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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 32168 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 38480 | . 2 ⊢ (𝜑 → ∃𝑞 ∈ 𝐴 (𝑞 ⊆ 𝑈 ∧ ¬ 𝑞 ⊆ 𝑇)) |
8 | ancom 460 | . . . 4 ⊢ ((𝑞 ⊆ 𝑈 ∧ ¬ 𝑞 ⊆ 𝑇) ↔ (¬ 𝑞 ⊆ 𝑇 ∧ 𝑞 ⊆ 𝑈)) | |
9 | lrelat.p | . . . . . 6 ⊢ ⊕ = (LSSum‘𝑊) | |
10 | 3 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑊 ∈ LMod) |
11 | 1 | lsssssubg 20836 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
12 | 10, 11 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑆 ⊆ (SubGrp‘𝑊)) |
13 | 4 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑇 ∈ 𝑆) |
14 | 12, 13 | sseldd 3980 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑇 ∈ (SubGrp‘𝑊)) |
15 | simpr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ 𝐴) | |
16 | 1, 2, 10, 15 | lsatlssel 38464 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ 𝑆) |
17 | 12, 16 | sseldd 3980 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ (SubGrp‘𝑊)) |
18 | 9, 14, 17 | lssnle 19623 | . . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (¬ 𝑞 ⊆ 𝑇 ↔ 𝑇 ⊊ (𝑇 ⊕ 𝑞))) |
19 | 6 | pssssd 4094 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ⊆ 𝑈) |
20 | 19 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑇 ⊆ 𝑈) |
21 | 20 | biantrurd 532 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 ⊆ 𝑈 ↔ (𝑇 ⊆ 𝑈 ∧ 𝑞 ⊆ 𝑈))) |
22 | 5 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑈 ∈ 𝑆) |
23 | 12, 22 | sseldd 3980 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑈 ∈ (SubGrp‘𝑊)) |
24 | 9 | lsmlub 19613 | . . . . . . 7 ⊢ ((𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑞 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → ((𝑇 ⊆ 𝑈 ∧ 𝑞 ⊆ 𝑈) ↔ (𝑇 ⊕ 𝑞) ⊆ 𝑈)) |
25 | 14, 17, 23, 24 | syl3anc 1369 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝑇 ⊆ 𝑈 ∧ 𝑞 ⊆ 𝑈) ↔ (𝑇 ⊕ 𝑞) ⊆ 𝑈)) |
26 | 21, 25 | bitrd 279 | . . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 ⊆ 𝑈 ↔ (𝑇 ⊕ 𝑞) ⊆ 𝑈)) |
27 | 18, 26 | anbi12d 631 | . . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((¬ 𝑞 ⊆ 𝑇 ∧ 𝑞 ⊆ 𝑈) ↔ (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈))) |
28 | 8, 27 | bitrid 283 | . . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝑞 ⊆ 𝑈 ∧ ¬ 𝑞 ⊆ 𝑇) ↔ (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈))) |
29 | 28 | rexbidva 3172 | . 2 ⊢ (𝜑 → (∃𝑞 ∈ 𝐴 (𝑞 ⊆ 𝑈 ∧ ¬ 𝑞 ⊆ 𝑇) ↔ ∃𝑞 ∈ 𝐴 (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈))) |
30 | 7, 29 | mpbid 231 | 1 ⊢ (𝜑 → ∃𝑞 ∈ 𝐴 (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∃wrex 3066 ⊆ wss 3945 ⊊ wpss 3946 ‘cfv 6543 (class class class)co 7415 SubGrpcsubg 19069 LSSumclsm 19583 LModclmod 20737 LSubSpclss 20809 LSAtomsclsa 38441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-0g 17417 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-submnd 18735 df-grp 18887 df-minusg 18888 df-sbg 18889 df-subg 19072 df-lsm 19585 df-mgp 20069 df-ur 20116 df-ring 20169 df-lmod 20739 df-lss 20810 df-lsp 20850 df-lsatoms 38443 |
This theorem is referenced by: lcvat 38497 |
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