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Mirrors > Home > MPE Home > Th. List > Mathboxes > lrelat | Structured version Visualization version GIF version |
Description: Subspaces are relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 29774 analog.) (Contributed by NM, 11-Jan-2015.) |
Ref | Expression |
---|---|
lrelat.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lrelat.p | ⊢ ⊕ = (LSSum‘𝑊) |
lrelat.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lrelat.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lrelat.t | ⊢ (𝜑 → 𝑇 ∈ 𝑆) |
lrelat.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lrelat.l | ⊢ (𝜑 → 𝑇 ⊊ 𝑈) |
Ref | Expression |
---|---|
lrelat | ⊢ (𝜑 → ∃𝑞 ∈ 𝐴 (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lrelat.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
2 | lrelat.a | . . 3 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
3 | lrelat.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
4 | lrelat.t | . . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑆) | |
5 | lrelat.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
6 | lrelat.l | . . 3 ⊢ (𝜑 → 𝑇 ⊊ 𝑈) | |
7 | 1, 2, 3, 4, 5, 6 | lpssat 35087 | . 2 ⊢ (𝜑 → ∃𝑞 ∈ 𝐴 (𝑞 ⊆ 𝑈 ∧ ¬ 𝑞 ⊆ 𝑇)) |
8 | ancom 454 | . . . 4 ⊢ ((𝑞 ⊆ 𝑈 ∧ ¬ 𝑞 ⊆ 𝑇) ↔ (¬ 𝑞 ⊆ 𝑇 ∧ 𝑞 ⊆ 𝑈)) | |
9 | lrelat.p | . . . . . 6 ⊢ ⊕ = (LSSum‘𝑊) | |
10 | 3 | adantr 474 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑊 ∈ LMod) |
11 | 1 | lsssssubg 19324 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
12 | 10, 11 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑆 ⊆ (SubGrp‘𝑊)) |
13 | 4 | adantr 474 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑇 ∈ 𝑆) |
14 | 12, 13 | sseldd 3828 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑇 ∈ (SubGrp‘𝑊)) |
15 | simpr 479 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ 𝐴) | |
16 | 1, 2, 10, 15 | lsatlssel 35071 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ 𝑆) |
17 | 12, 16 | sseldd 3828 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ (SubGrp‘𝑊)) |
18 | 9, 14, 17 | lssnle 18445 | . . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (¬ 𝑞 ⊆ 𝑇 ↔ 𝑇 ⊊ (𝑇 ⊕ 𝑞))) |
19 | 6 | pssssd 3932 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ⊆ 𝑈) |
20 | 19 | adantr 474 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑇 ⊆ 𝑈) |
21 | 20 | biantrurd 528 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 ⊆ 𝑈 ↔ (𝑇 ⊆ 𝑈 ∧ 𝑞 ⊆ 𝑈))) |
22 | 5 | adantr 474 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑈 ∈ 𝑆) |
23 | 12, 22 | sseldd 3828 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑈 ∈ (SubGrp‘𝑊)) |
24 | 9 | lsmlub 18436 | . . . . . . 7 ⊢ ((𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑞 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → ((𝑇 ⊆ 𝑈 ∧ 𝑞 ⊆ 𝑈) ↔ (𝑇 ⊕ 𝑞) ⊆ 𝑈)) |
25 | 14, 17, 23, 24 | syl3anc 1494 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝑇 ⊆ 𝑈 ∧ 𝑞 ⊆ 𝑈) ↔ (𝑇 ⊕ 𝑞) ⊆ 𝑈)) |
26 | 21, 25 | bitrd 271 | . . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 ⊆ 𝑈 ↔ (𝑇 ⊕ 𝑞) ⊆ 𝑈)) |
27 | 18, 26 | anbi12d 624 | . . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((¬ 𝑞 ⊆ 𝑇 ∧ 𝑞 ⊆ 𝑈) ↔ (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈))) |
28 | 8, 27 | syl5bb 275 | . . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝑞 ⊆ 𝑈 ∧ ¬ 𝑞 ⊆ 𝑇) ↔ (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈))) |
29 | 28 | rexbidva 3259 | . 2 ⊢ (𝜑 → (∃𝑞 ∈ 𝐴 (𝑞 ⊆ 𝑈 ∧ ¬ 𝑞 ⊆ 𝑇) ↔ ∃𝑞 ∈ 𝐴 (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈))) |
30 | 7, 29 | mpbid 224 | 1 ⊢ (𝜑 → ∃𝑞 ∈ 𝐴 (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∃wrex 3118 ⊆ wss 3798 ⊊ wpss 3799 ‘cfv 6127 (class class class)co 6910 SubGrpcsubg 17946 LSSumclsm 18407 LModclmod 19226 LSubSpclss 19295 LSAtomsclsa 35048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-0g 16462 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-submnd 17696 df-grp 17786 df-minusg 17787 df-sbg 17788 df-subg 17949 df-lsm 18409 df-mgp 18851 df-ur 18863 df-ring 18910 df-lmod 19228 df-lss 19296 df-lsp 19338 df-lsatoms 35050 |
This theorem is referenced by: lcvat 35104 |
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