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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem9 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 41179. Baer p. 45, line 4: "...so that x would consequently belong to Fy." (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) |
mapdpglem4.g0 | ⊢ (𝜑 → 𝑔 = 0 ) |
Ref | Expression |
---|---|
mapdpglem9 | ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌})) |
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 40583 | . . 3 ⊢ (𝜑 → 𝑈 ∈ LMod) |
5 | mapdpglem.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
6 | mapdpglem.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
7 | mapdpglem.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
8 | eqid 2728 | . . . 4 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
9 | mapdpglem.s | . . . 4 ⊢ − = (-g‘𝑈) | |
10 | 7, 8, 9 | lmodvnpcan 20799 | . . 3 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝑋 − 𝑌)(+g‘𝑈)𝑌) = 𝑋) |
11 | 4, 5, 6, 10 | syl3anc 1369 | . 2 ⊢ (𝜑 → ((𝑋 − 𝑌)(+g‘𝑈)𝑌) = 𝑋) |
12 | eqid 2728 | . . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
13 | mapdpglem.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑈) | |
14 | 7, 12, 13 | lspsncl 20861 | . . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
15 | 4, 6, 14 | syl2anc 583 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) |
16 | mapdpglem.m | . . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
17 | mapdpglem.c | . . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
18 | mapdpglem1.p | . . . . 5 ⊢ ⊕ = (LSSum‘𝐶) | |
19 | mapdpglem2.j | . . . . 5 ⊢ 𝐽 = (LSpan‘𝐶) | |
20 | mapdpglem3.f | . . . . 5 ⊢ 𝐹 = (Base‘𝐶) | |
21 | mapdpglem3.te | . . . . 5 ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) | |
22 | mapdpglem3.a | . . . . 5 ⊢ 𝐴 = (Scalar‘𝑈) | |
23 | mapdpglem3.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐴) | |
24 | mapdpglem3.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝐶) | |
25 | mapdpglem3.r | . . . . 5 ⊢ 𝑅 = (-g‘𝐶) | |
26 | mapdpglem3.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
27 | mapdpglem3.e | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) | |
28 | mapdpglem4.q | . . . . 5 ⊢ 𝑄 = (0g‘𝑈) | |
29 | mapdpglem.ne | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
30 | mapdpglem4.jt | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) | |
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 | mapdpglem4.g0 | . . . . 5 ⊢ (𝜑 → 𝑔 = 0 ) | |
37 | 1, 16, 2, 7, 9, 13, 17, 3, 5, 6, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36 | mapdpglem8 41152 | . . . 4 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ⊆ (𝑁‘{𝑌})) |
38 | 7, 9 | lmodvsubcl 20790 | . . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉) |
39 | 4, 5, 6, 38 | syl3anc 1369 | . . . . 5 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝑉) |
40 | 7, 13 | lspsnid 20877 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ (𝑋 − 𝑌) ∈ 𝑉) → (𝑋 − 𝑌) ∈ (𝑁‘{(𝑋 − 𝑌)})) |
41 | 4, 39, 40 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ (𝑁‘{(𝑋 − 𝑌)})) |
42 | 37, 41 | sseldd 3981 | . . 3 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ (𝑁‘{𝑌})) |
43 | 7, 13 | lspsnid 20877 | . . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ (𝑁‘{𝑌})) |
44 | 4, 6, 43 | syl2anc 583 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑌})) |
45 | 8, 12 | lssvacl 20827 | . . 3 ⊢ (((𝑈 ∈ LMod ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) ∧ ((𝑋 − 𝑌) ∈ (𝑁‘{𝑌}) ∧ 𝑌 ∈ (𝑁‘{𝑌}))) → ((𝑋 − 𝑌)(+g‘𝑈)𝑌) ∈ (𝑁‘{𝑌})) |
46 | 4, 15, 42, 44, 45 | syl22anc 838 | . 2 ⊢ (𝜑 → ((𝑋 − 𝑌)(+g‘𝑈)𝑌) ∈ (𝑁‘{𝑌})) |
47 | 11, 46 | eqeltrrd 2830 | 1 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 {csn 4629 ‘cfv 6548 (class class class)co 7420 Basecbs 17180 +gcplusg 17233 Scalarcsca 17236 ·𝑠 cvsca 17237 0gc0g 17421 -gcsg 18892 LSSumclsm 19589 LModclmod 20743 LSubSpclss 20815 LSpanclspn 20855 HLchlt 38822 LHypclh 39457 DVecHcdvh 40551 LCDualclcd 41059 mapdcmpd 41097 |
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 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-riotaBAD 38425 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-tpos 8232 df-undef 8279 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-sca 17249 df-vsca 17250 df-0g 17423 df-mre 17566 df-mrc 17567 df-acs 17569 df-proset 18287 df-poset 18305 df-plt 18322 df-lub 18338 df-glb 18339 df-join 18340 df-meet 18341 df-p0 18417 df-p1 18418 df-lat 18424 df-clat 18491 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-grp 18893 df-minusg 18894 df-sbg 18895 df-subg 19078 df-cntz 19268 df-oppg 19297 df-lsm 19591 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-oppr 20273 df-dvdsr 20296 df-unit 20297 df-invr 20327 df-dvr 20340 df-drng 20626 df-lmod 20745 df-lss 20816 df-lsp 20856 df-lvec 20988 df-lsatoms 38448 df-lshyp 38449 df-lcv 38491 df-lfl 38530 df-lkr 38558 df-ldual 38596 df-oposet 38648 df-ol 38650 df-oml 38651 df-covers 38738 df-ats 38739 df-atl 38770 df-cvlat 38794 df-hlat 38823 df-llines 38971 df-lplanes 38972 df-lvols 38973 df-lines 38974 df-psubsp 38976 df-pmap 38977 df-padd 39269 df-lhyp 39461 df-laut 39462 df-ldil 39577 df-ltrn 39578 df-trl 39632 df-tgrp 40216 df-tendo 40228 df-edring 40230 df-dveca 40476 df-disoa 40502 df-dvech 40552 df-dib 40612 df-dic 40646 df-dih 40702 df-doch 40821 df-djh 40868 df-lcdual 41060 df-mapd 41098 |
This theorem is referenced by: mapdpglem10 41154 |
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