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Mirrors > Home > MPE Home > Th. List > Mathboxes > zlmodzxzldep | Structured version Visualization version GIF version |
Description: { A , B } is a linearly dependent set within the ℤ-module ℤ × ℤ (see example in [Roman] p. 112). (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.) |
Ref | Expression |
---|---|
zlmodzxzldep.z | ⊢ 𝑍 = (ℤring freeLMod {0, 1}) |
zlmodzxzldep.a | ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} |
zlmodzxzldep.b | ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} |
Ref | Expression |
---|---|
zlmodzxzldep | ⊢ {𝐴, 𝐵} linDepS 𝑍 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zlmodzxzldep.z | . . . 4 ⊢ 𝑍 = (ℤring freeLMod {0, 1}) | |
2 | zlmodzxzldep.a | . . . 4 ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} | |
3 | zlmodzxzldep.b | . . . 4 ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} | |
4 | eqid 2731 | . . . 4 ⊢ {〈𝐴, 2〉, 〈𝐵, -3〉} = {〈𝐴, 2〉, 〈𝐵, -3〉} | |
5 | 1, 2, 3, 4 | zlmodzxzldeplem1 46829 | . . 3 ⊢ {〈𝐴, 2〉, 〈𝐵, -3〉} ∈ (ℤ ↑m {𝐴, 𝐵}) |
6 | 1, 2, 3, 4 | zlmodzxzldeplem2 46830 | . . . 4 ⊢ {〈𝐴, 2〉, 〈𝐵, -3〉} finSupp 0 |
7 | 1, 2, 3, 4 | zlmodzxzldeplem3 46831 | . . . 4 ⊢ ({〈𝐴, 2〉, 〈𝐵, -3〉} ( linC ‘𝑍){𝐴, 𝐵}) = (0g‘𝑍) |
8 | 1, 2, 3, 4 | zlmodzxzldeplem4 46832 | . . . 4 ⊢ ∃𝑦 ∈ {𝐴, 𝐵} ({〈𝐴, 2〉, 〈𝐵, -3〉}‘𝑦) ≠ 0 |
9 | 6, 7, 8 | 3pm3.2i 1339 | . . 3 ⊢ ({〈𝐴, 2〉, 〈𝐵, -3〉} finSupp 0 ∧ ({〈𝐴, 2〉, 〈𝐵, -3〉} ( linC ‘𝑍){𝐴, 𝐵}) = (0g‘𝑍) ∧ ∃𝑦 ∈ {𝐴, 𝐵} ({〈𝐴, 2〉, 〈𝐵, -3〉}‘𝑦) ≠ 0) |
10 | breq1 5144 | . . . . 5 ⊢ (𝑥 = {〈𝐴, 2〉, 〈𝐵, -3〉} → (𝑥 finSupp 0 ↔ {〈𝐴, 2〉, 〈𝐵, -3〉} finSupp 0)) | |
11 | oveq1 7400 | . . . . . 6 ⊢ (𝑥 = {〈𝐴, 2〉, 〈𝐵, -3〉} → (𝑥( linC ‘𝑍){𝐴, 𝐵}) = ({〈𝐴, 2〉, 〈𝐵, -3〉} ( linC ‘𝑍){𝐴, 𝐵})) | |
12 | 11 | eqeq1d 2733 | . . . . 5 ⊢ (𝑥 = {〈𝐴, 2〉, 〈𝐵, -3〉} → ((𝑥( linC ‘𝑍){𝐴, 𝐵}) = (0g‘𝑍) ↔ ({〈𝐴, 2〉, 〈𝐵, -3〉} ( linC ‘𝑍){𝐴, 𝐵}) = (0g‘𝑍))) |
13 | fveq1 6877 | . . . . . . 7 ⊢ (𝑥 = {〈𝐴, 2〉, 〈𝐵, -3〉} → (𝑥‘𝑦) = ({〈𝐴, 2〉, 〈𝐵, -3〉}‘𝑦)) | |
14 | 13 | neeq1d 2999 | . . . . . 6 ⊢ (𝑥 = {〈𝐴, 2〉, 〈𝐵, -3〉} → ((𝑥‘𝑦) ≠ 0 ↔ ({〈𝐴, 2〉, 〈𝐵, -3〉}‘𝑦) ≠ 0)) |
