![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dochexmidlem5 | Structured version Visualization version GIF version |
Description: Lemma for dochexmid 37259. (Contributed by NM, 15-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) |
dochexmidlem5.pp | ⊢ (𝜑 → 𝑝 ∈ 𝐴) |
dochexmidlem5.z | ⊢ 0 = (0g‘𝑈) |
dochexmidlem5.m | ⊢ 𝑀 = (𝑋 ⊕ 𝑝) |
dochexmidlem5.xn | ⊢ (𝜑 → 𝑋 ≠ { 0 }) |
dochexmidlem5.pl | ⊢ (𝜑 → ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) |
Ref | Expression |
---|---|
dochexmidlem5 | ⊢ (𝜑 → (( ⊥ ‘𝑋) ∩ 𝑀) = { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dochexmidlem5.pl | . 2 ⊢ (𝜑 → ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) | |
2 | dochexmidlem1.s | . . . . . 6 ⊢ 𝑆 = (LSubSp‘𝑈) | |
3 | dochexmidlem5.z | . . . . . 6 ⊢ 0 = (0g‘𝑈) | |
4 | dochexmidlem1.a | . . . . . 6 ⊢ 𝐴 = (LSAtoms‘𝑈) | |
5 | dochexmidlem1.h | . . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | dochexmidlem1.u | . . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
7 | dochexmidlem1.k | . . . . . . . 8 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
8 | 5, 6, 7 | dvhlmod 36901 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod) |
9 | 8 | adantr 472 | . . . . . 6 ⊢ ((𝜑 ∧ (( ⊥ ‘𝑋) ∩ 𝑀) ≠ { 0 }) → 𝑈 ∈ LMod) |
10 | dochexmidlem1.x | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | |
11 | dochexmidlem1.v | . . . . . . . . . . 11 ⊢ 𝑉 = (Base‘𝑈) | |
12 | 11, 2 | lssss 19139 | . . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ⊆ 𝑉) |
13 | 10, 12 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ 𝑉) |
14 | dochexmidlem1.o | . . . . . . . . . 10 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
15 | 5, 6, 11, 2, 14 | dochlss 37145 | . . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ∈ 𝑆) |
16 | 7, 13, 15 | syl2anc 696 | . . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ 𝑆) |
17 | dochexmidlem5.m | . . . . . . . . 9 ⊢ 𝑀 = (𝑋 ⊕ 𝑝) | |
18 | dochexmidlem5.pp | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑝 ∈ 𝐴) | |
19 | 2, 4, 8, 18 | lsatlssel 34787 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑝 ∈ 𝑆) |
20 | dochexmidlem1.p | . . . . . . . . . . 11 ⊢ ⊕ = (LSSum‘𝑈) | |
21 | 2, 20 | lsmcl 19285 | . . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑆 ∧ 𝑝 ∈ 𝑆) → (𝑋 ⊕ 𝑝) ∈ 𝑆) |
22 | 8, 10, 19, 21 | syl3anc 1477 | . . . . . . . . 9 ⊢ (𝜑 → (𝑋 ⊕ 𝑝) ∈ 𝑆) |
23 | 17, 22 | syl5eqel 2843 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ 𝑆) |
24 | 2 | lssincl 19167 | . . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ ( ⊥ ‘𝑋) ∈ 𝑆 ∧ 𝑀 ∈ 𝑆) → (( ⊥ ‘𝑋) ∩ 𝑀) ∈ 𝑆) |
25 | 8, 16, 23, 24 | syl3anc 1477 | . . . . . . 7 ⊢ (𝜑 → (( ⊥ ‘𝑋) ∩ 𝑀) ∈ 𝑆) |
26 | 25 | adantr 472 | . . . . . 6 ⊢ ((𝜑 ∧ (( ⊥ ‘𝑋) ∩ 𝑀) ≠ { 0 }) → (( ⊥ ‘𝑋) ∩ 𝑀) ∈ 𝑆) |
27 | simpr 479 | . . . . . 6 ⊢ ((𝜑 ∧ (( ⊥ ‘𝑋) ∩ 𝑀) ≠ { 0 }) → (( ⊥ ‘𝑋) ∩ 𝑀) ≠ { 0 }) | |
28 | 2, 3, 4, 9, 26, 27 | lssatomic 34801 | . . . . 5 ⊢ ((𝜑 ∧ (( ⊥ ‘𝑋) ∩ 𝑀) ≠ { 0 }) → ∃𝑞 ∈ 𝐴 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) |
29 | 28 | ex 449 | . . . 4 ⊢ (𝜑 → ((( ⊥ ‘𝑋) ∩ 𝑀) ≠ { 0 } → ∃𝑞 ∈ 𝐴 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀))) |
