . 5 ⊢ (𝜑 → 𝑌 ≠ 𝑄) | |
17 | eldifsn 4711 | . . . . 5 ⊢ (𝑌 ∈ (𝑉 ∖ {𝑄}) ↔ (𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 𝑄)) | |
18 | 15, 16, 17 | sylanbrc 583 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ {𝑄})) |
19 | 11, 12, 13, 10, 14, 18 | lsatlspsn 36009 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSAtoms‘𝑈)) |
20 | 4, 8, 9, 10, 5, 3, 6, 19 | mapdat 38683 | . 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 38706 | . 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 38705 | . 2 ⊢ (𝜑 → 𝐸 ≠ (0g‘𝐶)) |
42 | 1, 2, 3, 7, 20, 40, 41 | lsatel 36021 | 1 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐸})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∖ cdif 3930 {csn 4557 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 Scalarcsca 16556 ·𝑠 cvsca 16557 0gc0g 16701 -gcsg 18043 LSSumclsm 18688 invrcinvr 19350 LSpanclspn 19672 LSAtomsclsa 35990 HLchlt 36366 LHypclh 37000 DVecHcdvh 38094 LCDualclcd 38602 mapdcmpd 38640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-riotaBAD 35969 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-undef 7928 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 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 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-0g 16703 df-mre 16845 df-mrc 16846 df-acs 16848 df-proset 17526 df-poset 17544 df-plt 17556 df-lub 17572 df-glb 17573 df-join 17574 df-meet 17575 df-p0 17637 df-p1 17638 df-lat 17644 df-clat 17706 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-cntz 18385 df-oppg 18412 df-lsm 18690 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-drng 19433 df-lmod 19565 df-lss 19633 df-lsp 19673 df-lvec 19804 df-lsatoms 35992 df-lshyp 35993 df-lcv 36035 df-lfl 36074 df-lkr 36102 df-ldual 36140 df-oposet 36192 df-ol 36194 df-oml 36195 df-covers 36282 df-ats 36283 df-atl 36314 df-cvlat 36338 df-hlat 36367 df-llines 36514 df-lplanes 36515 df-lvols 36516 df-lines 36517 df-psubsp 36519 df-pmap 36520 df-padd 36812 df-lhyp 37004 df-laut 37005 df-ldil 37120 df-ltrn 37121 df-trl 37175 df-tgrp 37759 df-tendo 37771 df-edring 37773 df-dveca 38019 df-disoa 38045 df-dvech 38095 df-dib 38155 df-dic 38189 df-dih 38245 df-doch 38364 df-djh 38411 df-lcdual 38603 df-mapd 38641 |
This theorem is referenced by: mapdpglem23 38710 |
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