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 36656 | . . . 4 ⊢ LFinGen = {𝑎 ∈ LMod ∣ (Base‘𝑎) ∈ ((LSpan‘𝑎) “ (𝒫 (Base‘𝑎) ∩ Fin))} | |
2 | 1 | eleq2i 2680 | . . 3 ⊢ (𝑊 ∈ LFinGen ↔ 𝑊 ∈ {𝑎 ∈ LMod ∣ (Base‘𝑎) ∈ ((LSpan‘𝑎) “ (𝒫 (Base‘𝑎) ∩ Fin))}) |
3 | fveq2 6103 | . . . . 5 ⊢ (𝑎 = 𝑊 → (Base‘𝑎) = (Base‘𝑊)) | |
4 | fveq2 6103 | . . . . . . 7 ⊢ (𝑎 = 𝑊 → (LSpan‘𝑎) = (LSpan‘𝑊)) | |
5 | islmodfg.n | . . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑊) | |
6 | 4, 5 | syl6eqr 2662 | . . . . . 6 ⊢ (𝑎 = 𝑊 → (LSpan‘𝑎) = 𝑁) |
7 | 3 | pweqd 4113 | . . . . . . 7 ⊢ (𝑎 = 𝑊 → 𝒫 (Base‘𝑎) = 𝒫 (Base‘𝑊)) |
8 | 7 | ineq1d 3775 | . . . . . 6 ⊢ (𝑎 = 𝑊 → (𝒫 (Base‘𝑎) ∩ Fin) = (𝒫 (Base‘𝑊) ∩ Fin)) |
9 | 6, 8 | imaeq12d 5386 | . . . . 5 ⊢ (𝑎 = 𝑊 → ((LSpan‘𝑎) “ (𝒫 (Base‘𝑎) ∩ Fin)) = (𝑁 “ (𝒫 (Base‘𝑊) ∩ Fin))) |
10 | 3, 9 | eleq12d 2682 | . . . 4 ⊢ (𝑎 = 𝑊 → ((Base‘𝑎) ∈ ((LSpan‘𝑎) “ (𝒫 (Base‘𝑎) ∩ Fin)) ↔ (Base‘𝑊) ∈ (𝑁 “ (𝒫 (Base‘𝑊) ∩ Fin)))) |
11 | 10 | elrab3 3332 | . . 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 271 | . 2 ⊢ (𝑊 ∈ LMod → (𝑊 ∈ LFinGen ↔ (Base‘𝑊) ∈ (𝑁 “ (𝒫 (Base‘𝑊) ∩ Fin)))) |
13 | eqid 2610 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
14 | eqid 2610 | . . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
15 | 13, 14, 5 | lspf 18795 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝑁:𝒫 (Base‘𝑊)⟶(LSubSp‘𝑊)) |
16 | ffn 5958 | . . . . 5 ⊢ (𝑁:𝒫 (Base‘𝑊)⟶(LSubSp‘𝑊) → 𝑁 Fn 𝒫 (Base‘𝑊)) | |
17 | 15, 16 | syl 17 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝑁 Fn 𝒫 (Base‘𝑊)) |
18 | inss1 3795 | . . . 4 ⊢ (𝒫 (Base‘𝑊) ∩ Fin) ⊆ 𝒫 (Base‘𝑊) | |
19 | fvelimab 6163 | . . . 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))) | |
20 | 17, 18, 19 | sylancl 693 | . . 3 ⊢ (𝑊 ∈ LMod → ((Base‘𝑊) ∈ (𝑁 “ (𝒫 (Base‘𝑊) ∩ Fin)) ↔ ∃𝑏 ∈ (𝒫 (Base‘𝑊) ∩ Fin)(𝑁‘𝑏) = (Base‘𝑊))) |
21 | elin 3758 | . . . . . . 7 ⊢ (𝑏 ∈ (𝒫 (Base‘𝑊) ∩ Fin) ↔ (𝑏 ∈ 𝒫 (Base‘𝑊) ∧ 𝑏 ∈ Fin)) | |
