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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 | ⊢ 𝐾 = (LSpan‘𝑊) |
lindfind2.l | ⊢ 𝐿 = (Scalar‘𝑊) |
Ref | Expression |
---|---|
lindsind2 | ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) → ¬ 𝐸 ∈ (𝐾‘(𝐹 ∖ {𝐸}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1133 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) → (𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing)) | |
2 | linds2 20500 | . . . 4 ⊢ (𝐹 ∈ (LIndS‘𝑊) → ( I ↾ 𝐹) LIndF 𝑊) | |
3 | 2 | 3ad2ant2 1131 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) → ( I ↾ 𝐹) LIndF 𝑊) |
4 | dmresi 5888 | . . . . . 6 ⊢ dom ( I ↾ 𝐹) = 𝐹 | |
5 | 4 | eleq2i 2881 | . . . . 5 ⊢ (𝐸 ∈ dom ( I ↾ 𝐹) ↔ 𝐸 ∈ 𝐹) |
6 | 5 | biimpri 231 | . . . 4 ⊢ (𝐸 ∈ 𝐹 → 𝐸 ∈ dom ( I ↾ 𝐹)) |
7 | 6 | 3ad2ant3 1132 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) → 𝐸 ∈ dom ( I ↾ 𝐹)) |
8 | lindfind2.k | . . . 4 ⊢ 𝐾 = (LSpan‘𝑊) | |
9 | lindfind2.l | . . . 4 ⊢ 𝐿 = (Scalar‘𝑊) | |
10 | 8, 9 | lindfind2 20507 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ ( I ↾ 𝐹) LIndF 𝑊 ∧ 𝐸 ∈ dom ( I ↾ 𝐹)) → ¬ (( I ↾ 𝐹)‘𝐸) ∈ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝐸})))) |
11 | 1, 3, 7, 10 | syl3anc 1368 | . 2 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) → ¬ (( I ↾ 𝐹)‘𝐸) ∈ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝐸})))) |
12 | fvresi 6912 | . . . 4 ⊢ (𝐸 ∈ 𝐹 → (( I ↾ 𝐹)‘𝐸) = 𝐸) | |
13 | 4 | difeq1i 4046 | . . . . . . . 8 ⊢ (dom ( I ↾ 𝐹) ∖ {𝐸}) = (𝐹 ∖ {𝐸}) |
14 | 13 | imaeq2i 5894 | . . . . . . 7 ⊢ (( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝐸})) = (( I ↾ 𝐹) “ (𝐹 ∖ {𝐸})) |
15 | difss 4059 | . . . . . . . 8 ⊢ (𝐹 ∖ {𝐸}) ⊆ 𝐹 | |
16 | resiima 5911 | . . . . . . . 8 ⊢ ((𝐹 ∖ {𝐸}) ⊆ 𝐹 → (( I ↾ 𝐹) “ (𝐹 ∖ {𝐸})) = (𝐹 ∖ {𝐸})) | |
17 | 15, 16 | ax-mp 5 | . . . . . . 7 ⊢ (( I ↾ 𝐹) “ (𝐹 ∖ {𝐸})) = (𝐹 ∖ {𝐸}) |
18 | 14, 17 | eqtri 2821 | . . . . . 6 ⊢ (( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝐸})) = (𝐹 ∖ {𝐸}) |
19 | 18 | fveq2i 6648 | . . . . 5 ⊢ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝐸}))) = (𝐾‘(𝐹 ∖ {𝐸})) |
20 | 19 | a1i 11 | . . . 4 ⊢ (𝐸 ∈ 𝐹 → (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝐸}))) = (𝐾‘(𝐹 ∖ {𝐸}))) |
21 | 12, 20 | eleq12d 2884 | . . 3 ⊢ (𝐸 ∈ 𝐹 → ((( I ↾ 𝐹)‘𝐸) ∈ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝐸}))) ↔ 𝐸 ∈ (𝐾‘(𝐹 ∖ {𝐸})))) |
22 | 21 | 3ad2ant3 1132 | . 2 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) → ((( I ↾ 𝐹)‘𝐸) ∈ (𝐾‘(( I ↾ 𝐹) “ (dom ( I ↾ 𝐹) ∖ {𝐸}))) ↔ 𝐸 ∈ (𝐾‘(𝐹 ∖ {𝐸})))) |
23 | 11, 22 | mtbid 327 | 1 ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) → ¬ 𝐸 ∈ (𝐾‘(𝐹 ∖ {𝐸}))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 ⊆ wss 3881 {csn 4525 class class class wbr 5030 I cid 5424 dom cdm 5519 ↾ cres 5521 “ cima 5522 ‘cfv 6324 Scalarcsca 16560 LModclmod 19627 LSpanclspn 19736 NzRingcnzr 20023 LIndF clindf 20493 LIndSclinds 20494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mgp 19233 df-ur 19245 df-ring 19292 df-lmod 19629 df-nzr 20024 df-lindf 20495 df-linds 20496 |
This theorem is referenced by: islinds4 20524 lindsadd 35050 lindsdom 35051 lindsenlbs 35052 aacllem 45329 |
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