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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 11095 | . . . . . . . 8 ⊢ 3 ∈ ℂ | |
2 | 1 | 2timesi 11147 | . . . . . . 7 ⊢ (2 · 3) = (3 + 3) |
3 | 3p3e6 11161 | . . . . . . 7 ⊢ (3 + 3) = 6 | |
4 | 2, 3 | eqtri 2644 | . . . . . 6 ⊢ (2 · 3) = 6 |
5 | 3t2e6 11179 | . . . . . 6 ⊢ (3 · 2) = 6 | |
6 | 4, 5 | oveq12i 6662 | . . . . 5 ⊢ ((2 · 3) − (3 · 2)) = (6 − 6) |
7 | 6cn 11102 | . . . . . 6 ⊢ 6 ∈ ℂ | |
8 | 7 | subidi 10352 | . . . . 5 ⊢ (6 − 6) = 0 |
9 | 6, 8 | eqtri 2644 | . . . 4 ⊢ ((2 · 3) − (3 · 2)) = 0 |
10 | 9 | opeq2i 4406 | . . 3 ⊢ 〈0, ((2 · 3) − (3 · 2))〉 = 〈0, 0〉 |
11 | 2t6m3t4e0 42126 | . . . 4 ⊢ ((2 · 6) − (3 · 4)) = 0 | |
12 | 11 | opeq2i 4406 | . . 3 ⊢ 〈1, ((2 · 6) − (3 · 4))〉 = 〈1, 0〉 |
13 | 10, 12 | preq12i 4273 | . 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 6661 | . . . . 5 ⊢ (2 ∙ 𝐴) = (2 ∙ {〈0, 3〉, 〈1, 6〉}) |
16 | 2z 11409 | . . . . . 6 ⊢ 2 ∈ ℤ | |
17 | 3z 11410 | . . . . . 6 ⊢ 3 ∈ ℤ | |
18 | 6nn 11189 | . . . . . . 7 ⊢ 6 ∈ ℕ | |
19 | 18 | nnzi 11401 | . . . . . 6 ⊢ 6 ∈ ℤ |
20 | zlmodzxzequa.z | . . . . . . 7 ⊢ 𝑍 = (ℤring freeLMod {0, 1}) | |
21 | zlmodzxzequa.t | . . . . . . 7 ⊢ ∙ = ( ·𝑠 ‘𝑍) | |
22 | 20, 21 | zlmodzxzscm 42135 | . . . . . 6 ⊢ ((2 ∈ ℤ ∧ 3 ∈ ℤ ∧ 6 ∈ ℤ) → (2 ∙ {〈0, 3〉, 〈1, 6〉}) = {〈0, (2 · 3)〉, 〈1, (2 · 6)〉}) |
23 | 16, 17, 19, 22 | mp3an 1424 | . . . . 5 ⊢ (2 ∙ {〈0, 3〉, 〈1, 6〉}) = {〈0, (2 · 3)〉, 〈1, (2 · 6)〉} |
24 | 15, 23 | eqtri 2644 | . . . 4 ⊢ (2 ∙ 𝐴) = {〈0, (2 · 3)〉, 〈1, (2 · 6)〉} |
25 | zlmodzxzequa.b | . . . . . 6 ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} | |
26 | 25 | oveq2i 6661 | . . . . 5 ⊢ (3 ∙ 𝐵) = (3 ∙ {〈0, 2〉, 〈1, 4〉}) |
27 | 4z 11411 | . . . . . 6 ⊢ 4 ∈ ℤ | |
28 | 20, 21 | zlmodzxzscm 42135 | . . . . . 6 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ ∧ 4 ∈ ℤ) → (3 ∙ {〈0, 2〉, 〈1, 4〉}) = {〈0, (3 · 2)〉, 〈1, (3 · 4)〉}) |
29 | 17, 16, 27, 28 | mp3an 1424 | . . . . 5 ⊢ (3 ∙ {〈0, 2〉, 〈1, 4〉}) = {〈0, (3 · 2)〉, 〈1, (3 · 4)〉} |
30 | 26, 29 | eqtri 2644 | . . . 4 ⊢ (3 ∙ 𝐵) = {〈0, (3 · 2)〉, 〈1, (3 · 4)〉} |
31 | 24, 30 | oveq12i 6662 | . . 3 ⊢ ((2 ∙ 𝐴) − (3 ∙ 𝐵)) = ({〈0, (2 · 3)〉, 〈1, (2 · 6)〉} − {〈0, (3 · 2)〉, 〈1, (3 · 4)〉}) |
32 | zmulcl 11426 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ 3 ∈ ℤ) → (2 · 3) ∈ ℤ) | |
33 | 16, 17, 32 | mp2an 708 | . . . 4 ⊢ (2 · 3) ∈ ℤ |
34 | zmulcl 11426 | . . . . 5 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ) → (3 · 2) ∈ ℤ) | |
35 | 17, 16, 34 | mp2an 708 | . . . 4 ⊢ (3 · 2) ∈ ℤ |
36 | zmulcl 11426 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ 6 ∈ ℤ) → (2 · 6) ∈ ℤ) | |
37 | 16, 19, 36 | mp2an 708 | . . . 4 ⊢ (2 · 6) ∈ ℤ |
38 | zmulcl 11426 | . . . . 5 ⊢ ((3 ∈ ℤ ∧ 4 ∈ ℤ) → (3 · 4) ∈ ℤ) | |
39 | 17, 27, 38 | mp2an 708 | . . . 4 ⊢ (3 · 4) ∈ ℤ |
40 | zlmodzxzequa.m | . . . . 5 ⊢ − = (-g‘𝑍) | |
41 | 20, 40 | zlmodzxzsub 42138 | . . . 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 709 | . . 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 2644 | . 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 2654 | 1 ⊢ ((2 ∙ 𝐴) − (3 ∙ 𝐵)) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 {cpr 4179 〈cop 4183 ‘cfv 5888 (class class class)co 6650 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 − cmin 10266 2c2 11070 3c3 11071 4c4 11072 6c6 11074 ℤcz 11377 ·𝑠 cvsca 15945 -gcsg 17424 ℤringzring 19818 freeLMod cfrlm 20090 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-0g 16102 df-prds 16108 df-pws 16110 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cmn 18195 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-subrg 18778 df-lmod 18865 df-lss 18933 df-sra 19172 df-rgmod 19173 df-cnfld 19747 df-zring 19819 df-dsmm 20076 df-frlm 20091 |
This theorem is referenced by: (None) |
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