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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem2a | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 40158. (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})))) |
Ref | Expression |
---|---|
mapdpglem2a | ⊢ (𝜑 → 𝑡 ∈ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpglem.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | mapdpglem.c | . . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
3 | mapdpglem.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | 1, 2, 3 | lcdlmod 40044 | . . 3 ⊢ (𝜑 → 𝐶 ∈ LMod) |
5 | mapdpglem.m | . . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
6 | mapdpglem.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
7 | eqid 2736 | . . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
8 | eqid 2736 | . . . 4 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
9 | 1, 6, 3 | dvhlmod 39562 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
10 | mapdpglem.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
11 | mapdpglem.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑈) | |
12 | mapdpglem.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈) | |
13 | 11, 7, 12 | lspsncl 20434 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
14 | 9, 10, 13 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
15 | 1, 5, 6, 7, 2, 8, 3, 14 | mapdcl2 40108 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) ∈ (LSubSp‘𝐶)) |
16 | mapdpglem.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
17 | 11, 7, 12 | lspsncl 20434 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
18 | 9, 16, 17 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
19 | 1, 5, 6, 7, 2, 8, 3, 18 | mapdcl2 40108 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶)) |
20 | mapdpglem1.p | . . . 4 ⊢ ⊕ = (LSSum‘𝐶) | |
21 | 8, 20 | lsmcl 20540 | . . 3 ⊢ ((𝐶 ∈ LMod ∧ (𝑀‘(𝑁‘{𝑋})) ∈ (LSubSp‘𝐶) ∧ (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶)) → ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))) ∈ (LSubSp‘𝐶)) |
22 | 4, 15, 19, 21 | syl3anc 1371 | . 2 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))) ∈ (LSubSp‘𝐶)) |
23 | mapdpglem3.te | . 2 ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) | |
24 | mapdpglem3.f | . . 3 ⊢ 𝐹 = (Base‘𝐶) | |
25 | 24, 8 | lssel 20394 | . 2 ⊢ ((((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))) ∈ (LSubSp‘𝐶) ∧ 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) → 𝑡 ∈ 𝐹) |
26 | 22, 23, 25 | syl2anc 584 | 1 ⊢ (𝜑 → 𝑡 ∈ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {csn 4585 ‘cfv 6494 (class class class)co 7354 Basecbs 17080 -gcsg 18747 LSSumclsm 19412 LModclmod 20318 LSubSpclss 20388 LSpanclspn 20428 HLchlt 37801 LHypclh 38436 DVecHcdvh 39530 LCDualclcd 40038 mapdcmpd 40076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 ax-riotaBAD 37404 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 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 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-of 7614 df-om 7800 df-1st 7918 df-2nd 7919 df-tpos 8154 df-undef 8201 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-1o 8409 df-er 8645 df-map 8764 df-en 8881 df-dom 8882 df-sdom 8883 df-fin 8884 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-nn 12151 df-2 12213 df-3 12214 df-4 12215 df-5 12216 df-6 12217 df-n0 12411 df-z 12497 df-uz 12761 df-fz 13422 df-struct 17016 df-sets 17033 df-slot 17051 df-ndx 17063 df-base 17081 df-ress 17110 df-plusg 17143 df-mulr 17144 df-sca 17146 df-vsca 17147 df-0g 17320 df-mre 17463 df-mrc 17464 df-acs 17466 df-proset 18181 df-poset 18199 df-plt 18216 df-lub 18232 df-glb 18233 df-join 18234 df-meet 18235 df-p0 18311 df-p1 18312 df-lat 18318 df-clat 18385 df-mgm 18494 df-sgrp 18543 df-mnd 18554 df-submnd 18599 df-grp 18748 df-minusg 18749 df-sbg 18750 df-subg 18921 df-cntz 19093 df-oppg 19120 df-lsm 19414 df-cmn 19560 df-abl 19561 df-mgp 19893 df-ur 19910 df-ring 19962 df-oppr 20045 df-dvdsr 20066 df-unit 20067 df-invr 20097 df-dvr 20108 df-drng 20183 df-lmod 20320 df-lss 20389 df-lsp 20429 df-lvec 20560 df-lsatoms 37427 df-lshyp 37428 df-lcv 37470 df-lfl 37509 df-lkr 37537 df-ldual 37575 df-oposet 37627 df-ol 37629 df-oml 37630 df-covers 37717 df-ats 37718 df-atl 37749 df-cvlat 37773 df-hlat 37802 df-llines 37950 df-lplanes 37951 df-lvols 37952 df-lines 37953 df-psubsp 37955 df-pmap 37956 df-padd 38248 df-lhyp 38440 df-laut 38441 df-ldil 38556 df-ltrn 38557 df-trl 38611 df-tgrp 39195 df-tendo 39207 df-edring 39209 df-dveca 39455 df-disoa 39481 df-dvech 39531 df-dib 39591 df-dic 39625 df-dih 39681 df-doch 39800 df-djh 39847 df-lcdual 40039 df-mapd 40077 |
This theorem is referenced by: mapdpglem5N 40129 mapdpglem22 40145 |
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