30 | dochexmidlem1.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈) | |
31 | 7 | 3ad2ant1 1128 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
32 | 10 | 3ad2ant1 1128 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑋 ∈ 𝑆) |
33 | 18 | 3ad2ant1 1128 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑝 ∈ 𝐴) |
34 | simp2 1132 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑞 ∈ 𝐴) | |
35 | dochexmidlem5.xn | . . . . . . 7 ⊢ (𝜑 → 𝑋 ≠ { 0 }) | |
36 | 35 | 3ad2ant1 1128 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑋 ≠ { 0 }) |
37 | simp3 1133 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) | |
38 | 5, 14, 6, 11, 2, 30, 20, 4, 31, 32, 33, 34, 3, 17, 36, 37 | dochexmidlem4 37254 | . . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) |
39 | 38 | rexlimdv3a 3171 | . . . 4 ⊢ (𝜑 → (∃𝑞 ∈ 𝐴 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀) → 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))) |
40 | 29, 39 | syld 47 | . . 3 ⊢ (𝜑 → ((( ⊥ ‘𝑋) ∩ 𝑀) ≠ { 0 } → 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))) |
41 | 40 | necon1bd 2950 | . 2 ⊢ (𝜑 → (¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)) → (( ⊥ ‘𝑋) ∩ 𝑀) = { 0 })) |
42 | 1, 41 | mpd 15 | 1 ⊢ (𝜑 → (( ⊥ ‘𝑋) ∩ 𝑀) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 ∃wrex 3051 ∩ cin 3714 ⊆ wss 3715 {csn 4321 ‘cfv 6049 (class class class)co 6813 Basecbs 16059 0gc0g 16302 LSSumclsm 18249 LModclmod 19065 LSubSpclss 19134 LSpanclspn 19173 LSAtomsclsa 34764 HLchlt 35140 LHypclh 35773 DVecHcdvh 36869 ocHcoch 37138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-riotaBAD 34742 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-tpos 7521 df-undef 7568 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-map 8025 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-n0 11485 df-z 11570 df-uz 11880 df-fz 12520 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-sets 16066 df-ress 16067 df-plusg 16156 df-mulr 16157 df-sca 16159 df-vsca 16160 df-0g 16304 df-mre 16448 df-mrc 16449 df-acs 16451 df-preset 17129 df-poset 17147 df-plt 17159 df-lub 17175 df-glb 17176 df-join 17177 df-meet 17178 df-p0 17240 df-p1 17241 df-lat 17247 df-clat 17309 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-submnd 17537 df-grp 17626 df-minusg 17627 df-sbg 17628 df-subg 17792 df-cntz 17950 df-oppg 17976 df-lsm 18251 df-cmn 18395 df-abl 18396 df-mgp 18690 df-ur 18702 df-ring 18749 df-oppr 18823 df-dvdsr 18841 df-unit 18842 df-invr 18872 df-dvr 18883 df-drng 18951 df-lmod 19067 df-lss 19135 df-lsp 19174 df-lvec 19305 df-lsatoms 34766 df-lcv 34809 df-oposet 34966 df-ol 34968 df-oml 34969 df-covers 35056 df-ats 35057 df-atl 35088 df-cvlat 35112 df-hlat 35141 df-llines 35287 df-lplanes 35288 df-lvols 35289 df-lines 35290 df-psubsp 35292 df-pmap 35293 df-padd 35585 df-lhyp 35777 df-laut 35778 df-ldil 35893 df-ltrn 35894 df-trl 35949 df-tendo 36545 df-edring 36547 df-disoa 36820 df-dvech 36870 df-dib 36930 df-dic 36964 df-dih 37020 df-doch 37139 |
This theorem is referenced by: dochexmidlem6 37256 |
Copyright terms: Public domain | W3C validator |