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Mirrors > Home > MPE Home > Th. List > lmisfree | Structured version Visualization version GIF version |
Description: A module has a basis iff it is isomorphic to a free module. In settings where isomorphic objects are not distinguished, it is common to define "free module" as any module with a basis; thus for instance lbsex 19213 might be described as "every vector space is free." (Contributed by Stefan O'Rear, 26-Feb-2015.) |
Ref | Expression |
---|---|
lmisfree.j | ⊢ 𝐽 = (LBasis‘𝑊) |
lmisfree.f | ⊢ 𝐹 = (Scalar‘𝑊) |
Ref | Expression |
---|---|
lmisfree | ⊢ (𝑊 ∈ LMod → (𝐽 ≠ ∅ ↔ ∃𝑘 𝑊 ≃𝑚 (𝐹 freeLMod 𝑘))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 3964 | . . 3 ⊢ (𝐽 ≠ ∅ ↔ ∃𝑗 𝑗 ∈ 𝐽) | |
2 | vex 3234 | . . . . . . . 8 ⊢ 𝑗 ∈ V | |
3 | 2 | enref 8030 | . . . . . . 7 ⊢ 𝑗 ≈ 𝑗 |
4 | lmisfree.f | . . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊) | |
5 | lmisfree.j | . . . . . . . 8 ⊢ 𝐽 = (LBasis‘𝑊) | |
6 | 4, 5 | lbslcic 20228 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑗 ∈ 𝐽 ∧ 𝑗 ≈ 𝑗) → 𝑊 ≃𝑚 (𝐹 freeLMod 𝑗)) |
7 | 3, 6 | mp3an3 1453 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑗 ∈ 𝐽) → 𝑊 ≃𝑚 (𝐹 freeLMod 𝑗)) |
8 | oveq2 6698 | . . . . . . . 8 ⊢ (𝑘 = 𝑗 → (𝐹 freeLMod 𝑘) = (𝐹 freeLMod 𝑗)) | |
9 | 8 | breq2d 4697 | . . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝑊 ≃𝑚 (𝐹 freeLMod 𝑘) ↔ 𝑊 ≃𝑚 (𝐹 freeLMod 𝑗))) |
10 | 2, 9 | spcev 3331 | . . . . . 6 ⊢ (𝑊 ≃𝑚 (𝐹 freeLMod 𝑗) → ∃𝑘 𝑊 ≃𝑚 (𝐹 freeLMod 𝑘)) |
11 | 7, 10 | syl 17 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑗 ∈ 𝐽) → ∃𝑘 𝑊 ≃𝑚 (𝐹 freeLMod 𝑘)) |
12 | 11 | ex 449 | . . . 4 ⊢ (𝑊 ∈ LMod → (𝑗 ∈ 𝐽 → ∃𝑘 𝑊 ≃𝑚 (𝐹 freeLMod 𝑘))) |
13 | 12 | exlimdv 1901 | . . 3 ⊢ (𝑊 ∈ LMod → (∃𝑗 𝑗 ∈ 𝐽 → ∃𝑘 𝑊 ≃𝑚 (𝐹 freeLMod 𝑘))) |
14 | 1, 13 | syl5bi 232 | . 2 ⊢ (𝑊 ∈ LMod → (𝐽 ≠ ∅ → ∃𝑘 𝑊 ≃𝑚 (𝐹 freeLMod 𝑘))) |
15 | lmicsym 19120 | . . . 4 ⊢ (𝑊 ≃𝑚 (𝐹 freeLMod 𝑘) → (𝐹 freeLMod 𝑘) ≃𝑚 𝑊) | |
16 | lmiclcl 19118 | . . . . 5 ⊢ (𝑊 ≃𝑚 (𝐹 freeLMod 𝑘) → 𝑊 ∈ LMod) | |
17 | 4 | lmodring 18919 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
18 | vex 3234 | . . . . . . 7 ⊢ 𝑘 ∈ V | |
19 | eqid 2651 | . . . . . . . 8 ⊢ (𝐹 freeLMod 𝑘) = (𝐹 freeLMod 𝑘) | |
