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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem13 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 41688. (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) |
mapdpglem12.g0 | ⊢ (𝜑 → 𝑧 = (0g‘𝐶)) |
Ref | Expression |
---|---|
mapdpglem13 | ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ⊆ (𝑁‘{𝑋})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpglem4.jt | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) | |
2 | eqid 2734 | . . . 4 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶) | |
3 | mapdpglem2.j | . . . 4 ⊢ 𝐽 = (LSpan‘𝐶) | |
4 | mapdpglem.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | mapdpglem.c | . . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
6 | mapdpglem.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | 4, 5, 6 | lcdlmod 41574 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod) |
8 | mapdpglem.m | . . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
9 | mapdpglem.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
10 | eqid 2734 | . . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
11 | 4, 9, 6 | dvhlmod 41092 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod) |
12 | mapdpglem.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
13 | mapdpglem.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑈) | |
14 | mapdpglem.n | . . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑈) | |
15 | 13, 10, 14 | lspsncl 20992 | . . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
16 | 11, 12, 15 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
17 | 4, 8, 9, 10, 5, 2, 6, 16 | mapdcl2 41638 | . . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) ∈ (LSubSp‘𝐶)) |
18 | mapdpglem.s | . . . . 5 ⊢ − = (-g‘𝑈) | |
19 | mapdpglem.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
20 | mapdpglem1.p | . . . . 5 ⊢ ⊕ = (LSSum‘𝐶) | |
21 | mapdpglem3.f | . . . . 5 ⊢ 𝐹 = (Base‘𝐶) | |
22 | mapdpglem3.te | . . . . 5 ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) | |
23 | mapdpglem3.a | . . . . 5 ⊢ 𝐴 = (Scalar‘𝑈) | |
24 | mapdpglem3.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐴) | |
25 | mapdpglem3.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝐶) | |
26 | mapdpglem3.r | . . . . 5 ⊢ 𝑅 = (-g‘𝐶) | |
27 | mapdpglem3.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
28 | mapdpglem3.e | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) | |
29 | mapdpglem4.q | . . . . 5 ⊢ 𝑄 = (0g‘𝑈) | |
30 | mapdpglem.ne | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
31 | mapdpglem4.z | . . . . 5 ⊢ 0 = (0g‘𝐴) | |
32 | mapdpglem4.g4 | . . . . 5 ⊢ (𝜑 → 𝑔 ∈ 𝐵) | |
33 | mapdpglem4.z4 | . . . . 5 ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) | |
34 | mapdpglem4.t4 | . . . . 5 ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) | |
35 | mapdpglem4.xn | . . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝑄) | |
36 | mapdpglem12.yn | . . . . 5 ⊢ (𝜑 → 𝑌 ≠ 𝑄) | |
37 | mapdpglem12.g0 | . . . . 5 ⊢ (𝜑 → 𝑧 = (0g‘𝐶)) | |
38 | 4, 8, 9, 13, 18, 14, 5, 6, 12, 19, 20, 3, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 1, 31, 32, 33, 34, 35, 36, 37 | mapdpglem12 41665 | . . . 4 ⊢ (𝜑 → 𝑡 ∈ (𝑀‘(𝑁‘{𝑋}))) |
39 | 2, 3, 7, 17, 38 | ellspsn5 21011 | . . 3 ⊢ (𝜑 → (𝐽‘{𝑡}) ⊆ (𝑀‘(𝑁‘{𝑋}))) |
40 | 1, 39 | eqsstrd 4033 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) ⊆ (𝑀‘(𝑁‘{𝑋}))) |
41 | 13, 18 | lmodvsubcl 20921 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉) |
42 | 11, 12, 19, 41 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝑉) |
43 | 13, 10, 14 | lspsncl 20992 | . . . 4 ⊢ ((𝑈 ∈ LMod ∧ (𝑋 − 𝑌) ∈ 𝑉) → (𝑁‘{(𝑋 − 𝑌)}) ∈ (LSubSp‘𝑈)) |
44 | 11, 42, 43 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ∈ (LSubSp‘𝑈)) |
45 | 4, 9, 10, 8, 6, 44, 16 | mapdord 41620 | . 2 ⊢ (𝜑 → ((𝑀‘(𝑁‘{(𝑋 − 𝑌)})) ⊆ (𝑀‘(𝑁‘{𝑋})) ↔ (𝑁‘{(𝑋 − 𝑌)}) ⊆ (𝑁‘{𝑋}))) |
46 | 40, 45 | mpbid 232 | 1 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ⊆ (𝑁‘{𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ⊆ wss 3962 {csn 4630 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17485 -gcsg 18965 LSSumclsm 19666 LModclmod 20874 LSubSpclss 20946 LSpanclspn 20986 HLchlt 39331 LHypclh 39966 DVecHcdvh 41060 LCDualclcd 41568 mapdcmpd 41606 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-riotaBAD 38934 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-tpos 8249 df-undef 8296 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-0g 17487 df-mre 17630 df-mrc 17631 df-acs 17633 df-proset 18351 df-poset 18370 df-plt 18387 df-lub 18403 df-glb 18404 df-join 18405 df-meet 18406 df-p0 18482 df-p1 18483 df-lat 18489 df-clat 18556 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-subg 19153 df-cntz 19347 df-oppg 19376 df-lsm 19668 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-dvr 20417 df-nzr 20529 df-rlreg 20710 df-domn 20711 df-drng 20747 df-lmod 20876 df-lss 20947 df-lsp 20987 df-lvec 21119 df-lsatoms 38957 df-lshyp 38958 df-lcv 39000 df-lfl 39039 df-lkr 39067 df-ldual 39105 df-oposet 39157 df-ol 39159 df-oml 39160 df-covers 39247 df-ats 39248 df-atl 39279 df-cvlat 39303 df-hlat 39332 df-llines 39480 df-lplanes 39481 df-lvols 39482 df-lines 39483 df-psubsp 39485 df-pmap 39486 df-padd 39778 df-lhyp 39970 df-laut 39971 df-ldil 40086 df-ltrn 40087 df-trl 40141 df-tgrp 40725 df-tendo 40737 df-edring 40739 df-dveca 40985 df-disoa 41011 df-dvech 41061 df-dib 41121 df-dic 41155 df-dih 41211 df-doch 41330 df-djh 41377 df-lcdual 41569 df-mapd 41607 |
This theorem is referenced by: mapdpglem14 41667 |
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