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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > islssfgi | Structured version Visualization version GIF version |
Description: Finitely spanned subspaces are finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
Ref | Expression |
---|---|
islssfgi.n | ⊢ 𝑁 = (LSpan‘𝑊) |
islssfgi.v | ⊢ 𝑉 = (Base‘𝑊) |
islssfgi.x | ⊢ 𝑋 = (𝑊 ↾s (𝑁‘𝐵)) |
Ref | Expression |
---|---|
islssfgi | ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ⊆ 𝑉 ∧ 𝐵 ∈ Fin) → 𝑋 ∈ LFinGen) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islssfgi.v | . . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊) | |
2 | 1 | fvexi 6903 | . . . . . . 7 ⊢ 𝑉 ∈ V |
3 | 2 | elpw2 5345 | . . . . . 6 ⊢ (𝐵 ∈ 𝒫 𝑉 ↔ 𝐵 ⊆ 𝑉) |
4 | 3 | biimpri 227 | . . . . 5 ⊢ (𝐵 ⊆ 𝑉 → 𝐵 ∈ 𝒫 𝑉) |
5 | 4 | 3ad2ant2 1135 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ⊆ 𝑉 ∧ 𝐵 ∈ Fin) → 𝐵 ∈ 𝒫 𝑉) |
6 | simp3 1139 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ⊆ 𝑉 ∧ 𝐵 ∈ Fin) → 𝐵 ∈ Fin) | |
7 | 5, 6 | elind 4194 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ⊆ 𝑉 ∧ 𝐵 ∈ Fin) → 𝐵 ∈ (𝒫 𝑉 ∩ Fin)) |
8 | eqid 2733 | . . 3 ⊢ (𝑁‘𝐵) = (𝑁‘𝐵) | |
9 | fveqeq2 6898 | . . . 4 ⊢ (𝑎 = 𝐵 → ((𝑁‘𝑎) = (𝑁‘𝐵) ↔ (𝑁‘𝐵) = (𝑁‘𝐵))) | |
10 | 9 | rspcev 3613 | . . 3 ⊢ ((𝐵 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝑁‘𝐵) = (𝑁‘𝐵)) → ∃𝑎 ∈ (𝒫 𝑉 ∩ Fin)(𝑁‘𝑎) = (𝑁‘𝐵)) |
11 | 7, 8, 10 | sylancl 587 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ⊆ 𝑉 ∧ 𝐵 ∈ Fin) → ∃𝑎 ∈ (𝒫 𝑉 ∩ Fin)(𝑁‘𝑎) = (𝑁‘𝐵)) |
12 | simp1 1137 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ⊆ 𝑉 ∧ 𝐵 ∈ Fin) → 𝑊 ∈ LMod) | |
13 | eqid 2733 | . . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
14 | islssfgi.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑊) | |
15 | 1, 13, 14 | lspcl 20580 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ⊆ 𝑉) → (𝑁‘𝐵) ∈ (LSubSp‘𝑊)) |
16 | 15 | 3adant3 1133 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ⊆ 𝑉 ∧ 𝐵 ∈ Fin) → (𝑁‘𝐵) ∈ (LSubSp‘𝑊)) |
17 | islssfgi.x | . . . 4 ⊢ 𝑋 = (𝑊 ↾s (𝑁‘𝐵)) | |
18 | 17, 13, 14, 1 | islssfg2 41799 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘𝐵) ∈ (LSubSp‘𝑊)) → (𝑋 ∈ LFinGen ↔ ∃𝑎 ∈ (𝒫 𝑉 ∩ Fin)(𝑁‘𝑎) = (𝑁‘𝐵))) |
19 | 12, 16, 18 | syl2anc 585 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ⊆ 𝑉 ∧ 𝐵 ∈ Fin) → (𝑋 ∈ LFinGen ↔ ∃𝑎 ∈ (𝒫 𝑉 ∩ Fin)(𝑁‘𝑎) = (𝑁‘𝐵))) |
20 | 11, 19 | mpbird 257 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ⊆ 𝑉 ∧ 𝐵 ∈ Fin) → 𝑋 ∈ LFinGen) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 ∩ cin 3947 ⊆ wss 3948 𝒫 cpw 4602 ‘cfv 6541 (class class class)co 7406 Fincfn 8936 Basecbs 17141 ↾s cress 17170 LModclmod 20464 LSubSpclss 20535 LSpanclspn 20575 LFinGenclfig 41795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 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 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-sca 17210 df-vsca 17211 df-0g 17384 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-grp 18819 df-minusg 18820 df-sbg 18821 df-subg 18998 df-mgp 19983 df-ur 20000 df-ring 20052 df-lmod 20466 df-lss 20536 df-lsp 20576 df-lfig 41796 |
This theorem is referenced by: lsmfgcl 41802 lmhmfgima 41812 lmhmfgsplit 41814 |
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