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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 2728 | . . . . . 6 β’ (Baseβπ) = (Baseβπ) | |
2 | islindf3.l | . . . . . 6 β’ πΏ = (Scalarβπ) | |
3 | 1, 2 | lindff1 21768 | . . . . 5 β’ ((π β LMod β§ πΏ β NzRing β§ πΉ LIndF π) β πΉ:dom πΉβ1-1β(Baseβπ)) |
4 | 3 | 3expa 1115 | . . . 4 β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ LIndF π) β πΉ:dom πΉβ1-1β(Baseβπ)) |
5 | ssv 4006 | . . . 4 β’ (Baseβπ) β V | |
6 | f1ss 6804 | . . . 4 β’ ((πΉ:dom πΉβ1-1β(Baseβπ) β§ (Baseβπ) β V) β πΉ:dom πΉβ1-1βV) | |
7 | 4, 5, 6 | sylancl 584 | . . 3 β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ LIndF π) β πΉ:dom πΉβ1-1βV) |
8 | lindfrn 21769 | . . . 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 510 | . 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 6855 | . . . . 5 β’ (πΉ:dom πΉβ1-1βV β πΉ:dom πΉβ1-1-ontoβran πΉ) | |
14 | f1of1 6843 | . . . . 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 21773 | . . 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 1368 | . 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 394 = wceq 1533 β wcel 2098 Vcvv 3473 β wss 3949 class class class wbr 5152 dom cdm 5682 ran crn 5683 β1-1βwf1 6550 β1-1-ontoβwf1o 6552 βcfv 6553 Basecbs 17189 Scalarcsca 17245 NzRingcnzr 20465 LModclmod 20757 LIndF clindf 21752 LIndSclinds 21753 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-plusg 17255 df-0g 17432 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-grp 18907 df-mgp 20089 df-ur 20136 df-ring 20189 df-nzr 20466 df-lmod 20759 df-lss 20830 df-lsp 20870 df-lindf 21754 df-linds 21755 |
This theorem is referenced by: lindflbs 33127 aacllem 48330 |
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