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| Mirrors > Home > MPE Home > Th. List > Mathboxes > zlmodzxzequa | Structured version Visualization version GIF version | ||
| Description: Example of an equation within the ℤ-module ℤ × ℤ (see example in [Roman] p. 112 for a linearly dependent set). (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.) |
| Ref | Expression |
|---|---|
| zlmodzxzequa.z | ⊢ 𝑍 = (ℤring freeLMod {0, 1}) |
| zlmodzxzequa.o | ⊢ 0 = {〈0, 0〉, 〈1, 0〉} |
| zlmodzxzequa.t | ⊢ ∙ = ( ·𝑠 ‘𝑍) |
| zlmodzxzequa.m | ⊢ − = (-g‘𝑍) |
| zlmodzxzequa.a | ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} |
| zlmodzxzequa.b | ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} |
| Ref | Expression |
|---|---|
| zlmodzxzequa | ⊢ ((2 ∙ 𝐴) − (3 ∙ 𝐵)) = 0 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 3cn 12347 | . . . . . . . 8 ⊢ 3 ∈ ℂ | |
| 2 | 1 | 2timesi 12404 | . . . . . . 7 ⊢ (2 · 3) = (3 + 3) |
| 3 | 3p3e6 12418 | . . . . . . 7 ⊢ (3 + 3) = 6 | |
| 4 | 2, 3 | eqtri 2765 | . . . . . 6 ⊢ (2 · 3) = 6 |
| 5 | 3t2e6 12432 | . . . . . 6 ⊢ (3 · 2) = 6 | |
| 6 | 4, 5 | oveq12i 7443 | . . . . 5 ⊢ ((2 · 3) − (3 · 2)) = (6 − 6) |
| 7 | 6cn 12357 | . . . . . 6 ⊢ 6 ∈ ℂ | |
| 8 | 7 | subidi 11580 | . . . . 5 ⊢ (6 − 6) = 0 |
| 9 | 6, 8 | eqtri 2765 | . . . 4 ⊢ ((2 · 3) − (3 · 2)) = 0 |
| 10 | 9 | opeq2i 4877 | . . 3 ⊢ 〈0, ((2 · 3) − (3 · 2))〉 = 〈0, 0〉 |
| 11 | 2t6m3t4e0 48264 | . . . 4 ⊢ ((2 · 6) − (3 · 4)) = 0 | |
| 12 | 11 | opeq2i 4877 | . . 3 ⊢ 〈1, ((2 · 6) − (3 · 4))〉 = 〈1, 0〉 |
| 13 | 10, 12 | preq12i 4738 | . 2 ⊢ {〈0, ((2 · 3) − (3 · 2))〉, 〈1, ((2 · 6) − (3 · 4))〉} = {〈0, 0〉, 〈1, 0〉} |
| 14 | zlmodzxzequa.a | . . . . . 6 ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} | |
| 15 | 14 | oveq2i 7442 | . . . . 5 ⊢ (2 ∙ 𝐴) = (2 ∙ {〈0, 3〉, 〈1, 6〉}) |
| 16 | 2z 12649 | . . . . . 6 ⊢ 2 ∈ ℤ | |
| 17 | 3z 12650 | . . . . . 6 ⊢ 3 ∈ ℤ | |
| 18 | 6nn 12355 | . . . . . . 7 ⊢ 6 ∈ ℕ | |
| 19 | 18 | nnzi 12641 | . . . . . 6 ⊢ 6 ∈ ℤ |
| 20 | zlmodzxzequa.z | . . . . . . 7 ⊢ 𝑍 = (ℤring freeLMod {0, 1}) | |
| 21 | zlmodzxzequa.t | . . . . . . 7 ⊢ ∙ = ( ·𝑠 ‘𝑍) | |
| 22 | 20, 21 | zlmodzxzscm 48273 | . . . . . 6 ⊢ ((2 ∈ ℤ ∧ 3 ∈ ℤ ∧ 6 ∈ ℤ) → (2 ∙ {〈0, 3〉, 〈1, 6〉}) = {〈0, (2 · 3)〉, 〈1, (2 · 6)〉}) |
| 23 | 16, 17, 19, 22 | mp3an 1463 | . . . . 5 ⊢ (2 ∙ {〈0, 3〉, 〈1, 6〉}) = {〈0, (2 · 3)〉, 〈1, (2 · 6)〉} |
| 24 | 15, 23 | eqtri 2765 | . . . 4 ⊢ (2 ∙ 𝐴) = {〈0, (2 · 3)〉, 〈1, (2 · 6)〉} |
| 25 | zlmodzxzequa.b | . . . . . 6 ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} | |
| 26 | 25 | oveq2i 7442 | . . . . 5 ⊢ (3 ∙ 𝐵) = (3 ∙ {〈0, 2〉, 〈1, 4〉}) |
| 27 | 4z 12651 | . . . . . 6 ⊢ 4 ∈ ℤ | |
