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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsmcv2 | Structured version Visualization version GIF version |
Description: Subspace sum has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (spansncv2 30070 analog.) (Contributed by NM, 10-Jan-2015.) |
Ref | Expression |
---|---|
lsmcv2.v | ⊢ 𝑉 = (Base‘𝑊) |
lsmcv2.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lsmcv2.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lsmcv2.p | ⊢ ⊕ = (LSSum‘𝑊) |
lsmcv2.c | ⊢ 𝐶 = ( ⋖L ‘𝑊) |
lsmcv2.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lsmcv2.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lsmcv2.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lsmcv2.l | ⊢ (𝜑 → ¬ (𝑁‘{𝑋}) ⊆ 𝑈) |
Ref | Expression |
---|---|
lsmcv2 | ⊢ (𝜑 → 𝑈𝐶(𝑈 ⊕ (𝑁‘{𝑋}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsmcv2.l | . . 3 ⊢ (𝜑 → ¬ (𝑁‘{𝑋}) ⊆ 𝑈) | |
2 | lsmcv2.p | . . . 4 ⊢ ⊕ = (LSSum‘𝑊) | |
3 | lsmcv2.w | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
4 | lveclmod 19878 | . . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod) |
6 | lsmcv2.s | . . . . . . 7 ⊢ 𝑆 = (LSubSp‘𝑊) | |
7 | 6 | lsssssubg 19730 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
8 | 5, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊)) |
9 | lsmcv2.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
10 | 8, 9 | sseldd 3968 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
11 | lsmcv2.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
12 | lsmcv2.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑊) | |
13 | lsmcv2.n | . . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑊) | |
14 | 12, 6, 13 | lspsncl 19749 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ 𝑆) |
15 | 5, 11, 14 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝑆) |
16 | 8, 15 | sseldd 3968 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) |
17 | 2, 10, 16 | lssnle 18800 | . . 3 ⊢ (𝜑 → (¬ (𝑁‘{𝑋}) ⊆ 𝑈 ↔ 𝑈 ⊊ (𝑈 ⊕ (𝑁‘{𝑋})))) |
18 | 1, 17 | mpbid 234 | . 2 ⊢ (𝜑 → 𝑈 ⊊ (𝑈 ⊕ (𝑁‘{𝑋}))) |
19 | 3simpa 1144 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ (𝑈 ⊊ 𝑥 ∧ 𝑥 ⊆ (𝑈 ⊕ (𝑁‘{𝑋})))) → (𝜑 ∧ 𝑥 ∈ 𝑆)) | |
20 | simp3l 1197 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ (𝑈 ⊊ 𝑥 ∧ 𝑥 ⊆ (𝑈 ⊕ (𝑁‘{𝑋})))) → 𝑈 ⊊ 𝑥) | |
21 | simp3r 1198 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ (𝑈 ⊊ 𝑥 ∧ 𝑥 ⊆ (𝑈 ⊕ (𝑁‘{𝑋})))) → 𝑥 ⊆ (𝑈 ⊕ (𝑁‘{𝑋}))) | |
22 | 3 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑊 ∈ LVec) |
23 | 9 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑈 ∈ 𝑆) |
24 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆) | |
25 | 11 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑋 ∈ 𝑉) |
26 | 12, 6, 13, 2, 22, 23, 24, 25 | lsmcv 19913 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑈 ⊊ 𝑥 ∧ 𝑥 ⊆ (𝑈 ⊕ (𝑁‘{𝑋}))) → 𝑥 = (𝑈 ⊕ (𝑁‘{𝑋}))) |
27 | 19, 20, 21, 26 | syl3anc 1367 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ (𝑈 ⊊ 𝑥 ∧ 𝑥 ⊆ (𝑈 ⊕ (𝑁‘{𝑋})))) → 𝑥 = (𝑈 ⊕ (𝑁‘{𝑋}))) |
28 | 27 | 3exp 1115 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑆 → ((𝑈 ⊊ 𝑥 ∧ 𝑥 ⊆ (𝑈 ⊕ (𝑁‘{𝑋}))) → 𝑥 = (𝑈 ⊕ (𝑁‘{𝑋}))))) |
29 | 28 | ralrimiv 3181 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ((𝑈 ⊊ 𝑥 ∧ 𝑥 ⊆ (𝑈 ⊕ (𝑁‘{𝑋}))) → 𝑥 = (𝑈 ⊕ (𝑁‘{𝑋})))) |
30 | lsmcv2.c | . . 3 ⊢ 𝐶 = ( ⋖L ‘𝑊) | |
31 | 6, 2 | lsmcl 19855 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ (𝑁‘{𝑋}) ∈ 𝑆) → (𝑈 ⊕ (𝑁‘{𝑋})) ∈ 𝑆) |
32 | 5, 9, 15, 31 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑋})) ∈ 𝑆) |
33 | 6, 30, 3, 9, 32 | lcvbr2 36173 | . 2 ⊢ (𝜑 → (𝑈𝐶(𝑈 ⊕ (𝑁‘{𝑋})) ↔ (𝑈 ⊊ (𝑈 ⊕ (𝑁‘{𝑋})) ∧ ∀𝑥 ∈ 𝑆 ((𝑈 ⊊ 𝑥 ∧ 𝑥 ⊆ (𝑈 ⊕ (𝑁‘{𝑋}))) → 𝑥 = (𝑈 ⊕ (𝑁‘{𝑋})))))) |
34 | 18, 29, 33 | mpbir2and 711 | 1 ⊢ (𝜑 → 𝑈𝐶(𝑈 ⊕ (𝑁‘{𝑋}))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ⊆ wss 3936 ⊊ wpss 3937 {csn 4567 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 SubGrpcsubg 18273 LSSumclsm 18759 LModclmod 19634 LSubSpclss 19703 LSpanclspn 19743 LVecclvec 19874 ⋖L clcv 36169 |
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-tpos 7892 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-3 11702 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 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-cntz 18447 df-lsm 18761 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-drng 19504 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lvec 19875 df-lcv 36170 |
This theorem is referenced by: lcv1 36192 |
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