Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 | ⊢ 𝐾 = (LSpan‘𝑊) |
lindfind2.l | ⊢ 𝐿 = (Scalar‘𝑊) |
Ref | Expression |
---|---|
lindfind2 | ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) → ¬ (𝐹‘𝐸) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1l 1196 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) → 𝑊 ∈ LMod) | |
2 | simp2 1136 | . . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) → 𝐹 LIndF 𝑊) | |
3 | eqid 2738 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | 3 | lindff 21022 | . . . . 5 ⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ LMod) → 𝐹:dom 𝐹⟶(Base‘𝑊)) |
5 | 2, 1, 4 | syl2anc 584 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) → 𝐹:dom 𝐹⟶(Base‘𝑊)) |
6 | simp3 1137 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) → 𝐸 ∈ dom 𝐹) | |
7 | 5, 6 | ffvelrnd 6962 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) → (𝐹‘𝐸) ∈ (Base‘𝑊)) |
8 | lindfind2.l | . . . 4 ⊢ 𝐿 = (Scalar‘𝑊) | |
9 | eqid 2738 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
10 | eqid 2738 | . . . 4 ⊢ (1r‘𝐿) = (1r‘𝐿) | |
11 | 3, 8, 9, 10 | lmodvs1 20151 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝐹‘𝐸) ∈ (Base‘𝑊)) → ((1r‘𝐿)( ·𝑠 ‘𝑊)(𝐹‘𝐸)) = (𝐹‘𝐸)) |
12 | 1, 7, 11 | syl2anc 584 | . 2 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) → ((1r‘𝐿)( ·𝑠 ‘𝑊)(𝐹‘𝐸)) = (𝐹‘𝐸)) |
13 | nzrring 20532 | . . . . . 6 ⊢ (𝐿 ∈ NzRing → 𝐿 ∈ Ring) | |
14 | eqid 2738 | . . . . . . 7 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
15 | 14, 10 | ringidcl 19807 | . . . . . 6 ⊢ (𝐿 ∈ Ring → (1r‘𝐿) ∈ (Base‘𝐿)) |
16 | 13, 15 | syl 17 | . . . . 5 ⊢ (𝐿 ∈ NzRing → (1r‘𝐿) ∈ (Base‘𝐿)) |
17 | 16 | adantl 482 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) → (1r‘𝐿) ∈ (Base‘𝐿)) |
18 | 17 | 3ad2ant1 1132 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) → (1r‘𝐿) ∈ (Base‘𝐿)) |
19 | eqid 2738 | . . . . . 6 ⊢ (0g‘𝐿) = (0g‘𝐿) | |
20 | 10, 19 | nzrnz 20531 | . . . . 5 ⊢ (𝐿 ∈ NzRing → (1r‘𝐿) ≠ (0g‘𝐿)) |
21 | 20 | adantl 482 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) → (1r‘𝐿) ≠ (0g‘𝐿)) |
22 | 21 | 3ad2ant1 1132 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) → (1r‘𝐿) ≠ (0g‘𝐿)) |
23 | lindfind2.k | . . . 4 ⊢ 𝐾 = (LSpan‘𝑊) | |
24 | 9, 23, 8, 19, 14 | lindfind 21023 | . . 3 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ ((1r‘𝐿) ∈ (Base‘𝐿) ∧ (1r‘𝐿) ≠ (0g‘𝐿))) → ¬ ((1r‘𝐿)( ·𝑠 ‘𝑊)(𝐹‘𝐸)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))) |
25 | 2, 6, 18, 22, 24 | syl22anc 836 | . 2 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) → ¬ ((1r‘𝐿)( ·𝑠 ‘𝑊)(𝐹‘𝐸)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))) |
26 | 12, 25 | eqneltrrd 2859 | 1 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) → ¬ (𝐹‘𝐸) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∖ cdif 3884 {csn 4561 class class class wbr 5074 dom cdm 5589 “ cima 5592 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 Scalarcsca 16965 ·𝑠 cvsca 16966 0gc0g 17150 1rcur 19737 Ringcrg 19783 LModclmod 20123 LSpanclspn 20233 NzRingcnzr 20528 LIndF clindf 21011 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-plusg 16975 df-0g 17152 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mgp 19721 df-ur 19738 df-ring 19785 df-lmod 20125 df-nzr 20529 df-lindf 21013 |
This theorem is referenced by: lindsind2 21026 lindff1 21027 |
Copyright terms: Public domain | W3C validator |