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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 47270 | . . 3 ⊢ {⟨𝐴, 2⟩, ⟨𝐵, -3⟩} ∈ (ℤ ↑m {𝐴, 𝐵}) |
6 | 1, 2, 3, 4 | zlmodzxzldeplem2 47271 | . . . 4 ⊢ {⟨𝐴, 2⟩, ⟨𝐵, -3⟩} finSupp 0 |
7 | 1, 2, 3, 4 | zlmodzxzldeplem3 47272 | . . . 4 ⊢ ({⟨𝐴, 2⟩, ⟨𝐵, -3⟩} ( linC ‘𝑍){𝐴, 𝐵}) = (0g‘𝑍) |
8 | 1, 2, 3, 4 | zlmodzxzldeplem4 47273 | . . . 4 ⊢ ∃𝑦 ∈ {𝐴, 𝐵} ({⟨𝐴, 2⟩, ⟨𝐵, -3⟩}‘𝑦) ≠ 0 |
9 | 6, 7, 8 | 3pm3.2i 1338 | . . 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 5152 | . . . . 5 ⊢ (𝑥 = {⟨𝐴, 2⟩, ⟨𝐵, -3⟩} → (𝑥 finSupp 0 ↔ {⟨𝐴, 2⟩, ⟨𝐵, -3⟩} finSupp 0)) | |
11 | oveq1 7419 | . . . . . 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 6891 | . . . . . . 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 1435 | . . . 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 3613 | . . 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 689 | . 2 ⊢ ∃𝑥 ∈ (ℤ ↑m {𝐴, 𝐵})(𝑥 finSupp 0 ∧ (𝑥( linC ‘𝑍){𝐴, 𝐵}) = (0g‘𝑍) ∧ ∃𝑦 ∈ {𝐴, 𝐵} (𝑥‘𝑦) ≠ 0) |
19 | ovex 7445 | . . . 4 ⊢ (ℤring freeLMod {0, 1}) ∈ V | |
20 | 1, 19 | eqeltri 2828 | . . 3 ⊢ 𝑍 ∈ V |
21 | 3z 12600 | . . . . . 6 ⊢ 3 ∈ ℤ | |
22 | 6nn 12306 | . . . . . . 7 ⊢ 6 ∈ ℕ | |
23 | 22 | nnzi 12591 | . . . . . 6 ⊢ 6 ∈ ℤ |
24 | 1 | zlmodzxzel 47121 | . . . . . 6 ⊢ ((3 ∈ ℤ ∧ 6 ∈ ℤ) → {⟨0, 3⟩, ⟨1, 6⟩} ∈ (Base‘𝑍)) |
25 | 21, 23, 24 | mp2an 689 | . . . . 5 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} ∈ (Base‘𝑍) |
26 | 2, 25 | eqeltri 2828 | . . . 4 ⊢ 𝐴 ∈ (Base‘𝑍) |
27 | 2z 12599 | . . . . . 6 ⊢ 2 ∈ ℤ | |
28 | 4z 12601 | . . . . . 6 ⊢ 4 ∈ ℤ | |
29 | 1 | zlmodzxzel 47121 | . . . . . 6 ⊢ ((2 ∈ ℤ ∧ 4 ∈ ℤ) → {⟨0, 2⟩, ⟨1, 4⟩} ∈ (Base‘𝑍)) |
30 | 27, 28, 29 | mp2an 689 | . . . . 5 ⊢ {⟨0, 2⟩, ⟨1, 4⟩} ∈ (Base‘𝑍) |
31 | 3, 30 | eqeltri 2828 | . . . 4 ⊢ 𝐵 ∈ (Base‘𝑍) |
32 | prelpwi 5448 | . . . 4 ⊢ ((𝐴 ∈ (Base‘𝑍) ∧ 𝐵 ∈ (Base‘𝑍)) → {𝐴, 𝐵} ∈ 𝒫 (Base‘𝑍)) | |
33 | 26, 31, 32 | mp2an 689 | . . 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 47120 | . . . . 5 ⊢ (𝑍 ∈ LMod ∧ ℤring = (Scalar‘𝑍)) |
37 | 36 | simpri 485 | . . . 4 ⊢ ℤring = (Scalar‘𝑍) |
38 | zringbas 21225 | . . . 4 ⊢ ℤ = (Base‘ℤring) | |
39 | zring0 21230 | . . . 4 ⊢ 0 = (0g‘ℤring) | |
40 | 34, 35, 37, 38, 39 | islindeps 47223 | . . 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 689 | . 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 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∃wrex 3069 Vcvv 3473 𝒫 cpw 4603 {cpr 4631 ⟨cop 4635 class class class wbr 5149 ‘cfv 6544 (class class class)co 7412 ↑m cmap 8823 finSupp cfsupp 9364 0cc0 11113 1c1 11114 -cneg 11450 2c2 12272 3c3 12273 4c4 12274 6c6 12276 ℤcz 12563 Basecbs 17149 Scalarcsca 17205 0gc0g 17390 LModclmod 20615 ℤringczring 21218 freeLMod cfrlm 21521 linC clinc 47174 linDepS clindeps 47211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-addf 11192 ax-mulf 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7673 df-om 7859 df-1st 7978 df-2nd 7979 df-supp 8150 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-map 8825 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9365 df-sup 9440 df-oi 9508 df-card 9937 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-fz 13490 df-fzo 13633 df-seq 13972 df-hash 14296 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-0g 17392 df-gsum 17393 df-prds 17398 df-pws 17400 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-grp 18859 df-minusg 18860 df-sbg 18861 df-mulg 18988 df-subg 19040 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-subrng 20435 df-subrg 20460 df-lmod 20617 df-lss 20688 df-sra 20931 df-rgmod 20932 df-cnfld 21146 df-zring 21219 df-dsmm 21507 df-frlm 21522 df-linc 47176 df-lininds 47212 df-lindeps 47214 |
This theorem is referenced by: ldepsnlinc 47278 |
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