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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 | ⊢ 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 1190 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) → 𝑊 ∈ LMod) | |
2 | simp2 1130 | . . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) → 𝐹 LIndF 𝑊) | |
3 | eqid 2794 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | 3 | lindff 20641 | . . . . 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 1131 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) → 𝐸 ∈ dom 𝐹) | |
7 | 5, 6 | ffvelrnd 6720 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) → (𝐹‘𝐸) ∈ (Base‘𝑊)) |
8 | lindfind2.l | . . . 4 ⊢ 𝐿 = (Scalar‘𝑊) | |
9 | eqid 2794 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
10 | eqid 2794 | . . . 4 ⊢ (1r‘𝐿) = (1r‘𝐿) | |
11 | 3, 8, 9, 10 | lmodvs1 19352 | . . 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 19723 | . . . . . 6 ⊢ (𝐿 ∈ NzRing → 𝐿 ∈ Ring) | |
14 | eqid 2794 | . . . . . . 7 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
15 | 14, 10 | ringidcl 19008 | . . . . . 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 1126 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) → (1r‘𝐿) ∈ (Base‘𝐿)) |
19 | eqid 2794 | . . . . . 6 ⊢ (0g‘𝐿) = (0g‘𝐿) | |
20 | 10, 19 | nzrnz 19722 | . . . . 5 ⊢ (𝐿 ∈ NzRing → (1r‘𝐿) ≠ (0g‘𝐿)) |
21 | 20 | adantl 482 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) → (1r‘𝐿) ≠ (0g‘𝐿)) |
22 | 21 | 3ad2ant1 1126 | . . 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 20642 | . . 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 835 | . 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 2902 | 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 1080 = wceq 1522 ∈ wcel 2080 ≠ wne 2983 ∖ cdif 3858 {csn 4474 class class class wbr 4964 dom cdm 5446 “ cima 5449 ⟶wf 6224 ‘cfv 6228 (class class class)co 7019 Basecbs 16312 Scalarcsca 16397 ·𝑠 cvsca 16398 0gc0g 16542 1rcur 18941 Ringcrg 18987 LModclmod 19324 LSpanclspn 19433 NzRingcnzr 19719 LIndF clindf 20630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-cnex 10442 ax-resscn 10443 ax-1cn 10444 ax-icn 10445 ax-addcl 10446 ax-addrcl 10447 ax-mulcl 10448 ax-mulrcl 10449 ax-mulcom 10450 ax-addass 10451 ax-mulass 10452 ax-distr 10453 ax-i2m1 10454 ax-1ne0 10455 ax-1rid 10456 ax-rnegex 10457 ax-rrecex 10458 ax-cnre 10459 ax-pre-lttri 10460 ax-pre-lttrn 10461 ax-pre-ltadd 10462 ax-pre-mulgt0 10463 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-nel 3090 df-ral 3109 df-rex 3110 df-reu 3111 df-rmo 3112 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-iun 4829 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-pred 6026 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-riota 6980 df-ov 7022 df-oprab 7023 df-mpo 7024 df-om 7440 df-wrecs 7801 df-recs 7863 df-rdg 7901 df-er 8142 df-en 8361 df-dom 8362 df-sdom 8363 df-pnf 10526 df-mnf 10527 df-xr 10528 df-ltxr 10529 df-le 10530 df-sub 10721 df-neg 10722 df-nn 11489 df-2 11550 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-plusg 16407 df-0g 16544 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-mgp 18930 df-ur 18942 df-ring 18989 df-lmod 19326 df-nzr 19720 df-lindf 20632 |
This theorem is referenced by: lindsind2 20645 lindff1 20646 |
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