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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 1198 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) → 𝑊 ∈ LMod) | |
| 2 | simp2 1138 | . . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) → 𝐹 LIndF 𝑊) | |
| 3 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 4 | 3 | lindff 21835 | . . . . 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 1139 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) → 𝐸 ∈ dom 𝐹) | |
| 7 | 5, 6 | ffvelcdmd 7105 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) → (𝐹‘𝐸) ∈ (Base‘𝑊)) |
| 8 | lindfind2.l | . . . 4 ⊢ 𝐿 = (Scalar‘𝑊) | |
| 9 | eqid 2737 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
| 10 | eqid 2737 | . . . 4 ⊢ (1r‘𝐿) = (1r‘𝐿) | |
| 11 | 3, 8, 9, 10 | lmodvs1 20888 | . . 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 20516 | . . . . . 6 ⊢ (𝐿 ∈ NzRing → 𝐿 ∈ Ring) | |
| 14 | eqid 2737 | . . . . . . 7 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
| 15 | 14, 10 | ringidcl 20262 | . . . . . 6 ⊢ (𝐿 ∈ Ring → (1r‘𝐿) ∈ (Base‘𝐿)) |
| 16 | 13, 15 | syl 17 | . . . . 5 ⊢ (𝐿 ∈ NzRing → (1r‘𝐿) ∈ (Base‘𝐿)) |
| 17 | 16 | adantl 481 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) → (1r‘𝐿) ∈ (Base‘𝐿)) |
| 18 | 17 | 3ad2ant1 1134 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) → (1r‘𝐿) ∈ (Base‘𝐿)) |
| 19 | eqid 2737 | . . . . . 6 ⊢ (0g‘𝐿) = (0g‘𝐿) | |
| 20 | 10, 19 | nzrnz 20515 | . . . . 5 ⊢ (𝐿 ∈ NzRing → (1r‘𝐿) ≠ (0g‘𝐿)) |
| 21 | 20 | adantl 481 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) → (1r‘𝐿) ≠ (0g‘𝐿)) |
| 22 | 21 | 3ad2ant1 1134 | . . 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 21836 | . . 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 839 | . 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 2862 | 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 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 {csn 4626 class class class wbr 5143 dom cdm 5685 “ cima 5688 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 1rcur 20178 Ringcrg 20230 NzRingcnzr 20512 LModclmod 20858 LSpanclspn 20969 LIndF clindf 21824 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mgp 20138 df-ur 20179 df-ring 20232 df-nzr 20513 df-lmod 20860 df-lindf 21826 |
| This theorem is referenced by: lindsind2 21839 lindff1 21840 |
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