20 | eqid 2651 | . . . . . . . 8 ⊢ (𝐹 unitVec 𝑘) = (𝐹 unitVec 𝑘) | |
21 | eqid 2651 | . . . . . . . 8 ⊢ (LBasis‘(𝐹 freeLMod 𝑘)) = (LBasis‘(𝐹 freeLMod 𝑘)) | |
22 | 19, 20, 21 | frlmlbs 20184 | . . . . . . 7 ⊢ ((𝐹 ∈ Ring ∧ 𝑘 ∈ V) → ran (𝐹 unitVec 𝑘) ∈ (LBasis‘(𝐹 freeLMod 𝑘))) |
23 | 17, 18, 22 | sylancl 695 | . . . . . 6 ⊢ (𝑊 ∈ LMod → ran (𝐹 unitVec 𝑘) ∈ (LBasis‘(𝐹 freeLMod 𝑘))) |
24 | ne0i 3954 | . . . . . 6 ⊢ (ran (𝐹 unitVec 𝑘) ∈ (LBasis‘(𝐹 freeLMod 𝑘)) → (LBasis‘(𝐹 freeLMod 𝑘)) ≠ ∅) | |
25 | 23, 24 | syl 17 | . . . . 5 ⊢ (𝑊 ∈ LMod → (LBasis‘(𝐹 freeLMod 𝑘)) ≠ ∅) |
26 | 16, 25 | syl 17 | . . . 4 ⊢ (𝑊 ≃𝑚 (𝐹 freeLMod 𝑘) → (LBasis‘(𝐹 freeLMod 𝑘)) ≠ ∅) |
27 | 21, 5 | lmiclbs 20224 | . . . 4 ⊢ ((𝐹 freeLMod 𝑘) ≃𝑚 𝑊 → ((LBasis‘(𝐹 freeLMod 𝑘)) ≠ ∅ → 𝐽 ≠ ∅)) |
28 | 15, 26, 27 | sylc 65 | . . 3 ⊢ (𝑊 ≃𝑚 (𝐹 freeLMod 𝑘) → 𝐽 ≠ ∅) |
29 | 28 | exlimiv 1898 | . 2 ⊢ (∃𝑘 𝑊 ≃𝑚 (𝐹 freeLMod 𝑘) → 𝐽 ≠ ∅) |
30 | 14, 29 | impbid1 215 | 1 ⊢ (𝑊 ∈ LMod → (𝐽 ≠ ∅ ↔ ∃𝑘 𝑊 ≃𝑚 (𝐹 freeLMod 𝑘))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∃wex 1744 ∈ wcel 2030 ≠ wne 2823 Vcvv 3231 ∅c0 3948 class class class wbr 4685 ran crn 5144 ‘cfv 5926 (class class class)co 6690 ≈ cen 7994 Scalarcsca 15991 Ringcrg 18593 LModclmod 18911 ≃𝑚 clmic 19069 LBasisclbs 19122 freeLMod cfrlm 20138 unitVec cuvc 20169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-sup 8389 df-oi 8456 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-fz 12365 df-fzo 12505 df-seq 12842 df-hash 13158 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-hom 16013 df-cco 16014 df-0g 16149 df-gsum 16150 df-prds 16155 df-pws 16157 df-mre 16293 df-mrc 16294 df-acs 16296 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-mhm 17382 df-submnd 17383 df-grp 17472 df-minusg 17473 df-sbg 17474 df-mulg 17588 df-subg 17638 df-ghm 17705 df-cntz 17796 df-cmn 18241 df-abl 18242 df-mgp 18536 df-ur 18548 df-ring 18595 df-subrg 18826 df-lmod 18913 df-lss 18981 df-lsp 19020 df-lmhm 19070 df-lmim 19071 df-lmic 19072 df-lbs 19123 df-sra 19220 df-rgmod 19221 df-nzr 19306 df-dsmm 20124 df-frlm 20139 df-uvc 20170 df-lindf 20193 df-linds 20194 |
This theorem is referenced by: lvecisfrlm 20230 |
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