15 | 14 | rexbidv 3177 | . . . . 5 ⊢ (𝑥 = {〈𝐴, 2〉, 〈𝐵, -3〉} → (∃𝑦 ∈ {𝐴, 𝐵} (𝑥‘𝑦) ≠ 0 ↔ ∃𝑦 ∈ {𝐴, 𝐵} ({〈𝐴, 2〉, 〈𝐵, -3〉}‘𝑦) ≠ 0)) |
16 | 10, 12, 15 | 3anbi123d 1436 | . . . 4 ⊢ (𝑥 = {〈𝐴, 2〉, 〈𝐵, -3〉} → ((𝑥 finSupp 0 ∧ (𝑥( linC ‘𝑍){𝐴, 𝐵}) = (0g‘𝑍) ∧ ∃𝑦 ∈ {𝐴, 𝐵} (𝑥‘𝑦) ≠ 0) ↔ ({〈𝐴, 2〉, 〈𝐵, -3〉} finSupp 0 ∧ ({〈𝐴, 2〉, 〈𝐵, -3〉} ( linC ‘𝑍){𝐴, 𝐵}) = (0g‘𝑍) ∧ ∃𝑦 ∈ {𝐴, 𝐵} ({〈𝐴, 2〉, 〈𝐵, -3〉}‘𝑦) ≠ 0))) |
17 | 16 | rspcev 3609 | . . 3 ⊢ (({〈𝐴, 2〉, 〈𝐵, -3〉} ∈ (ℤ ↑m {𝐴, 𝐵}) ∧ ({〈𝐴, 2〉, 〈𝐵, -3〉} finSupp 0 ∧ ({〈𝐴, 2〉, 〈𝐵, -3〉} ( linC ‘𝑍){𝐴, 𝐵}) = (0g‘𝑍) ∧ ∃𝑦 ∈ {𝐴, 𝐵} ({〈𝐴, 2〉, 〈𝐵, -3〉}‘𝑦) ≠ 0)) → ∃𝑥 ∈ (ℤ ↑m {𝐴, 𝐵})(𝑥 finSupp 0 ∧ (𝑥( linC ‘𝑍){𝐴, 𝐵}) = (0g‘𝑍) ∧ ∃𝑦 ∈ {𝐴, 𝐵} (𝑥‘𝑦) ≠ 0)) |
18 | 5, 9, 17 | mp2an 690 | . 2 ⊢ ∃𝑥 ∈ (ℤ ↑m {𝐴, 𝐵})(𝑥 finSupp 0 ∧ (𝑥( linC ‘𝑍){𝐴, 𝐵}) = (0g‘𝑍) ∧ ∃𝑦 ∈ {𝐴, 𝐵} (𝑥‘𝑦) ≠ 0) |
19 | ovex 7426 | . . . 4 ⊢ (ℤring freeLMod {0, 1}) ∈ V | |
20 | 1, 19 | eqeltri 2828 | . . 3 ⊢ 𝑍 ∈ V |
21 | 3z 12577 | . . . . . 6 ⊢ 3 ∈ ℤ | |
22 | 6nn 12283 | . . . . . . 7 ⊢ 6 ∈ ℕ | |
23 | 22 | nnzi 12568 | . . . . . 6 ⊢ 6 ∈ ℤ |
24 | 1 | zlmodzxzel 46679 | . . . . . 6 ⊢ ((3 ∈ ℤ ∧ 6 ∈ ℤ) → {〈0, 3〉, 〈1, 6〉} ∈ (Base‘𝑍)) |
25 | 21, 23, 24 | mp2an 690 | . . . . 5 ⊢ {〈0, 3〉, 〈1, 6〉} ∈ (Base‘𝑍) |
26 | 2, 25 | eqeltri 2828 | . . . 4 ⊢ 𝐴 ∈ (Base‘𝑍) |
27 | 2z 12576 | . . . . . 6 ⊢ 2 ∈ ℤ | |
28 | 4z 12578 | . . . . . 6 ⊢ 4 ∈ ℤ | |
29 | 1 | zlmodzxzel 46679 | . . . . . 6 ⊢ ((2 ∈ ℤ ∧ 4 ∈ ℤ) → {〈0, 2〉, 〈1, 4〉} ∈ (Base‘𝑍)) |
30 | 27, 28, 29 | mp2an 690 | . . . . 5 ⊢ {〈0, 2〉, 〈1, 4〉} ∈ (Base‘𝑍) |
31 | 3, 30 | eqeltri 2828 | . . . 4 ⊢ 𝐵 ∈ (Base‘𝑍) |
32 | prelpwi 5440 | . . . 4 ⊢ ((𝐴 ∈ (Base‘𝑍) ∧ 𝐵 ∈ (Base‘𝑍)) → {𝐴, 𝐵} ∈ 𝒫 (Base‘𝑍)) | |
33 | 26, 31, 32 | mp2an 690 | . . 3 ⊢ {𝐴, 𝐵} ∈ 𝒫 (Base‘𝑍) |
34 | eqid 2731 | . . . 4 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
35 | eqid 2731 | . . . 4 ⊢ (0g‘𝑍) = (0g‘𝑍) | |
36 | 1 | zlmodzxzlmod 46678 | . . . . 5 ⊢ (𝑍 ∈ LMod ∧ ℤring = (Scalar‘𝑍)) |
37 | 36 | simpri 486 | . . . 4 ⊢ ℤring = (Scalar‘𝑍) |
38 | zringbas 20957 | . . . 4 ⊢ ℤ = (Base‘ℤring) | |
39 | zring0 20961 | . . . 4 ⊢ 0 = (0g‘ℤring) | |
