![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > lsmsatcv | Structured version Visualization version GIF version |
Description: Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 31339 analog.) Explicit atom version of lsmcv 20988. (Contributed by NM, 29-Oct-2014.) |
Ref | Expression |
---|---|
lsmsatcv.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lsmsatcv.p | ⊢ ⊕ = (LSSum‘𝑊) |
lsmsatcv.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lsmsatcv.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lsmsatcv.t | ⊢ (𝜑 → 𝑇 ∈ 𝑆) |
lsmsatcv.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lsmsatcv.x | ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
Ref | Expression |
---|---|
lsmsatcv | ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ 𝑄)) → 𝑈 = (𝑇 ⊕ 𝑄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsmsatcv.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
2 | lsmsatcv.x | . . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝐴) | |
3 | eqid 2731 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | eqid 2731 | . . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
5 | lsmsatcv.a | . . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
6 | 3, 4, 5 | islsati 38330 | . . . 4 ⊢ ((𝑊 ∈ LVec ∧ 𝑄 ∈ 𝐴) → ∃𝑣 ∈ (Base‘𝑊)𝑄 = ((LSpan‘𝑊)‘{𝑣})) |
7 | 1, 2, 6 | syl2anc 583 | . . 3 ⊢ (𝜑 → ∃𝑣 ∈ (Base‘𝑊)𝑄 = ((LSpan‘𝑊)‘{𝑣})) |
8 | lsmsatcv.s | . . . . . . . 8 ⊢ 𝑆 = (LSubSp‘𝑊) | |
9 | lsmsatcv.p | . . . . . . . 8 ⊢ ⊕ = (LSSum‘𝑊) | |
10 | 1 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑊 ∈ LVec) |
11 | lsmsatcv.t | . . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ 𝑆) | |
12 | 11 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑇 ∈ 𝑆) |
13 | lsmsatcv.u | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
14 | 13 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑈 ∈ 𝑆) |
15 | simpr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑣 ∈ (Base‘𝑊)) | |
16 | 3, 8, 4, 9, 10, 12, 14, 15 | lsmcv 20988 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣}))) → 𝑈 = (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣}))) |
17 | 16 | 3expib 1121 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊)) → ((𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣}))) → 𝑈 = (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣})))) |
18 | 17 | 3adant3 1131 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑣})) → ((𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣}))) → 𝑈 = (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣})))) |
19 | oveq2 7420 | . . . . . . . . 9 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑣}) → (𝑇 ⊕ 𝑄) = (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣}))) | |
20 | 19 | sseq2d 4014 | . . . . . . . 8 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑣}) → (𝑈 ⊆ (𝑇 ⊕ 𝑄) ↔ 𝑈 ⊆ (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣})))) |
21 | 20 | anbi2d 628 | . . . . . . 7 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑣}) → ((𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ 𝑄)) ↔ (𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣}))))) |
22 | 19 | eqeq2d 2742 | . . . . . . 7 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑣}) → (𝑈 = (𝑇 ⊕ 𝑄) ↔ 𝑈 = (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣})))) |
23 | 21, 22 | imbi12d 344 | . . . . . 6 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑣}) → (((𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ 𝑄)) → 𝑈 = (𝑇 ⊕ 𝑄)) ↔ ((𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣}))) → 𝑈 = (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣}))))) |
24 | 23 | 3ad2ant3 1134 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑣})) → (((𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ 𝑄)) → 𝑈 = (𝑇 ⊕ 𝑄)) ↔ ((𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣}))) → 𝑈 = (𝑇 ⊕ ((LSpan‘𝑊)‘{𝑣}))))) |
25 | 18, 24 | mpbird 257 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑣})) → ((𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ 𝑄)) → 𝑈 = (𝑇 ⊕ 𝑄))) |
26 | 25 | rexlimdv3a 3158 | . . 3 ⊢ (𝜑 → (∃𝑣 ∈ (Base‘𝑊)𝑄 = ((LSpan‘𝑊)‘{𝑣}) → ((𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ 𝑄)) → 𝑈 = (𝑇 ⊕ 𝑄)))) |
27 | 7, 26 | mpd 15 | . 2 ⊢ (𝜑 → ((𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ 𝑄)) → 𝑈 = (𝑇 ⊕ 𝑄))) |
28 | 27 | 3impib 1115 | 1 ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ 𝑄)) → 𝑈 = (𝑇 ⊕ 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ∃wrex 3069 ⊆ wss 3948 ⊊ wpss 3949 {csn 4628 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 LSSumclsm 19550 LSubSpclss 20774 LSpanclspn 20814 LVecclvec 20946 LSAtomsclsa 38310 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19046 df-lsm 19552 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-drng 20585 df-lmod 20704 df-lss 20775 df-lsp 20815 df-lvec 20947 df-lsatoms 38312 |
This theorem is referenced by: dochsat 40720 |
Copyright terms: Public domain | W3C validator |