| 28 | 20, 21 | zlmodzxzscm 48273 | . . . . . 6 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ ∧ 4 ∈ ℤ) → (3 ∙ {〈0, 2〉, 〈1, 4〉}) = {〈0, (3 · 2)〉, 〈1, (3 · 4)〉}) |
| 29 | 17, 16, 27, 28 | mp3an 1463 | . . . . 5 ⊢ (3 ∙ {〈0, 2〉, 〈1, 4〉}) = {〈0, (3 · 2)〉, 〈1, (3 · 4)〉} |
| 30 | 26, 29 | eqtri 2765 | . . . 4 ⊢ (3 ∙ 𝐵) = {〈0, (3 · 2)〉, 〈1, (3 · 4)〉} |
| 31 | 24, 30 | oveq12i 7443 | . . 3 ⊢ ((2 ∙ 𝐴) − (3 ∙ 𝐵)) = ({〈0, (2 · 3)〉, 〈1, (2 · 6)〉} − {〈0, (3 · 2)〉, 〈1, (3 · 4)〉}) |
| 32 | zmulcl 12666 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ 3 ∈ ℤ) → (2 · 3) ∈ ℤ) | |
| 33 | 16, 17, 32 | mp2an 692 | . . . 4 ⊢ (2 · 3) ∈ ℤ |
| 34 | zmulcl 12666 | . . . . 5 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ) → (3 · 2) ∈ ℤ) | |
| 35 | 17, 16, 34 | mp2an 692 | . . . 4 ⊢ (3 · 2) ∈ ℤ |
| 36 | zmulcl 12666 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ 6 ∈ ℤ) → (2 · 6) ∈ ℤ) | |
| 37 | 16, 19, 36 | mp2an 692 | . . . 4 ⊢ (2 · 6) ∈ ℤ |
| 38 | zmulcl 12666 | . . . . 5 ⊢ ((3 ∈ ℤ ∧ 4 ∈ ℤ) → (3 · 4) ∈ ℤ) | |
| 39 | 17, 27, 38 | mp2an 692 | . . . 4 ⊢ (3 · 4) ∈ ℤ |
| 40 | zlmodzxzequa.m | . . . . 5 ⊢ − = (-g‘𝑍) | |
| 41 | 20, 40 | zlmodzxzsub 48276 | . . . 4 ⊢ ((((2 · 3) ∈ ℤ ∧ (3 · 2) ∈ ℤ) ∧ ((2 · 6) ∈ ℤ ∧ (3 · 4) ∈ ℤ)) → ({〈0, (2 · 3)〉, 〈1, (2 · 6)〉} − {〈0, (3 · 2)〉, 〈1, (3 · 4)〉}) = {〈0, ((2 · 3) − (3 · 2))〉, 〈1, ((2 · 6) − (3 · 4))〉}) |
| 42 | 33, 35, 37, 39, 41 | mp4an 693 | . . 3 ⊢ ({〈0, (2 · 3)〉, 〈1, (2 · 6)〉} − {〈0, (3 · 2)〉, 〈1, (3 · 4)〉}) = {〈0, ((2 · 3) − (3 · 2))〉, 〈1, ((2 · 6) − (3 · 4))〉} |
| 43 | 31, 42 | eqtri 2765 | . 2 ⊢ ((2 ∙ 𝐴) − (3 ∙ 𝐵)) = {〈0, ((2 · 3) − (3 · 2))〉, 〈1, ((2 · 6) − (3 · 4))〉} |
| 44 | zlmodzxzequa.o | . 2 ⊢ 0 = {〈0, 0〉, 〈1, 0〉} | |
| 45 | 13, 43, 44 | 3eqtr4i 2775 | 1 ⊢ ((2 ∙ 𝐴) − (3 ∙ 𝐵)) = 0 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 {cpr 4628 〈cop 4632 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 − cmin 11492 2c2 12321 3c3 12322 4c4 12323 6c6 12325 ℤcz 12613 ·𝑠 cvsca 17301 -gcsg 18953 ℤringczring 21457 freeLMod cfrlm 21766 |
| 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-rep 5279 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 ax-addf 11234 ax-mulf 11235 |
| 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-tp 4631 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-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 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-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17486 df-prds 17492 df-pws 17494 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-subrng 20546 df-subrg 20570 df-lmod 20860 df-lss 20930 df-sra 21172 df-rgmod 21173 df-cnfld 21365 df-zring 21458 df-dsmm 21752 df-frlm 21767 |
| This theorem is referenced by: (None) |
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