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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochexmidlem2 | Structured version Visualization version GIF version |
Description: Lemma for dochexmid 41448. (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 41090 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
6 | dochexmidlem1.s | . . . . . 6 ⊢ 𝑆 = (LSubSp‘𝑈) | |
7 | 6 | lsssssubg 20948 | . . . . 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 3983 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (SubGrp‘𝑈)) |
11 | dochexmidlem1.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑈) | |
12 | 11, 6 | lssss 20926 | . . . . . 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 41334 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ∈ 𝑆) |
16 | 4, 13, 15 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ 𝑆) |
17 | 8, 16 | sseldd 3983 | . . 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 19676 | . . 3 ⊢ (((𝑋 ∈ (SubGrp‘𝑈) ∧ ( ⊥ ‘𝑋) ∈ (SubGrp‘𝑈)) ∧ (𝑟 ⊆ 𝑋 ∧ 𝑞 ⊆ ( ⊥ ‘𝑋))) → (𝑟 ⊕ 𝑞) ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) |
22 | 10, 17, 18, 19, 21 | syl22anc 839 | . 2 ⊢ (𝜑 → (𝑟 ⊕ 𝑞) ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) |
23 | 1, 22 | sstrd 3993 | 1 ⊢ (𝜑 → 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3950 ‘cfv 6559 (class class class)co 7429 Basecbs 17243 SubGrpcsubg 19134 LSSumclsm 19648 LModclmod 20850 LSubSpclss 20921 LSpanclspn 20961 LSAtomsclsa 38953 HLchlt 39329 LHypclh 39964 DVecHcdvh 41058 ocHcoch 41327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 ax-riotaBAD 38932 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-om 7884 df-1st 8010 df-2nd 8011 df-tpos 8247 df-undef 8294 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-1o 8502 df-er 8741 df-map 8864 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-nn 12263 df-2 12325 df-3 12326 df-4 12327 df-5 12328 df-6 12329 df-n0 12523 df-z 12610 df-uz 12875 df-fz 13544 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17244 df-ress 17271 df-plusg 17306 df-mulr 17307 df-sca 17309 df-vsca 17310 df-0g 17482 df-proset 18336 df-poset 18355 df-plt 18371 df-lub 18387 df-glb 18388 df-join 18389 df-meet 18390 df-p0 18466 df-p1 18467 df-lat 18473 df-clat 18540 df-mgm 18649 df-sgrp 18728 df-mnd 18744 df-submnd 18793 df-grp 18950 df-minusg 18951 df-sbg 18952 df-subg 19137 df-cntz 19331 df-lsm 19650 df-cmn 19796 df-abl 19797 df-mgp 20134 df-rng 20146 df-ur 20175 df-ring 20228 df-oppr 20326 df-dvdsr 20349 df-unit 20350 df-invr 20380 df-dvr 20393 df-drng 20723 df-lmod 20852 df-lss 20922 df-lsp 20962 df-lvec 21094 df-oposet 39155 df-ol 39157 df-oml 39158 df-covers 39245 df-ats 39246 df-atl 39277 df-cvlat 39301 df-hlat 39330 df-llines 39478 df-lplanes 39479 df-lvols 39480 df-lines 39481 df-psubsp 39483 df-pmap 39484 df-padd 39776 df-lhyp 39968 df-laut 39969 df-ldil 40084 df-ltrn 40085 df-trl 40139 df-tendo 40735 df-edring 40737 df-disoa 41009 df-dvech 41059 df-dib 41119 df-dic 41153 df-dih 41209 df-doch 41328 |
This theorem is referenced by: dochexmidlem3 41442 |
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