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Mirrors > Home > MPE Home > Th. List > islindf3 | Structured version Visualization version GIF version |
Description: In a nonzero ring, independent families can be equivalently characterized as renamings of independent sets. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
Ref | Expression |
---|---|
islindf3.l | β’ πΏ = (Scalarβπ) |
Ref | Expression |
---|---|
islindf3 | β’ ((π β LMod β§ πΏ β NzRing) β (πΉ LIndF π β (πΉ:dom πΉβ1-1βV β§ ran πΉ β (LIndSβπ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2732 | . . . . . 6 β’ (Baseβπ) = (Baseβπ) | |
2 | islindf3.l | . . . . . 6 β’ πΏ = (Scalarβπ) | |
3 | 1, 2 | lindff1 21366 | . . . . 5 β’ ((π β LMod β§ πΏ β NzRing β§ πΉ LIndF π) β πΉ:dom πΉβ1-1β(Baseβπ)) |
4 | 3 | 3expa 1118 | . . . 4 β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ LIndF π) β πΉ:dom πΉβ1-1β(Baseβπ)) |
5 | ssv 4005 | . . . 4 β’ (Baseβπ) β V | |
6 | f1ss 6790 | . . . 4 β’ ((πΉ:dom πΉβ1-1β(Baseβπ) β§ (Baseβπ) β V) β πΉ:dom πΉβ1-1βV) | |
7 | 4, 5, 6 | sylancl 586 | . . 3 β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ LIndF π) β πΉ:dom πΉβ1-1βV) |
8 | lindfrn 21367 | . . . 4 β’ ((π β LMod β§ πΉ LIndF π) β ran πΉ β (LIndSβπ)) | |
9 | 8 | adantlr 713 | . . 3 β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ LIndF π) β ran πΉ β (LIndSβπ)) |
10 | 7, 9 | jca 512 | . 2 β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ LIndF π) β (πΉ:dom πΉβ1-1βV β§ ran πΉ β (LIndSβπ))) |
11 | simpll 765 | . . 3 β’ (((π β LMod β§ πΏ β NzRing) β§ (πΉ:dom πΉβ1-1βV β§ ran πΉ β (LIndSβπ))) β π β LMod) | |
12 | simprr 771 | . . 3 β’ (((π β LMod β§ πΏ β NzRing) β§ (πΉ:dom πΉβ1-1βV β§ ran πΉ β (LIndSβπ))) β ran πΉ β (LIndSβπ)) | |
13 | f1f1orn 6841 | . . . . 5 β’ (πΉ:dom πΉβ1-1βV β πΉ:dom πΉβ1-1-ontoβran πΉ) | |
14 | f1of1 6829 | . . . . 5 β’ (πΉ:dom πΉβ1-1-ontoβran πΉ β πΉ:dom πΉβ1-1βran πΉ) | |
15 | 13, 14 | syl 17 | . . . 4 β’ (πΉ:dom πΉβ1-1βV β πΉ:dom πΉβ1-1βran πΉ) |
16 | 15 | ad2antrl 726 | . . 3 β’ (((π β LMod β§ πΏ β NzRing) β§ (πΉ:dom πΉβ1-1βV β§ ran πΉ β (LIndSβπ))) β πΉ:dom πΉβ1-1βran πΉ) |
17 | f1linds 21371 | . . 3 β’ ((π β LMod β§ ran πΉ β (LIndSβπ) β§ πΉ:dom πΉβ1-1βran πΉ) β πΉ LIndF π) | |
18 | 11, 12, 16, 17 | syl3anc 1371 | . 2 β’ (((π β LMod β§ πΏ β NzRing) β§ (πΉ:dom πΉβ1-1βV β§ ran πΉ β (LIndSβπ))) β πΉ LIndF π) |
19 | 10, 18 | impbida 799 | 1 β’ ((π β LMod β§ πΏ β NzRing) β (πΉ LIndF π β (πΉ:dom πΉβ1-1βV β§ ran πΉ β (LIndSβπ)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 396 = wceq 1541 β wcel 2106 Vcvv 3474 β wss 3947 class class class wbr 5147 dom cdm 5675 ran crn 5676 β1-1βwf1 6537 β1-1-ontoβwf1o 6539 βcfv 6540 Basecbs 17140 Scalarcsca 17196 NzRingcnzr 20283 LModclmod 20463 LIndF clindf 21350 LIndSclinds 21351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-mgp 19982 df-ur 19999 df-ring 20051 df-nzr 20284 df-lmod 20465 df-lss 20535 df-lsp 20575 df-lindf 21352 df-linds 21353 |
This theorem is referenced by: lindflbs 32483 aacllem 47801 |
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