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Mirrors > Home > MPE Home > Th. List > Mathboxes > l1cvat | Structured version Visualization version GIF version |
Description: Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in [Crawley] p. 112. (1cvrat 36606 analog.) (Contributed by NM, 11-Jan-2015.) |
Ref | Expression |
---|---|
l1cvat.v | ⊢ 𝑉 = (Base‘𝑊) |
l1cvat.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
l1cvat.p | ⊢ ⊕ = (LSSum‘𝑊) |
l1cvat.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
l1cvat.c | ⊢ 𝐶 = ( ⋖L ‘𝑊) |
l1cvat.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
l1cvat.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
l1cvat.q | ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
l1cvat.r | ⊢ (𝜑 → 𝑅 ∈ 𝐴) |
l1cvat.n | ⊢ (𝜑 → 𝑄 ≠ 𝑅) |
l1cvat.l | ⊢ (𝜑 → 𝑈𝐶𝑉) |
l1cvat.m | ⊢ (𝜑 → ¬ 𝑄 ⊆ 𝑈) |
Ref | Expression |
---|---|
l1cvat | ⊢ (𝜑 → ((𝑄 ⊕ 𝑅) ∩ 𝑈) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | l1cvat.w | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
2 | lveclmod 19872 | . . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
3 | 1, 2 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) |
4 | lmodabl 19675 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ Abel) |
6 | l1cvat.s | . . . . . . . 8 ⊢ 𝑆 = (LSubSp‘𝑊) | |
7 | 6 | lsssssubg 19724 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
8 | 3, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊)) |
9 | l1cvat.a | . . . . . . 7 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
10 | l1cvat.q | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ 𝐴) | |
11 | 6, 9, 3, 10 | lsatlssel 36127 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ 𝑆) |
12 | 8, 11 | sseldd 3968 | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊)) |
13 | l1cvat.r | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ 𝐴) | |
14 | 6, 9, 3, 13 | lsatlssel 36127 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ 𝑆) |
15 | 8, 14 | sseldd 3968 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊)) |
16 | l1cvat.p | . . . . . 6 ⊢ ⊕ = (LSSum‘𝑊) | |
17 | 16 | lsmcom 18972 | . . . . 5 ⊢ ((𝑊 ∈ Abel ∧ 𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄)) |
18 | 5, 12, 15, 17 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄)) |
19 | 18 | ineq1d 4188 | . . 3 ⊢ (𝜑 → ((𝑄 ⊕ 𝑅) ∩ 𝑈) = ((𝑅 ⊕ 𝑄) ∩ 𝑈)) |
20 | incom 4178 | . . 3 ⊢ ((𝑅 ⊕ 𝑄) ∩ 𝑈) = (𝑈 ∩ (𝑅 ⊕ 𝑄)) | |
21 | 19, 20 | syl6eq 2872 | . 2 ⊢ (𝜑 → ((𝑄 ⊕ 𝑅) ∩ 𝑈) = (𝑈 ∩ (𝑅 ⊕ 𝑄))) |
22 | l1cvat.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
23 | l1cvat.n | . . . 4 ⊢ (𝜑 → 𝑄 ≠ 𝑅) | |
24 | 23 | necomd 3071 | . . 3 ⊢ (𝜑 → 𝑅 ≠ 𝑄) |
25 | l1cvat.m | . . 3 ⊢ (𝜑 → ¬ 𝑄 ⊆ 𝑈) | |
26 | l1cvat.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
27 | 26, 9, 3, 13 | lsatssv 36128 | . . . 4 ⊢ (𝜑 → 𝑅 ⊆ 𝑉) |
28 | l1cvat.c | . . . . 5 ⊢ 𝐶 = ( ⋖L ‘𝑊) | |
29 | l1cvat.l | . . . . 5 ⊢ (𝜑 → 𝑈𝐶𝑉) | |
30 | 26, 6, 16, 9, 28, 1, 22, 10, 29, 25 | l1cvpat 36184 | . . . 4 ⊢ (𝜑 → (𝑈 ⊕ 𝑄) = 𝑉) |
31 | 27, 30 | sseqtrrd 4008 | . . 3 ⊢ (𝜑 → 𝑅 ⊆ (𝑈 ⊕ 𝑄)) |
32 | 6, 16, 9, 1, 22, 13, 10, 24, 25, 31 | lsatcvat3 36182 | . 2 ⊢ (𝜑 → (𝑈 ∩ (𝑅 ⊕ 𝑄)) ∈ 𝐴) |
33 | 21, 32 | eqeltrd 2913 | 1 ⊢ (𝜑 → ((𝑄 ⊕ 𝑅) ∩ 𝑈) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∩ cin 3935 ⊆ wss 3936 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 SubGrpcsubg 18267 LSSumclsm 18753 Abelcabl 18901 LModclmod 19628 LSubSpclss 19697 LVecclvec 19868 LSAtomsclsa 36104 ⋖L clcv 36148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3497 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 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-0g 16709 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-cntz 18441 df-oppg 18468 df-lsm 18755 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-drng 19498 df-lmod 19630 df-lss 19698 df-lsp 19738 df-lvec 19869 df-lsatoms 36106 df-lshyp 36107 df-lcv 36149 |
This theorem is referenced by: lshpat 36186 |
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