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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 1137 | . . 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 21233 | . . . 4 β’ (πΉ β (LIndSβπ) β ( I βΎ πΉ) LIndF π) | |
3 | 2 | 3ad2ant2 1135 | . . 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 6006 | . . . . . 6 β’ dom ( I βΎ πΉ) = πΉ | |
5 | 4 | eleq2i 2826 | . . . . 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 1136 | . . 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 21240 | . . 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 1372 | . 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 7120 | . . . 4 β’ (πΈ β πΉ β (( I βΎ πΉ)βπΈ) = πΈ) | |
13 | 4 | difeq1i 4079 | . . . . . . . 8 β’ (dom ( I βΎ πΉ) β {πΈ}) = (πΉ β {πΈ}) |
14 | 13 | imaeq2i 6012 | . . . . . . 7 β’ (( I βΎ πΉ) β (dom ( I βΎ πΉ) β {πΈ})) = (( I βΎ πΉ) β (πΉ β {πΈ})) |
15 | difss 4092 | . . . . . . . 8 β’ (πΉ β {πΈ}) β πΉ | |
16 | resiima 6029 | . . . . . . . 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 2761 | . . . . . 6 β’ (( I βΎ πΉ) β (dom ( I βΎ πΉ) β {πΈ})) = (πΉ β {πΈ}) |
19 | 18 | fveq2i 6846 | . . . . 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 2828 | . . 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 1136 | . 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 324 | 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β cdif 3908 β wss 3911 {csn 4587 class class class wbr 5106 I cid 5531 dom cdm 5634 βΎ cres 5636 β cima 5637 βcfv 6497 Scalarcsca 17141 LModclmod 20336 LSpanclspn 20447 NzRingcnzr 20743 LIndF clindf 21226 LIndSclinds 21227 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-mgp 19902 df-ur 19919 df-ring 19971 df-lmod 20338 df-nzr 20744 df-lindf 21228 df-linds 21229 |
This theorem is referenced by: islinds4 21257 lindsadd 36117 lindsdom 36118 lindsenlbs 36119 aacllem 47334 |
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