22 | islmodfg.b | . . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑊) | |
23 | 22 | eqcomi 2619 | . . . . . . . . . 10 ⊢ (Base‘𝑊) = 𝐵 |
24 | 23 | pweqi 4112 | . . . . . . . . 9 ⊢ 𝒫 (Base‘𝑊) = 𝒫 𝐵 |
25 | 24 | eleq2i 2680 | . . . . . . . 8 ⊢ (𝑏 ∈ 𝒫 (Base‘𝑊) ↔ 𝑏 ∈ 𝒫 𝐵) |
26 | 25 | anbi1i 727 | . . . . . . 7 ⊢ ((𝑏 ∈ 𝒫 (Base‘𝑊) ∧ 𝑏 ∈ Fin) ↔ (𝑏 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ Fin)) |
27 | 21, 26 | bitri 263 | . . . . . 6 ⊢ (𝑏 ∈ (𝒫 (Base‘𝑊) ∩ Fin) ↔ (𝑏 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ Fin)) |
28 | 23 | eqeq2i 2622 | . . . . . 6 ⊢ ((𝑁‘𝑏) = (Base‘𝑊) ↔ (𝑁‘𝑏) = 𝐵) |
29 | 27, 28 | anbi12i 729 | . . . . 5 ⊢ ((𝑏 ∈ (𝒫 (Base‘𝑊) ∩ Fin) ∧ (𝑁‘𝑏) = (Base‘𝑊)) ↔ ((𝑏 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ Fin) ∧ (𝑁‘𝑏) = 𝐵)) |
30 | anass 679 | . . . . 5 ⊢ (((𝑏 ∈ 𝒫 𝐵 ∧ 𝑏 ∈ Fin) ∧ (𝑁‘𝑏) = 𝐵) ↔ (𝑏 ∈ 𝒫 𝐵 ∧ (𝑏 ∈ Fin ∧ (𝑁‘𝑏) = 𝐵))) | |
31 | 29, 30 | bitri 263 | . . . 4 ⊢ ((𝑏 ∈ (𝒫 (Base‘𝑊) ∩ Fin) ∧ (𝑁‘𝑏) = (Base‘𝑊)) ↔ (𝑏 ∈ 𝒫 𝐵 ∧ (𝑏 ∈ Fin ∧ (𝑁‘𝑏) = 𝐵))) |
32 | 31 | rexbii2 3021 | . . 3 ⊢ (∃𝑏 ∈ (𝒫 (Base‘𝑊) ∩ Fin)(𝑁‘𝑏) = (Base‘𝑊) ↔ ∃𝑏 ∈ 𝒫 𝐵(𝑏 ∈ Fin ∧ (𝑁‘𝑏) = 𝐵)) |
33 | 20, 32 | syl6bb 275 | . 2 ⊢ (𝑊 ∈ LMod → ((Base‘𝑊) ∈ (𝑁 “ (𝒫 (Base‘𝑊) ∩ Fin)) ↔ ∃𝑏 ∈ 𝒫 𝐵(𝑏 ∈ Fin ∧ (𝑁‘𝑏) = 𝐵))) |
34 | 12, 33 | bitrd 267 | 1 ⊢ (𝑊 ∈ LMod → (𝑊 ∈ LFinGen ↔ ∃𝑏 ∈ 𝒫 𝐵(𝑏 ∈ Fin ∧ (𝑁‘𝑏) = 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 {crab 2900 ∩ cin 3539 ⊆ wss 3540 𝒫 cpw 4108 “ cima 5041 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 Fincfn 7841 Basecbs 15695 LModclmod 18686 LSubSpclss 18753 LSpanclspn 18792 LFinGenclfig 36655 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mgp 18313 df-ur 18325 df-ring 18372 df-lmod 18688 df-lss 18754 df-lsp 18793 df-lfig 36656 |
This theorem is referenced by: islssfg 36658 lnrfg 36708 |
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