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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem14 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 41405. (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 |
---|---|
mapdpglem14 | ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑋})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpglem.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | mapdpglem.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | mapdpglem.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | 1, 2, 3 | dvhlmod 40809 | . . 3 ⊢ (𝜑 → 𝑈 ∈ LMod) |
5 | mapdpglem.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
6 | mapdpglem.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
7 | mapdpglem.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
8 | eqid 2726 | . . . 4 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
9 | mapdpglem.s | . . . 4 ⊢ − = (-g‘𝑈) | |
10 | 7, 8, 9 | lmodvnpcan 20892 | . . 3 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → ((𝑌 − 𝑋)(+g‘𝑈)𝑋) = 𝑌) |
11 | 4, 5, 6, 10 | syl3anc 1368 | . 2 ⊢ (𝜑 → ((𝑌 − 𝑋)(+g‘𝑈)𝑋) = 𝑌) |
12 | eqid 2726 | . . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
13 | mapdpglem.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑈) | |
14 | 7, 12, 13 | lspsncl 20954 | . . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
15 | 4, 6, 14 | syl2anc 582 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) |
16 | lmodgrp 20843 | . . . . . 6 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Grp) | |
17 | 4, 16 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ Grp) |
18 | eqid 2726 | . . . . . 6 ⊢ (invg‘𝑈) = (invg‘𝑈) | |
19 | 7, 9, 18 | grpinvsub 19016 | . . . . 5 ⊢ ((𝑈 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((invg‘𝑈)‘(𝑋 − 𝑌)) = (𝑌 − 𝑋)) |
20 | 17, 6, 5, 19 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → ((invg‘𝑈)‘(𝑋 − 𝑌)) = (𝑌 − 𝑋)) |
21 | mapdpglem.m | . . . . . . 7 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
22 | mapdpglem.c | . . . . . . 7 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
23 | mapdpglem1.p | . . . . . . 7 ⊢ ⊕ = (LSSum‘𝐶) | |
24 | mapdpglem2.j | . . . . . . 7 ⊢ 𝐽 = (LSpan‘𝐶) | |
25 | mapdpglem3.f | . . . . . . 7 ⊢ 𝐹 = (Base‘𝐶) | |
26 | mapdpglem3.te | . . . . . . 7 ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) | |
27 | mapdpglem3.a | . . . . . . 7 ⊢ 𝐴 = (Scalar‘𝑈) | |
28 | mapdpglem3.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐴) | |
29 | mapdpglem3.t | . . . . . . 7 ⊢ · = ( ·𝑠 ‘𝐶) | |
30 | mapdpglem3.r | . . . . . . 7 ⊢ 𝑅 = (-g‘𝐶) | |
31 | mapdpglem3.g | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
32 | mapdpglem3.e | . . . . . . 7 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) | |
33 | mapdpglem4.q | . . . . . . 7 ⊢ 𝑄 = (0g‘𝑈) | |
34 | mapdpglem.ne | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
35 | mapdpglem4.jt | . . . . . . 7 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) | |
36 | mapdpglem4.z | . . . . . . 7 ⊢ 0 = (0g‘𝐴) | |
37 | mapdpglem4.g4 | . . . . . . 7 ⊢ (𝜑 → 𝑔 ∈ 𝐵) | |
38 | mapdpglem4.z4 | . . . . . . 7 ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) | |
39 | mapdpglem4.t4 | . . . . . . 7 ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) | |
40 | mapdpglem4.xn | . . . . . . 7 ⊢ (𝜑 → 𝑋 ≠ 𝑄) | |
41 | mapdpglem12.yn | . . . . . . 7 ⊢ (𝜑 → 𝑌 ≠ 𝑄) | |
