Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > islmodfg | Structured version Visualization version GIF version |
Description: Property of a finitely generated left module. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
Ref | Expression |
---|---|
islmodfg.b | ⊢ 𝐵 = (Base‘𝑊) |
islmodfg.n | ⊢ 𝑁 = (LSpan‘𝑊) |
Ref | Expression |
---|---|
islmodfg | ⊢ (𝑊 ∈ LMod → (𝑊 ∈ LFinGen ↔ ∃𝑏 ∈ 𝒫 𝐵(𝑏 ∈ Fin ∧ (𝑁‘𝑏) = 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-lfig 39548 | . . . 4 ⊢ LFinGen = {𝑎 ∈ LMod ∣ (Base‘𝑎) ∈ ((LSpan‘𝑎) “ (𝒫 (Base‘𝑎) ∩ Fin))} | |
2 | 1 | eleq2i 2904 | . . 3 ⊢ (𝑊 ∈ LFinGen ↔ 𝑊 ∈ {𝑎 ∈ LMod ∣ (Base‘𝑎) ∈ ((LSpan‘𝑎) “ (𝒫 (Base‘𝑎) ∩ Fin))}) |
3 | fveq2 6664 | . . . . 5 ⊢ (𝑎 = 𝑊 → (Base‘𝑎) = (Base‘𝑊)) | |
4 | fveq2 6664 | . . . . . . 7 ⊢ (𝑎 = 𝑊 → (LSpan‘𝑎) = (LSpan‘𝑊)) | |
5 | islmodfg.n | . . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑊) | |
6 | 4, 5 | syl6eqr 2874 | . . . . . 6 ⊢ (𝑎 = 𝑊 → (LSpan‘𝑎) = 𝑁) |
7 | 3 | pweqd 4542 | . . . . . . 7 ⊢ (𝑎 = 𝑊 → 𝒫 (Base‘𝑎) = 𝒫 (Base‘𝑊)) |
8 | 7 | ineq1d 4187 | . . . . . 6 ⊢ (𝑎 = 𝑊 → (𝒫 (Base‘𝑎) ∩ Fin) = (𝒫 (Base‘𝑊) ∩ Fin)) |
9 | 6, 8 | imaeq12d 5924 | . . . . 5 ⊢ (𝑎 = 𝑊 → ((LSpan‘𝑎) “ (𝒫 (Base‘𝑎) ∩ Fin)) = (𝑁 “ (𝒫 (Base‘𝑊) ∩ Fin))) |
10 | 3, 9 | eleq12d 2907 | . . . 4 ⊢ (𝑎 = 𝑊 → ((Base‘𝑎) ∈ ((LSpan‘𝑎) “ (𝒫 (Base‘𝑎) ∩ Fin)) ↔ (Base‘𝑊) ∈ (𝑁 “ (𝒫 (Base‘𝑊) ∩ Fin)))) |
11 | 10 | elrab3 3680 | . . 3 ⊢ (𝑊 ∈ LMod → (𝑊 ∈ {𝑎 ∈ LMod ∣ (Base‘𝑎) ∈ ((LSpan‘𝑎) “ (𝒫 (Base‘𝑎) ∩ Fin))} ↔ (Base‘𝑊) ∈ (𝑁 “ (𝒫 (Base‘𝑊) ∩ Fin)))) |
12 | 2, 11 | syl5bb 284 | . 2 ⊢ (𝑊 ∈ LMod → (𝑊 ∈ LFinGen ↔ (Base‘𝑊) ∈ (𝑁 “ (𝒫 (Base‘𝑊) ∩ Fin)))) |
13 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
14 | eqid 2821 | . . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
15 | 13, 14, 5 | lspf 19677 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝑁:𝒫 (Base‘𝑊)⟶(LSubSp‘𝑊)) |
16 | 15 | ffnd 6509 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝑁 Fn 𝒫 (Base‘𝑊)) |
17 | inss1 4204 | . . . 4 ⊢ (𝒫 (Base‘𝑊) ∩ Fin) ⊆ 𝒫 (Base‘𝑊) | |
18 | fvelimab 6731 | . . . 4 ⊢ ((𝑁 Fn 𝒫 (Base‘𝑊) ∧ (𝒫 (Base‘𝑊) ∩ Fin) ⊆ 𝒫 (Base‘𝑊)) → ((Base‘𝑊) ∈ (𝑁 “ (𝒫 (Base‘𝑊) ∩ Fin)) ↔ ∃𝑏 ∈ (𝒫 (Base‘𝑊) ∩ Fin)(𝑁‘𝑏) = (Base‘𝑊))) | |
19 | 16, 17, 18 | sylancl 586 | . . 3 ⊢ (𝑊 ∈ LMod → ((Base‘𝑊) ∈ (𝑁 “ (𝒫 (Base‘𝑊) ∩ Fin)) ↔ ∃𝑏 ∈ (𝒫 (Base‘𝑊) ∩ Fin)(𝑁‘𝑏) = (Base‘𝑊))) |
