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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 30141 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 36164 | . 2 ⊢ (𝜑 → ∃𝑞 ∈ 𝐴 (𝑞 ⊆ 𝑈 ∧ ¬ 𝑞 ⊆ 𝑇)) |
8 | ancom 463 | . . . 4 ⊢ ((𝑞 ⊆ 𝑈 ∧ ¬ 𝑞 ⊆ 𝑇) ↔ (¬ 𝑞 ⊆ 𝑇 ∧ 𝑞 ⊆ 𝑈)) | |
9 | lrelat.p | . . . . . 6 ⊢ ⊕ = (LSSum‘𝑊) | |
10 | 3 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑊 ∈ LMod) |
11 | 1 | lsssssubg 19730 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
12 | 10, 11 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑆 ⊆ (SubGrp‘𝑊)) |
13 | 4 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑇 ∈ 𝑆) |
14 | 12, 13 | sseldd 3968 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑇 ∈ (SubGrp‘𝑊)) |
15 | simpr 487 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ 𝐴) | |
16 | 1, 2, 10, 15 | lsatlssel 36148 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ 𝑆) |
17 | 12, 16 | sseldd 3968 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ (SubGrp‘𝑊)) |
18 | 9, 14, 17 | lssnle 18800 | . . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (¬ 𝑞 ⊆ 𝑇 ↔ 𝑇 ⊊ (𝑇 ⊕ 𝑞))) |
19 | 6 | pssssd 4074 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ⊆ 𝑈) |
20 | 19 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑇 ⊆ 𝑈) |
21 | 20 | biantrurd 535 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 ⊆ 𝑈 ↔ (𝑇 ⊆ 𝑈 ∧ 𝑞 ⊆ 𝑈))) |
22 | 5 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑈 ∈ 𝑆) |
23 | 12, 22 | sseldd 3968 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝑈 ∈ (SubGrp‘𝑊)) |
24 | 9 | lsmlub 18790 | . . . . . . 7 ⊢ ((𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑞 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → ((𝑇 ⊆ 𝑈 ∧ 𝑞 ⊆ 𝑈) ↔ (𝑇 ⊕ 𝑞) ⊆ 𝑈)) |
25 | 14, 17, 23, 24 | syl3anc 1367 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝑇 ⊆ 𝑈 ∧ 𝑞 ⊆ 𝑈) ↔ (𝑇 ⊕ 𝑞) ⊆ 𝑈)) |
26 | 21, 25 | bitrd 281 | . . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (𝑞 ⊆ 𝑈 ↔ (𝑇 ⊕ 𝑞) ⊆ 𝑈)) |
27 | 18, 26 | anbi12d 632 | . . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((¬ 𝑞 ⊆ 𝑇 ∧ 𝑞 ⊆ 𝑈) ↔ (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈))) |
28 | 8, 27 | syl5bb 285 | . . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → ((𝑞 ⊆ 𝑈 ∧ ¬ 𝑞 ⊆ 𝑇) ↔ (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈))) |
29 | 28 | rexbidva 3296 | . 2 ⊢ (𝜑 → (∃𝑞 ∈ 𝐴 (𝑞 ⊆ 𝑈 ∧ ¬ 𝑞 ⊆ 𝑇) ↔ ∃𝑞 ∈ 𝐴 (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈))) |
30 | 7, 29 | mpbid 234 | 1 ⊢ (𝜑 → ∃𝑞 ∈ 𝐴 (𝑇 ⊊ (𝑇 ⊕ 𝑞) ∧ (𝑇 ⊕ 𝑞) ⊆ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 ⊆ wss 3936 ⊊ wpss 3937 ‘cfv 6355 (class class class)co 7156 SubGrpcsubg 18273 LSSumclsm 18759 LModclmod 19634 LSubSpclss 19703 LSAtomsclsa 36125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 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 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-lsm 18761 df-mgp 19240 df-ur 19252 df-ring 19299 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lsatoms 36127 |
This theorem is referenced by: lcvat 36181 |
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