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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem20 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 41188. (Contributed by NM, 11-Mar-2015.) |
Ref | Expression |
---|---|
lcfrlem17.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcfrlem17.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lcfrlem17.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcfrlem17.v | ⊢ 𝑉 = (Base‘𝑈) |
lcfrlem17.p | ⊢ + = (+g‘𝑈) |
lcfrlem17.z | ⊢ 0 = (0g‘𝑈) |
lcfrlem17.n | ⊢ 𝑁 = (LSpan‘𝑈) |
lcfrlem17.a | ⊢ 𝐴 = (LSAtoms‘𝑈) |
lcfrlem17.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lcfrlem17.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
lcfrlem17.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
lcfrlem17.ne | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
lcfrlem20.e | ⊢ (𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) |
Ref | Expression |
---|---|
lcfrlem20 | ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfrlem17.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
2 | lcfrlem17.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑈) | |
3 | eqid 2725 | . . . 4 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈) | |
4 | lcfrlem17.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | lcfrlem17.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
6 | lcfrlem17.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | 4, 5, 6 | dvhlmod 40713 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
8 | lcfrlem17.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
9 | 8 | eldifad 3956 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
10 | lcfrlem17.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
11 | 10 | eldifad 3956 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
12 | 1, 2, 3, 7, 9, 11 | lsmpr 20986 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌}))) |
13 | 12 | ineq1d 4209 | . 2 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) = (((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))) |
14 | eqid 2725 | . . 3 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
15 | eqid 2725 | . . 3 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈) | |
16 | lcfrlem17.a | . . 3 ⊢ 𝐴 = (LSAtoms‘𝑈) | |
17 | 4, 5, 6 | dvhlvec 40712 | . . 3 ⊢ (𝜑 → 𝑈 ∈ LVec) |
18 | lcfrlem17.o | . . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
19 | lcfrlem17.z | . . . 4 ⊢ 0 = (0g‘𝑈) | |
20 | lcfrlem17.p | . . . . 5 ⊢ + = (+g‘𝑈) | |
21 | lcfrlem17.ne | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
22 | 4, 18, 5, 1, 20, 19, 2, 16, 6, 8, 10, 21 | lcfrlem17 41162 | . . . 4 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝑉 ∖ { 0 })) |
23 | 4, 18, 5, 1, 19, 15, 6, 22 | dochsnshp 41056 | . . 3 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ (LSHyp‘𝑈)) |
24 | 1, 2, 19, 16, 7, 8 | lsatlspsn 38595 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝐴) |
25 | 1, 2, 19, 16, 7, 10 | lsatlspsn 38595 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ 𝐴) |
26 | lcfrlem20.e | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})) | |
27 | 1, 20 | lmodvacl 20770 | . . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉) |
28 | 7, 9, 11, 27 | syl3anc 1368 | . . . . . . 7 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉) |
29 | 28 | snssd 4814 | . . . . . 6 ⊢ (𝜑 → {(𝑋 + 𝑌)} ⊆ 𝑉) |
30 | 4, 5, 1, 14, 18 | dochlss 40957 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {(𝑋 + 𝑌)} ⊆ 𝑉) → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ (LSubSp‘𝑈)) |
31 | 6, 29, 30 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ (LSubSp‘𝑈)) |
32 | 1, 14, 2, 7, 31, 9 | lspsnel5 20891 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}) ↔ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘{(𝑋 + 𝑌)}))) |
33 | 26, 32 | mtbid 323 | . . 3 ⊢ (𝜑 → ¬ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘{(𝑋 + 𝑌)})) |
34 | 14, 3, 15, 16, 17, 23, 24, 25, 21, 33 | lshpat 38658 | . 2 ⊢ (𝜑 → (((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴) |
35 | 13, 34 | eqeltrd 2825 | 1 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∖ cdif 3941 ∩ cin 3943 ⊆ wss 3944 {csn 4630 {cpr 4632 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 +gcplusg 17236 0gc0g 17424 LSSumclsm 19601 LModclmod 20755 LSubSpclss 20827 LSpanclspn 20867 LSAtomsclsa 38576 LSHypclsh 38577 HLchlt 38952 LHypclh 39587 DVecHcdvh 40681 ocHcoch 40950 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-riotaBAD 38555 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 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 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 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 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-sca 17252 df-vsca 17253 df-0g 17426 df-mre 17569 df-mrc 17570 df-acs 17572 df-proset 18290 df-poset 18308 df-plt 18325 df-lub 18341 df-glb 18342 df-join 18343 df-meet 18344 df-p0 18420 df-p1 18421 df-lat 18427 df-clat 18494 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-grp 18901 df-minusg 18902 df-sbg 18903 df-subg 19086 df-cntz 19280 df-oppg 19309 df-lsm 19603 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-dvr 20352 df-drng 20638 df-lmod 20757 df-lss 20828 df-lsp 20868 df-lvec 21000 df-lsatoms 38578 df-lshyp 38579 df-lcv 38621 df-oposet 38778 df-ol 38780 df-oml 38781 df-covers 38868 df-ats 38869 df-atl 38900 df-cvlat 38924 df-hlat 38953 df-llines 39101 df-lplanes 39102 df-lvols 39103 df-lines 39104 df-psubsp 39106 df-pmap 39107 df-padd 39399 df-lhyp 39591 df-laut 39592 df-ldil 39707 df-ltrn 39708 df-trl 39762 df-tgrp 40346 df-tendo 40358 df-edring 40360 df-dveca 40606 df-disoa 40632 df-dvech 40682 df-dib 40742 df-dic 40776 df-dih 40832 df-doch 40951 df-djh 40998 |
This theorem is referenced by: lcfrlem21 41166 |
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