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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochexmidlem2 | Structured version Visualization version GIF version |
Description: Lemma for dochexmid 37627. (Contributed by NM, 14-Jan-2015.) |
Ref | Expression |
---|---|
dochexmidlem1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dochexmidlem1.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
dochexmidlem1.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dochexmidlem1.v | ⊢ 𝑉 = (Base‘𝑈) |
dochexmidlem1.s | ⊢ 𝑆 = (LSubSp‘𝑈) |
dochexmidlem1.n | ⊢ 𝑁 = (LSpan‘𝑈) |
dochexmidlem1.p | ⊢ ⊕ = (LSSum‘𝑈) |
dochexmidlem1.a | ⊢ 𝐴 = (LSAtoms‘𝑈) |
dochexmidlem1.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dochexmidlem1.x | ⊢ (𝜑 → 𝑋 ∈ 𝑆) |
dochexmidlem2.pp | ⊢ (𝜑 → 𝑝 ∈ 𝐴) |
dochexmidlem2.qq | ⊢ (𝜑 → 𝑞 ∈ 𝐴) |
dochexmidlem2.rr | ⊢ (𝜑 → 𝑟 ∈ 𝐴) |
dochexmidlem2.ql | ⊢ (𝜑 → 𝑞 ⊆ ( ⊥ ‘𝑋)) |
dochexmidlem2.rl | ⊢ (𝜑 → 𝑟 ⊆ 𝑋) |
dochexmidlem2.pl | ⊢ (𝜑 → 𝑝 ⊆ (𝑟 ⊕ 𝑞)) |
Ref | Expression |
---|---|
dochexmidlem2 | ⊢ (𝜑 → 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dochexmidlem2.pl | . 2 ⊢ (𝜑 → 𝑝 ⊆ (𝑟 ⊕ 𝑞)) | |
2 | dochexmidlem1.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | dochexmidlem1.u | . . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | dochexmidlem1.k | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
5 | 2, 3, 4 | dvhlmod 37269 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
6 | dochexmidlem1.s | . . . . . 6 ⊢ 𝑆 = (LSubSp‘𝑈) | |
7 | 6 | lsssssubg 19357 | . . . . 5 ⊢ (𝑈 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑈)) |
8 | 5, 7 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑈)) |
9 | dochexmidlem1.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | |
10 | 8, 9 | sseldd 3822 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (SubGrp‘𝑈)) |
11 | dochexmidlem1.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑈) | |
12 | 11, 6 | lssss 19333 | . . . . . 6 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ⊆ 𝑉) |
13 | 9, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑋 ⊆ 𝑉) |
14 | dochexmidlem1.o | . . . . . 6 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
15 | 2, 3, 11, 6, 14 | dochlss 37513 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ∈ 𝑆) |
16 | 4, 13, 15 | syl2anc 579 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ 𝑆) |
17 | 8, 16 | sseldd 3822 | . . 3 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ (SubGrp‘𝑈)) |
18 | dochexmidlem2.rl | . . 3 ⊢ (𝜑 → 𝑟 ⊆ 𝑋) | |
19 | dochexmidlem2.ql | . . 3 ⊢ (𝜑 → 𝑞 ⊆ ( ⊥ ‘𝑋)) | |
20 | dochexmidlem1.p | . . . 4 ⊢ ⊕ = (LSSum‘𝑈) | |
21 | 20 | lsmless12 18464 | . . 3 ⊢ (((𝑋 ∈ (SubGrp‘𝑈) ∧ ( ⊥ ‘𝑋) ∈ (SubGrp‘𝑈)) ∧ (𝑟 ⊆ 𝑋 ∧ 𝑞 ⊆ ( ⊥ ‘𝑋))) → (𝑟 ⊕ 𝑞) ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) |
22 | 10, 17, 18, 19, 21 | syl22anc 829 | . 2 ⊢ (𝜑 → (𝑟 ⊕ 𝑞) ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) |
23 | 1, 22 | sstrd 3831 | 1 ⊢ (𝜑 → 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ⊆ wss 3792 ‘cfv 6137 (class class class)co 6924 Basecbs 16259 SubGrpcsubg 17976 LSSumclsm 18437 LModclmod 19259 LSubSpclss 19328 LSpanclspn 19370 LSAtomsclsa 35133 HLchlt 35509 LHypclh 36143 DVecHcdvh 37237 ocHcoch 37506 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-riotaBAD 35112 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-tpos 7636 df-undef 7683 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-n0 11647 df-z 11733 df-uz 11997 df-fz 12648 df-struct 16261 df-ndx 16262 df-slot 16263 df-base 16265 df-sets 16266 df-ress 16267 df-plusg 16355 df-mulr 16356 df-sca 16358 df-vsca 16359 df-0g 16492 df-proset 17318 df-poset 17336 df-plt 17348 df-lub 17364 df-glb 17365 df-join 17366 df-meet 17367 df-p0 17429 df-p1 17430 df-lat 17436 df-clat 17498 df-mgm 17632 df-sgrp 17674 df-mnd 17685 df-submnd 17726 df-grp 17816 df-minusg 17817 df-sbg 17818 df-subg 17979 df-cntz 18137 df-lsm 18439 df-cmn 18585 df-abl 18586 df-mgp 18881 df-ur 18893 df-ring 18940 df-oppr 19014 df-dvdsr 19032 df-unit 19033 df-invr 19063 df-dvr 19074 df-drng 19145 df-lmod 19261 df-lss 19329 df-lsp 19371 df-lvec 19502 df-oposet 35335 df-ol 35337 df-oml 35338 df-covers 35425 df-ats 35426 df-atl 35457 df-cvlat 35481 df-hlat 35510 df-llines 35657 df-lplanes 35658 df-lvols 35659 df-lines 35660 df-psubsp 35662 df-pmap 35663 df-padd 35955 df-lhyp 36147 df-laut 36148 df-ldil 36263 df-ltrn 36264 df-trl 36318 df-tendo 36914 df-edring 36916 df-disoa 37188 df-dvech 37238 df-dib 37298 df-dic 37332 df-dih 37388 df-doch 37507 |
This theorem is referenced by: dochexmidlem3 37621 |
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