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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatcvat | Structured version Visualization version GIF version |
Description: A nonzero subspace less than the sum of two atoms is an atom. (atcvati 28629 analog.) (Contributed by NM, 10-Jan-2015.) |
Ref | Expression |
---|---|
lsatcvat.o | ⊢ 0 = (0g‘𝑊) |
lsatcvat.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lsatcvat.p | ⊢ ⊕ = (LSSum‘𝑊) |
lsatcvat.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lsatcvat.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lsatcvat.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lsatcvat.q | ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
lsatcvat.r | ⊢ (𝜑 → 𝑅 ∈ 𝐴) |
lsatcvat.n | ⊢ (𝜑 → 𝑈 ≠ { 0 }) |
lsatcvat.l | ⊢ (𝜑 → 𝑈 ⊊ (𝑄 ⊕ 𝑅)) |
Ref | Expression |
---|---|
lsatcvat | ⊢ (𝜑 → 𝑈 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatcvat.o | . . 3 ⊢ 0 = (0g‘𝑊) | |
2 | lsatcvat.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | lsatcvat.p | . . 3 ⊢ ⊕ = (LSSum‘𝑊) | |
4 | lsatcvat.a | . . 3 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
5 | lsatcvat.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
6 | 5 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑊 ∈ LVec) |
7 | lsatcvat.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
8 | 7 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ∈ 𝑆) |
9 | lsatcvat.q | . . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝐴) | |
10 | 9 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑄 ∈ 𝐴) |
11 | lsatcvat.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝐴) | |
12 | 11 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑅 ∈ 𝐴) |
13 | lsatcvat.n | . . . 4 ⊢ (𝜑 → 𝑈 ≠ { 0 }) | |
14 | 13 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ≠ { 0 }) |
15 | lsatcvat.l | . . . 4 ⊢ (𝜑 → 𝑈 ⊊ (𝑄 ⊕ 𝑅)) | |
16 | 15 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ⊊ (𝑄 ⊕ 𝑅)) |
17 | simpr 476 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → ¬ 𝑄 ⊆ 𝑈) | |
18 | 1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 17 | lsatcvatlem 33354 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ∈ 𝐴) |
19 | 5 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑊 ∈ LVec) |
20 | 7 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ∈ 𝑆) |
21 | 11 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑅 ∈ 𝐴) |
22 | 9 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑄 ∈ 𝐴) |
23 | 13 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ≠ { 0 }) |
24 | lveclmod 18927 | . . . . . . . . 9 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
25 | 5, 24 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LMod) |
26 | lmodabl 18733 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
27 | 25, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Abel) |
28 | 2 | lsssssubg 18779 | . . . . . . . . 9 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
29 | 25, 28 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊)) |
30 | 2, 4, 25, 9 | lsatlssel 33302 | . . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝑆) |
31 | 29, 30 | sseldd 3569 | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊)) |
32 | 2, 4, 25, 11 | lsatlssel 33302 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ 𝑆) |
33 | 29, 32 | sseldd 3569 | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊)) |
34 | 3 | lsmcom 18084 | . . . . . . 7 ⊢ ((𝑊 ∈ Abel ∧ 𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄)) |
35 | 27, 31, 33, 34 | syl3anc 1318 | . . . . . 6 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄)) |
36 | 35 | psseq2d 3662 | . . . . 5 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) ↔ 𝑈 ⊊ (𝑅 ⊕ 𝑄))) |
37 | 15, 36 | mpbid 221 | . . . 4 ⊢ (𝜑 → 𝑈 ⊊ (𝑅 ⊕ 𝑄)) |
38 | 37 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ⊊ (𝑅 ⊕ 𝑄)) |
39 | simpr 476 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → ¬ 𝑅 ⊆ 𝑈) | |
40 | 1, 2, 3, 4, 19, 20, 21, 22, 23, 38, 39 | lsatcvatlem 33354 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ∈ 𝐴) |
41 | 29, 7 | sseldd 3569 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
42 | 3 | lsmlub 17901 | . . . . . . 7 ⊢ ((𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (𝑄 ⊕ 𝑅) ⊆ 𝑈)) |
43 | 31, 33, 41, 42 | syl3anc 1318 | . . . . . 6 ⊢ (𝜑 → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (𝑄 ⊕ 𝑅) ⊆ 𝑈)) |
44 | ssnpss 3672 | . . . . . 6 ⊢ ((𝑄 ⊕ 𝑅) ⊆ 𝑈 → ¬ 𝑈 ⊊ (𝑄 ⊕ 𝑅)) | |
45 | 43, 44 | syl6bi 242 | . . . . 5 ⊢ (𝜑 → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) → ¬ 𝑈 ⊊ (𝑄 ⊕ 𝑅))) |
46 | 45 | con2d 128 | . . . 4 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) → ¬ (𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈))) |
47 | ianor 508 | . . . 4 ⊢ (¬ (𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈)) | |
48 | 46, 47 | syl6ib 240 | . . 3 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) → (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈))) |
49 | 15, 48 | mpd 15 | . 2 ⊢ (𝜑 → (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈)) |
50 | 18, 40, 49 | mpjaodan 823 | 1 ⊢ (𝜑 → 𝑈 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ⊆ wss 3540 ⊊ wpss 3541 {csn 4125 ‘cfv 5804 (class class class)co 6549 0gc0g 15923 SubGrpcsubg 17411 LSSumclsm 17872 Abelcabl 18017 LModclmod 18686 LSubSpclss 18753 LVecclvec 18923 LSAtomsclsa 33279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-0g 15925 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-cntz 17573 df-oppg 17599 df-lsm 17874 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-drng 18572 df-lmod 18688 df-lss 18754 df-lsp 18793 df-lvec 18924 df-lsatoms 33281 df-lcv 33324 |
This theorem is referenced by: lsatcvat2 33356 |
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