42 | mapdpglem12.g0 | . . . . . . 7 ⊢ (𝜑 → 𝑧 = (0g‘𝐶)) | |
43 | 1, 21, 2, 7, 9, 13, 22, 3, 6, 5, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42 | mapdpglem13 41383 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ⊆ (𝑁‘{𝑋})) |
44 | 7, 9 | lmodvsubcl 20883 | . . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉) |
45 | 4, 6, 5, 44 | syl3anc 1368 | . . . . . . 7 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝑉) |
46 | 7, 13 | lspsnid 20970 | . . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ (𝑋 − 𝑌) ∈ 𝑉) → (𝑋 − 𝑌) ∈ (𝑁‘{(𝑋 − 𝑌)})) |
47 | 4, 45, 46 | syl2anc 582 | . . . . . 6 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ (𝑁‘{(𝑋 − 𝑌)})) |
48 | 43, 47 | sseldd 3980 | . . . . 5 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ (𝑁‘{𝑋})) |
49 | 12, 18 | lssvnegcl 20933 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈) ∧ (𝑋 − 𝑌) ∈ (𝑁‘{𝑋})) → ((invg‘𝑈)‘(𝑋 − 𝑌)) ∈ (𝑁‘{𝑋})) |
50 | 4, 15, 48, 49 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → ((invg‘𝑈)‘(𝑋 − 𝑌)) ∈ (𝑁‘{𝑋})) |
51 | 20, 50 | eqeltrrd 2827 | . . 3 ⊢ (𝜑 → (𝑌 − 𝑋) ∈ (𝑁‘{𝑋})) |
52 | 7, 13 | lspsnid 20970 | . . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋})) |
53 | 4, 6, 52 | syl2anc 582 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑋})) |
54 | 8, 12 | lssvacl 20920 | . . 3 ⊢ (((𝑈 ∈ LMod ∧ (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈)) ∧ ((𝑌 − 𝑋) ∈ (𝑁‘{𝑋}) ∧ 𝑋 ∈ (𝑁‘{𝑋}))) → ((𝑌 − 𝑋)(+g‘𝑈)𝑋) ∈ (𝑁‘{𝑋})) |
55 | 4, 15, 51, 53, 54 | syl22anc 837 | . 2 ⊢ (𝜑 → ((𝑌 − 𝑋)(+g‘𝑈)𝑋) ∈ (𝑁‘{𝑋})) |
56 | 11, 55 | eqeltrrd 2827 | 1 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 {csn 4633 ‘cfv 6554 (class class class)co 7424 Basecbs 17213 +gcplusg 17266 Scalarcsca 17269 ·𝑠 cvsca 17270 0gc0g 17454 Grpcgrp 18928 invgcminusg 18929 -gcsg 18930 LSSumclsm 19632 LModclmod 20836 LSubSpclss 20908 LSpanclspn 20948 HLchlt 39048 LHypclh 39683 DVecHcdvh 40777 LCDualclcd 41285 mapdcmpd 41323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-riotaBAD 38651 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-tpos 8241 df-undef 8288 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-sca 17282 df-vsca 17283 df-0g 17456 df-mre 17599 df-mrc 17600 df-acs 17602 df-proset 18320 df-poset 18338 df-plt 18355 df-lub 18371 df-glb 18372 df-join 18373 df-meet 18374 df-p0 18450 df-p1 18451 df-lat 18457 df-clat 18524 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-grp 18931 df-minusg 18932 df-sbg 18933 df-subg 19117 df-cntz 19311 df-oppg 19340 df-lsm 19634 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-ring 20218 df-oppr 20316 df-dvdsr 20339 df-unit 20340 df-invr 20370 df-dvr 20383 df-nzr 20495 df-rlreg 20672 df-domn 20673 df-drng 20709 df-lmod 20838 df-lss 20909 df-lsp 20949 df-lvec 21081 df-lsatoms 38674 df-lshyp 38675 df-lcv 38717 df-lfl 38756 df-lkr 38784 df-ldual 38822 df-oposet 38874 df-ol 38876 df-oml 38877 df-covers 38964 df-ats 38965 df-atl 38996 df-cvlat 39020 df-hlat 39049 df-llines 39197 df-lplanes 39198 df-lvols 39199 df-lines 39200 df-psubsp 39202 df-pmap 39203 df-padd 39495 df-lhyp 39687 df-laut 39688 df-ldil 39803 df-ltrn 39804 df-trl 39858 df-tgrp 40442 df-tendo 40454 df-edring 40456 df-dveca 40702 df-disoa 40728 df-dvech 40778 df-dib 40838 df-dic 40872 df-dih 40928 df-doch 41047 df-djh 41094 df-lcdual 41286 df-mapd 41324 |
This theorem is referenced by: mapdpglem15 41385 |
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