40 | 34, 35, 37, 38, 39 | islindeps 46782 | . . 3 ⊢ ((𝑍 ∈ V ∧ {𝐴, 𝐵} ∈ 𝒫 (Base‘𝑍)) → ({𝐴, 𝐵} linDepS 𝑍 ↔ ∃𝑥 ∈ (ℤ ↑m {𝐴, 𝐵})(𝑥 finSupp 0 ∧ (𝑥( linC ‘𝑍){𝐴, 𝐵}) = (0g‘𝑍) ∧ ∃𝑦 ∈ {𝐴, 𝐵} (𝑥‘𝑦) ≠ 0))) |
41 | 20, 33, 40 | mp2an 690 | . 2 ⊢ ({𝐴, 𝐵} linDepS 𝑍 ↔ ∃𝑥 ∈ (ℤ ↑m {𝐴, 𝐵})(𝑥 finSupp 0 ∧ (𝑥( linC ‘𝑍){𝐴, 𝐵}) = (0g‘𝑍) ∧ ∃𝑦 ∈ {𝐴, 𝐵} (𝑥‘𝑦) ≠ 0)) |
42 | 18, 41 | mpbir 230 | 1 ⊢ {𝐴, 𝐵} linDepS 𝑍 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ∃wrex 3069 Vcvv 3473 𝒫 cpw 4596 {cpr 4624 〈cop 4628 class class class wbr 5141 ‘cfv 6532 (class class class)co 7393 ↑m cmap 8803 finSupp cfsupp 9344 0cc0 11092 1c1 11093 -cneg 11427 2c2 12249 3c3 12250 4c4 12251 6c6 12253 ℤcz 12540 Basecbs 17126 Scalarcsca 17182 0gc0g 17367 LModclmod 20420 ℤringczring 20951 freeLMod cfrlm 21234 linC clinc 46733 linDepS clindeps 46770 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-addf 11171 ax-mulf 11172 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 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 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-isom 6541 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-of 7653 df-om 7839 df-1st 7957 df-2nd 7958 df-supp 8129 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-map 8805 df-ixp 8875 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-fsupp 9345 df-sup 9419 df-oi 9487 df-card 9916 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-dec 12660 df-uz 12805 df-fz 13467 df-fzo 13610 df-seq 13949 df-hash 14273 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-0g 17369 df-gsum 17370 df-prds 17375 df-pws 17377 df-mre 17512 df-mrc 17513 df-acs 17515 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-submnd 18648 df-grp 18797 df-minusg 18798 df-sbg 18799 df-mulg 18923 df-subg 18975 df-cntz 19147 df-cmn 19614 df-abl 19615 df-mgp 19947 df-ur 19964 df-ring 20016 df-cring 20017 df-subrg 20310 df-lmod 20422 df-lss 20492 df-sra 20734 df-rgmod 20735 df-cnfld 20879 df-zring 20952 df-dsmm 21220 df-frlm 21235 df-linc 46735 df-lininds 46771 df-lindeps 46773 |
This theorem is referenced by: ldepsnlinc 46837 |
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