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Mirrors > Home > MPE Home > Th. List > lindsind2 | Structured version Visualization version GIF version |
Description: In a linearly independent set in a module over a nonzero ring, no element is contained in the span of any non-containing set. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
Ref | Expression |
---|---|
lindfind2.k | β’ πΎ = (LSpanβπ) |
lindfind2.l | β’ πΏ = (Scalarβπ) |
Ref | Expression |
---|---|
lindsind2 | β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ β (LIndSβπ) β§ πΈ β πΉ) β Β¬ πΈ β (πΎβ(πΉ β {πΈ}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1133 | . . 3 β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ β (LIndSβπ) β§ πΈ β πΉ) β (π β LMod β§ πΏ β NzRing)) | |
2 | linds2 21749 | . . . 4 β’ (πΉ β (LIndSβπ) β ( I βΎ πΉ) LIndF π) | |
3 | 2 | 3ad2ant2 1131 | . . 3 β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ β (LIndSβπ) β§ πΈ β πΉ) β ( I βΎ πΉ) LIndF π) |
4 | dmresi 6050 | . . . . . 6 β’ dom ( I βΎ πΉ) = πΉ | |
5 | 4 | eleq2i 2817 | . . . . 5 β’ (πΈ β dom ( I βΎ πΉ) β πΈ β πΉ) |
6 | 5 | biimpri 227 | . . . 4 β’ (πΈ β πΉ β πΈ β dom ( I βΎ πΉ)) |
7 | 6 | 3ad2ant3 1132 | . . 3 β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ β (LIndSβπ) β§ πΈ β πΉ) β πΈ β dom ( I βΎ πΉ)) |
8 | lindfind2.k | . . . 4 β’ πΎ = (LSpanβπ) | |
9 | lindfind2.l | . . . 4 β’ πΏ = (Scalarβπ) | |
10 | 8, 9 | lindfind2 21756 | . . 3 β’ (((π β LMod β§ πΏ β NzRing) β§ ( I βΎ πΉ) LIndF π β§ πΈ β dom ( I βΎ πΉ)) β Β¬ (( I βΎ πΉ)βπΈ) β (πΎβ(( I βΎ πΉ) β (dom ( I βΎ πΉ) β {πΈ})))) |
11 | 1, 3, 7, 10 | syl3anc 1368 | . 2 β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ β (LIndSβπ) β§ πΈ β πΉ) β Β¬ (( I βΎ πΉ)βπΈ) β (πΎβ(( I βΎ πΉ) β (dom ( I βΎ πΉ) β {πΈ})))) |
12 | fvresi 7178 | . . . 4 β’ (πΈ β πΉ β (( I βΎ πΉ)βπΈ) = πΈ) | |
13 | 4 | difeq1i 4110 | . . . . . . . 8 β’ (dom ( I βΎ πΉ) β {πΈ}) = (πΉ β {πΈ}) |
14 | 13 | imaeq2i 6056 | . . . . . . 7 β’ (( I βΎ πΉ) β (dom ( I βΎ πΉ) β {πΈ})) = (( I βΎ πΉ) β (πΉ β {πΈ})) |
15 | difss 4124 | . . . . . . . 8 β’ (πΉ β {πΈ}) β πΉ | |
16 | resiima 6074 | . . . . . . . 8 β’ ((πΉ β {πΈ}) β πΉ β (( I βΎ πΉ) β (πΉ β {πΈ})) = (πΉ β {πΈ})) | |
17 | 15, 16 | ax-mp 5 | . . . . . . 7 β’ (( I βΎ πΉ) β (πΉ β {πΈ})) = (πΉ β {πΈ}) |
18 | 14, 17 | eqtri 2753 | . . . . . 6 β’ (( I βΎ πΉ) β (dom ( I βΎ πΉ) β {πΈ})) = (πΉ β {πΈ}) |
19 | 18 | fveq2i 6895 | . . . . 5 β’ (πΎβ(( I βΎ πΉ) β (dom ( I βΎ πΉ) β {πΈ}))) = (πΎβ(πΉ β {πΈ})) |
20 | 19 | a1i 11 | . . . 4 β’ (πΈ β πΉ β (πΎβ(( I βΎ πΉ) β (dom ( I βΎ πΉ) β {πΈ}))) = (πΎβ(πΉ β {πΈ}))) |
21 | 12, 20 | eleq12d 2819 | . . 3 β’ (πΈ β πΉ β ((( I βΎ πΉ)βπΈ) β (πΎβ(( I βΎ πΉ) β (dom ( I βΎ πΉ) β {πΈ}))) β πΈ β (πΎβ(πΉ β {πΈ})))) |
22 | 21 | 3ad2ant3 1132 | . 2 β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ β (LIndSβπ) β§ πΈ β πΉ) β ((( I βΎ πΉ)βπΈ) β (πΎβ(( I βΎ πΉ) β (dom ( I βΎ πΉ) β {πΈ}))) β πΈ β (πΎβ(πΉ β {πΈ})))) |
23 | 11, 22 | mtbid 323 | 1 β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ β (LIndSβπ) β§ πΈ β πΉ) β Β¬ πΈ β (πΎβ(πΉ β {πΈ}))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β cdif 3936 β wss 3939 {csn 4624 class class class wbr 5143 I cid 5569 dom cdm 5672 βΎ cres 5674 β cima 5675 βcfv 6543 Scalarcsca 17235 NzRingcnzr 20455 LModclmod 20747 LSpanclspn 20859 LIndF clindf 21742 LIndSclinds 21743 |
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 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mgp 20079 df-ur 20126 df-ring 20179 df-nzr 20456 df-lmod 20749 df-lindf 21744 df-linds 21745 |
This theorem is referenced by: islinds4 21773 lindsadd 37143 lindsdom 37144 lindsenlbs 37145 aacllem 48346 |
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