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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem6 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 38841. Baer p. 45, line 4: "If g were 0, then t would be in (Fy)*..." (Contributed by NM, 18-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) |
mapdpglem4.g0 | ⊢ (𝜑 → 𝑔 = 0 ) |
Ref | Expression |
---|---|
mapdpglem6 | ⊢ (𝜑 → 𝑡 ∈ (𝑀‘(𝑁‘{𝑌}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpglem4.t4 | . 2 ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) | |
2 | mapdpglem.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | mapdpglem.c | . . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
4 | mapdpglem.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
5 | 2, 3, 4 | lcdlmod 38727 | . . 3 ⊢ (𝜑 → 𝐶 ∈ LMod) |
6 | mapdpglem.m | . . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
7 | mapdpglem.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
8 | eqid 2821 | . . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
9 | eqid 2821 | . . . 4 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
10 | 2, 7, 4 | dvhlmod 38245 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
11 | mapdpglem.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
12 | mapdpglem.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑈) | |
13 | mapdpglem.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈) | |
14 | 12, 8, 13 | lspsncl 19748 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
15 | 10, 11, 14 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
16 | 2, 6, 7, 8, 3, 9, 4, 15 | mapdcl2 38791 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶)) |
17 | mapdpglem4.g0 | . . . . . 6 ⊢ (𝜑 → 𝑔 = 0 ) | |
18 | 17 | oveq1d 7170 | . . . . 5 ⊢ (𝜑 → (𝑔 · 𝐺) = ( 0 · 𝐺)) |
19 | mapdpglem3.a | . . . . . 6 ⊢ 𝐴 = (Scalar‘𝑈) | |
20 | mapdpglem4.z | . . . . . 6 ⊢ 0 = (0g‘𝐴) | |
21 | mapdpglem3.f | . . . . . 6 ⊢ 𝐹 = (Base‘𝐶) | |
22 | mapdpglem3.t | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝐶) | |
23 | eqid 2821 | . . . . . 6 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
24 | mapdpglem3.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
25 | 2, 7, 19, 20, 3, 21, 22, 23, 4, 24 | lcd0vs 38750 | . . . . 5 ⊢ (𝜑 → ( 0 · 𝐺) = (0g‘𝐶)) |
26 | 18, 25 | eqtrd 2856 | . . . 4 ⊢ (𝜑 → (𝑔 · 𝐺) = (0g‘𝐶)) |
27 | 23, 9 | lss0cl 19717 | . . . . 5 ⊢ ((𝐶 ∈ LMod ∧ (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶)) → (0g‘𝐶) ∈ (𝑀‘(𝑁‘{𝑌}))) |
28 | 5, 16, 27 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (0g‘𝐶) ∈ (𝑀‘(𝑁‘{𝑌}))) |
29 | 26, 28 | eqeltrd 2913 | . . 3 ⊢ (𝜑 → (𝑔 · 𝐺) ∈ (𝑀‘(𝑁‘{𝑌}))) |
30 | mapdpglem4.z4 | . . 3 ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) | |
31 | mapdpglem3.r | . . . 4 ⊢ 𝑅 = (-g‘𝐶) | |
32 | 31, 9 | lssvsubcl 19714 | . . 3 ⊢ (((𝐶 ∈ LMod ∧ (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶)) ∧ ((𝑔 · 𝐺) ∈ (𝑀‘(𝑁‘{𝑌})) ∧ 𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))) → ((𝑔 · 𝐺)𝑅𝑧) ∈ (𝑀‘(𝑁‘{𝑌}))) |
33 | 5, 16, 29, 30, 32 | syl22anc 836 | . 2 ⊢ (𝜑 → ((𝑔 · 𝐺)𝑅𝑧) ∈ (𝑀‘(𝑁‘{𝑌}))) |
34 | 1, 33 | eqeltrd 2913 | 1 ⊢ (𝜑 → 𝑡 ∈ (𝑀‘(𝑁‘{𝑌}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 {csn 4566 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 Scalarcsca 16567 ·𝑠 cvsca 16568 0gc0g 16712 -gcsg 18104 LSSumclsm 18758 LModclmod 19633 LSubSpclss 19702 LSpanclspn 19742 HLchlt 36485 LHypclh 37119 DVecHcdvh 38213 LCDualclcd 38721 mapdcmpd 38759 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-riotaBAD 36088 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-tpos 7891 df-undef 7938 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-sca 16580 df-vsca 16581 df-0g 16714 df-mre 16856 df-mrc 16857 df-acs 16859 df-proset 17537 df-poset 17555 df-plt 17567 df-lub 17583 df-glb 17584 df-join 17585 df-meet 17586 df-p0 17648 df-p1 17649 df-lat 17655 df-clat 17717 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-grp 18105 df-minusg 18106 df-sbg 18107 df-subg 18275 df-cntz 18446 df-oppg 18473 df-lsm 18760 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-ring 19298 df-oppr 19372 df-dvdsr 19390 df-unit 19391 df-invr 19421 df-dvr 19432 df-drng 19503 df-lmod 19635 df-lss 19703 df-lsp 19743 df-lvec 19874 df-lsatoms 36111 df-lshyp 36112 df-lcv 36154 df-lfl 36193 df-lkr 36221 df-ldual 36259 df-oposet 36311 df-ol 36313 df-oml 36314 df-covers 36401 df-ats 36402 df-atl 36433 df-cvlat 36457 df-hlat 36486 df-llines 36633 df-lplanes 36634 df-lvols 36635 df-lines 36636 df-psubsp 36638 df-pmap 36639 df-padd 36931 df-lhyp 37123 df-laut 37124 df-ldil 37239 df-ltrn 37240 df-trl 37294 df-tgrp 37878 df-tendo 37890 df-edring 37892 df-dveca 38138 df-disoa 38164 df-dvech 38214 df-dib 38274 df-dic 38308 df-dih 38364 df-doch 38483 df-djh 38530 df-lcdual 38722 df-mapd 38760 |
This theorem is referenced by: mapdpglem8 38814 |
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