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| Mirrors > Home > MPE Home > Th. List > lindfind2 | Structured version Visualization version GIF version | ||
| Description: In a linearly independent family 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 |
|---|---|
| lindfind2 | β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ LIndF π β§ πΈ β dom πΉ) β Β¬ (πΉβπΈ) β (πΎβ(πΉ β (dom πΉ β {πΈ})))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp1l 1196 | . . 3 β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ LIndF π β§ πΈ β dom πΉ) β π β LMod) | |
| 2 | simp2 1136 | . . . . 5 β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ LIndF π β§ πΈ β dom πΉ) β πΉ LIndF π) | |
| 3 | eqid 2731 | . . . . . 6 β’ (Baseβπ) = (Baseβπ) | |
| 4 | 3 | lindff 21590 | . . . . 5 β’ ((πΉ LIndF π β§ π β LMod) β πΉ:dom πΉβΆ(Baseβπ)) |
| 5 | 2, 1, 4 | syl2anc 583 | . . . 4 β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ LIndF π β§ πΈ β dom πΉ) β πΉ:dom πΉβΆ(Baseβπ)) |
| 6 | simp3 1137 | . . . 4 β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ LIndF π β§ πΈ β dom πΉ) β πΈ β dom πΉ) | |
| 7 | 5, 6 | ffvelcdmd 7088 | . . 3 β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ LIndF π β§ πΈ β dom πΉ) β (πΉβπΈ) β (Baseβπ)) |
| 8 | lindfind2.l | . . . 4 β’ πΏ = (Scalarβπ) | |
| 9 | eqid 2731 | . . . 4 β’ ( Β·π βπ) = ( Β·π βπ) | |
| 10 | eqid 2731 | . . . 4 β’ (1rβπΏ) = (1rβπΏ) | |
| 11 | 3, 8, 9, 10 | lmodvs1 20645 | . . 3 β’ ((π β LMod β§ (πΉβπΈ) β (Baseβπ)) β ((1rβπΏ)( Β·π βπ)(πΉβπΈ)) = (πΉβπΈ)) |
| 12 | 1, 7, 11 | syl2anc 583 | . 2 β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ LIndF π β§ πΈ β dom πΉ) β ((1rβπΏ)( Β·π βπ)(πΉβπΈ)) = (πΉβπΈ)) |
| 13 | nzrring 20408 | . . . . . 6 β’ (πΏ β NzRing β πΏ β Ring) | |
| 14 | eqid 2731 | . . . . . . 7 β’ (BaseβπΏ) = (BaseβπΏ) | |
| 15 | 14, 10 | ringidcl 20155 | . . . . . 6 β’ (πΏ β Ring β (1rβπΏ) β (BaseβπΏ)) |
| 16 | 13, 15 | syl 17 | . . . . 5 β’ (πΏ β NzRing β (1rβπΏ) β (BaseβπΏ)) |
| 17 | 16 | adantl 481 | . . . 4 β’ ((π β LMod β§ πΏ β NzRing) β (1rβπΏ) β (BaseβπΏ)) |
| 18 | 17 | 3ad2ant1 1132 | . . 3 β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ LIndF π β§ πΈ β dom πΉ) β (1rβπΏ) β (BaseβπΏ)) |
| 19 | eqid 2731 | . . . . . 6 β’ (0gβπΏ) = (0gβπΏ) | |
| 20 | 10, 19 | nzrnz 20407 | . . . . 5 β’ (πΏ β NzRing β (1rβπΏ) β (0gβπΏ)) |
| 21 | 20 | adantl 481 | . . . 4 β’ ((π β LMod β§ πΏ β NzRing) β (1rβπΏ) β (0gβπΏ)) |
| 22 | 21 | 3ad2ant1 1132 | . . 3 β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ LIndF π β§ πΈ β dom πΉ) β (1rβπΏ) β (0gβπΏ)) |
| 23 | lindfind2.k | . . . 4 β’ πΎ = (LSpanβπ) | |
| 24 | 9, 23, 8, 19, 14 | lindfind 21591 | . . 3 β’ (((πΉ LIndF π β§ πΈ β dom πΉ) β§ ((1rβπΏ) β (BaseβπΏ) β§ (1rβπΏ) β (0gβπΏ))) β Β¬ ((1rβπΏ)( Β·π βπ)(πΉβπΈ)) β (πΎβ(πΉ β (dom πΉ β {πΈ})))) |
| 25 | 2, 6, 18, 22, 24 | syl22anc 836 | . 2 β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ LIndF π β§ πΈ β dom πΉ) β Β¬ ((1rβπΏ)( Β·π βπ)(πΉβπΈ)) β (πΎβ(πΉ β (dom πΉ β {πΈ})))) |
| 26 | 12, 25 | eqneltrrd 2853 | 1 β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ LIndF π β§ πΈ β dom πΉ) β Β¬ (πΉβπΈ) β (πΎβ(πΉ β (dom πΉ β {πΈ})))) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 β§ w3a 1086 = wceq 1540 β wcel 2105 β wne 2939 β cdif 3946 {csn 4629 class class class wbr 5149 dom cdm 5677 β cima 5680 βΆwf 6540 βcfv 6544 (class class class)co 7412 Basecbs 17149 Scalarcsca 17205 Β·π cvsca 17206 0gc0g 17390 1rcur 20076 Ringcrg 20128 NzRingcnzr 20404 LModclmod 20615 LSpanclspn 20727 LIndF clindf 21579 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 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 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-er 8706 df-en 8943 df-dom 8944 df-sdom 8945 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-0g 17392 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mgp 20030 df-ur 20077 df-ring 20130 df-nzr 20405 df-lmod 20617 df-lindf 21581 |
| This theorem is referenced by: lindsind2 21594 lindff1 21595 |
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