20 | elin 4168 | . . . . . . 7 ⊢ (𝑏 ∈ (𝒫 (Base‘𝑊) ∩ Fin) ↔ (𝑏 ∈ 𝒫 (Base‘𝑊) ∧ 𝑏 ∈ Fin)) | |
21 | islmodfg.b | . . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑊) | |
22 | 21 | eqcomi 2830 | . . . . . . . . . 10 ⊢ (Base‘𝑊) = 𝐵 |
23 | 22 | pweqi 4541 | . . . . . . . . 9 ⊢ 𝒫 (Base‘𝑊) = 𝒫 𝐵 |
24 | 23 | eleq2i 2904 | . . . . . . . 8 ⊢ (𝑏 ∈ 𝒫 (Base‘𝑊) ↔ 𝑏 ∈ 𝒫 𝐵) |
25 | 24 | anbi1i 623 | . . . . . . 7 ⊢ ((𝑏 ∈ 𝒫 (Base‘𝑊) ∧ 𝑏 ∈ Fin) ↔ (𝑏 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ Fin)) |
26 | 20, 25 | bitri 276 | . . . . . 6 ⊢ (𝑏 ∈ (𝒫 (Base‘𝑊) ∩ Fin) ↔ (𝑏 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ Fin)) |
27 | 22 | eqeq2i 2834 | . . . . . 6 ⊢ ((𝑁‘𝑏) = (Base‘𝑊) ↔ (𝑁‘𝑏) = 𝐵) |
28 | 26, 27 | anbi12i 626 | . . . . 5 ⊢ ((𝑏 ∈ (𝒫 (Base‘𝑊) ∩ Fin) ∧ (𝑁‘𝑏) = (Base‘𝑊)) ↔ ((𝑏 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ Fin) ∧ (𝑁‘𝑏) = 𝐵)) |
29 | anass 469 | . . . . 5 ⊢ (((𝑏 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ Fin) ∧ (𝑁‘𝑏) = 𝐵) ↔ (𝑏 ∈ 𝒫 𝐵 ∧ (𝑏 ∈ Fin ∧ (𝑁‘𝑏) = 𝐵))) | |
30 | 28, 29 | bitri 276 | . . . 4 ⊢ ((𝑏 ∈ (𝒫 (Base‘𝑊) ∩ Fin) ∧ (𝑁‘𝑏) = (Base‘𝑊)) ↔ (𝑏 ∈ 𝒫 𝐵 ∧ (𝑏 ∈ Fin ∧ (𝑁‘𝑏) = 𝐵))) |
31 | 30 | rexbii2 3245 | . . 3 ⊢ (∃𝑏 ∈ (𝒫 (Base‘𝑊) ∩ Fin)(𝑁‘𝑏) = (Base‘𝑊) ↔ ∃𝑏 ∈ 𝒫 𝐵(𝑏 ∈ Fin ∧ (𝑁‘𝑏) = 𝐵)) |
32 | 19, 31 | syl6bb 288 | . 2 ⊢ (𝑊 ∈ LMod → ((Base‘𝑊) ∈ (𝑁 “ (𝒫 (Base‘𝑊) ∩ Fin)) ↔ ∃𝑏 ∈ 𝒫 𝐵(𝑏 ∈ Fin ∧ (𝑁‘𝑏) = 𝐵))) |
33 | 12, 32 | bitrd 280 | 1 ⊢ (𝑊 ∈ LMod → (𝑊 ∈ LFinGen ↔ ∃𝑏 ∈ 𝒫 𝐵(𝑏 ∈ Fin ∧ (𝑁‘𝑏) = 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∃wrex 3139 {crab 3142 ∩ cin 3934 ⊆ wss 3935 𝒫 cpw 4537 “ cima 5552 Fn wfn 6344 ‘cfv 6349 Fincfn 8498 Basecbs 16473 LModclmod 19565 LSubSpclss 19634 LSpanclspn 19674 LFinGenclfig 39547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 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 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11628 df-2 11689 df-ndx 16476 df-slot 16477 df-base 16479 df-sets 16480 df-plusg 16568 df-0g 16705 df-mgm 17842 df-sgrp 17891 df-mnd 17902 df-grp 18046 df-minusg 18047 df-sbg 18048 df-mgp 19171 df-ur 19183 df-ring 19230 df-lmod 19567 df-lss 19635 df-lsp 19675 df-lfig 39548 |
This theorem is referenced by: islssfg 39550 lnrfg 39599 |
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