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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochexmidlem5 | Structured version Visualization version GIF version |
Description: Lemma for dochexmid 38619. (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 38261 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod) |
9 | 8 | adantr 483 | . . . . . 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 19708 | . . . . . . . . . 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 38505 | . . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ∈ 𝑆) |
16 | 7, 13, 15 | syl2anc 586 | . . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ 𝑆) |
17 | dochexmidlem5.m | . . . . . . . . 9 ⊢ 𝑀 = (𝑋 ⊕ 𝑝) | |
18 | dochexmidlem5.pp | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑝 ∈ 𝐴) | |
19 | 2, 4, 8, 18 | lsatlssel 36148 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑝 ∈ 𝑆) |
20 | dochexmidlem1.p | . . . . . . . . . . 11 ⊢ ⊕ = (LSSum‘𝑈) | |
21 | 2, 20 | lsmcl 19855 | . . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑆 ∧ 𝑝 ∈ 𝑆) → (𝑋 ⊕ 𝑝) ∈ 𝑆) |
22 | 8, 10, 19, 21 | syl3anc 1367 | . . . . . . . . 9 ⊢ (𝜑 → (𝑋 ⊕ 𝑝) ∈ 𝑆) |
23 | 17, 22 | eqeltrid 2917 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ 𝑆) |
24 | 2 | lssincl 19737 | . . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ ( ⊥ ‘𝑋) ∈ 𝑆 ∧ 𝑀 ∈ 𝑆) → (( ⊥ ‘𝑋) ∩ 𝑀) ∈ 𝑆) |
25 | 8, 16, 23, 24 | syl3anc 1367 | . . . . . . 7 ⊢ (𝜑 → (( ⊥ ‘𝑋) ∩ 𝑀) ∈ 𝑆) |
26 | 25 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ (( ⊥ ‘𝑋) ∩ 𝑀) ≠ { 0 }) → (( ⊥ ‘𝑋) ∩ 𝑀) ∈ 𝑆) |
27 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ (( ⊥ ‘𝑋) ∩ 𝑀) ≠ { 0 }) → (( ⊥ ‘𝑋) ∩ 𝑀) ≠ { 0 }) | |
28 | 2, 3, 4, 9, 26, 27 | lssatomic 36162 | . . . . 5 ⊢ ((𝜑 ∧ (( ⊥ ‘𝑋) ∩ 𝑀) ≠ { 0 }) → ∃𝑞 ∈ 𝐴 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) |
29 | 28 | ex 415 | . . . 4 ⊢ (𝜑 → ((( ⊥ ‘𝑋) ∩ 𝑀) ≠ { 0 } → ∃𝑞 ∈ 𝐴 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀))) |
30 | dochexmidlem1.n | . . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈) | |
31 | 7 | 3ad2ant1 1129 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
32 | 10 | 3ad2ant1 1129 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑋 ∈ 𝑆) |
33 | 18 | 3ad2ant1 1129 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑝 ∈ 𝐴) |
34 | simp2 1133 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑞 ∈ 𝐴) | |
35 | dochexmidlem5.xn | . . . . . . 7 ⊢ (𝜑 → 𝑋 ≠ { 0 }) | |
36 | 35 | 3ad2ant1 1129 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑋 ≠ { 0 }) |
37 | simp3 1134 | . . . . . 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 38614 | . . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) |
39 | 38 | rexlimdv3a 3286 | . . . 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 3034 | . 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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 ∩ cin 3935 ⊆ wss 3936 {csn 4567 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 0gc0g 16713 LSSumclsm 18759 LModclmod 19634 LSubSpclss 19703 LSpanclspn 19743 LSAtomsclsa 36125 HLchlt 36501 LHypclh 37135 DVecHcdvh 38229 ocHcoch 38498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-riotaBAD 36104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-undef 7939 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-0g 16715 df-mre 16857 df-mrc 16858 df-acs 16860 df-proset 17538 df-poset 17556 df-plt 17568 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-p0 17649 df-p1 17650 df-lat 17656 df-clat 17718 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-cntz 18447 df-oppg 18474 df-lsm 18761 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-drng 19504 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lvec 19875 df-lsatoms 36127 df-lcv 36170 df-oposet 36327 df-ol 36329 df-oml 36330 df-covers 36417 df-ats 36418 df-atl 36449 df-cvlat 36473 df-hlat 36502 df-llines 36649 df-lplanes 36650 df-lvols 36651 df-lines 36652 df-psubsp 36654 df-pmap 36655 df-padd 36947 df-lhyp 37139 df-laut 37140 df-ldil 37255 df-ltrn 37256 df-trl 37310 df-tendo 37906 df-edring 37908 df-disoa 38180 df-dvech 38230 df-dib 38290 df-dic 38324 df-dih 38380 df-doch 38499 |
This theorem is referenced by: dochexmidlem6 38616 |
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