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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 11719 | . . . . . . . 8 ⊢ 3 ∈ ℂ | |
2 | 1 | 2timesi 11776 | . . . . . . 7 ⊢ (2 · 3) = (3 + 3) |
3 | 3p3e6 11790 | . . . . . . 7 ⊢ (3 + 3) = 6 | |
4 | 2, 3 | eqtri 2844 | . . . . . 6 ⊢ (2 · 3) = 6 |
5 | 3t2e6 11804 | . . . . . 6 ⊢ (3 · 2) = 6 | |
6 | 4, 5 | oveq12i 7168 | . . . . 5 ⊢ ((2 · 3) − (3 · 2)) = (6 − 6) |
7 | 6cn 11729 | . . . . . 6 ⊢ 6 ∈ ℂ | |
8 | 7 | subidi 10957 | . . . . 5 ⊢ (6 − 6) = 0 |
9 | 6, 8 | eqtri 2844 | . . . 4 ⊢ ((2 · 3) − (3 · 2)) = 0 |
10 | 9 | opeq2i 4807 | . . 3 ⊢ 〈0, ((2 · 3) − (3 · 2))〉 = 〈0, 0〉 |
11 | 2t6m3t4e0 44416 | . . . 4 ⊢ ((2 · 6) − (3 · 4)) = 0 | |
12 | 11 | opeq2i 4807 | . . 3 ⊢ 〈1, ((2 · 6) − (3 · 4))〉 = 〈1, 0〉 |
13 | 10, 12 | preq12i 4674 | . 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 7167 | . . . . 5 ⊢ (2 ∙ 𝐴) = (2 ∙ {〈0, 3〉, 〈1, 6〉}) |
16 | 2z 12015 | . . . . . 6 ⊢ 2 ∈ ℤ | |
17 | 3z 12016 | . . . . . 6 ⊢ 3 ∈ ℤ | |
18 | 6nn 11727 | . . . . . . 7 ⊢ 6 ∈ ℕ | |
19 | 18 | nnzi 12007 | . . . . . 6 ⊢ 6 ∈ ℤ |
20 | zlmodzxzequa.z | . . . . . . 7 ⊢ 𝑍 = (ℤring freeLMod {0, 1}) | |
21 | zlmodzxzequa.t | . . . . . . 7 ⊢ ∙ = ( ·𝑠 ‘𝑍) | |
22 | 20, 21 | zlmodzxzscm 44425 | . . . . . 6 ⊢ ((2 ∈ ℤ ∧ 3 ∈ ℤ ∧ 6 ∈ ℤ) → (2 ∙ {〈0, 3〉, 〈1, 6〉}) = {〈0, (2 · 3)〉, 〈1, (2 · 6)〉}) |
23 | 16, 17, 19, 22 | mp3an 1457 | . . . . 5 ⊢ (2 ∙ {〈0, 3〉, 〈1, 6〉}) = {〈0, (2 · 3)〉, 〈1, (2 · 6)〉} |
24 | 15, 23 | eqtri 2844 | . . . 4 ⊢ (2 ∙ 𝐴) = {〈0, (2 · 3)〉, 〈1, (2 · 6)〉} |
25 | zlmodzxzequa.b | . . . . . 6 ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} | |
26 | 25 | oveq2i 7167 | . . . . 5 ⊢ (3 ∙ 𝐵) = (3 ∙ {〈0, 2〉, 〈1, 4〉}) |
27 | 4z 12017 | . . . . . 6 ⊢ 4 ∈ ℤ | |
28 | 20, 21 | zlmodzxzscm 44425 | . . . . . 6 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ ∧ 4 ∈ ℤ) → (3 ∙ {〈0, 2〉, 〈1, 4〉}) = {〈0, (3 · 2)〉, 〈1, (3 · 4)〉}) |
29 | 17, 16, 27, 28 | mp3an 1457 | . . . . 5 ⊢ (3 ∙ {〈0, 2〉, 〈1, 4〉}) = {〈0, (3 · 2)〉, 〈1, (3 · 4)〉} |
30 | 26, 29 | eqtri 2844 | . . . 4 ⊢ (3 ∙ 𝐵) = {〈0, (3 · 2)〉, 〈1, (3 · 4)〉} |
31 | 24, 30 | oveq12i 7168 | . . 3 ⊢ ((2 ∙ 𝐴) − (3 ∙ 𝐵)) = ({〈0, (2 · 3)〉, 〈1, (2 · 6)〉} − {〈0, (3 · 2)〉, 〈1, (3 · 4)〉}) |
32 | zmulcl 12032 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ 3 ∈ ℤ) → (2 · 3) ∈ ℤ) | |
33 | 16, 17, 32 | mp2an 690 | . . . 4 ⊢ (2 · 3) ∈ ℤ |
34 | zmulcl 12032 | . . . . 5 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ) → (3 · 2) ∈ ℤ) | |
35 | 17, 16, 34 | mp2an 690 | . . . 4 ⊢ (3 · 2) ∈ ℤ |
36 | zmulcl 12032 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ 6 ∈ ℤ) → (2 · 6) ∈ ℤ) | |
37 | 16, 19, 36 | mp2an 690 | . . . 4 ⊢ (2 · 6) ∈ ℤ |
38 | zmulcl 12032 | . . . . 5 ⊢ ((3 ∈ ℤ ∧ 4 ∈ ℤ) → (3 · 4) ∈ ℤ) | |
39 | 17, 27, 38 | mp2an 690 | . . . 4 ⊢ (3 · 4) ∈ ℤ |
40 | zlmodzxzequa.m | . . . . 5 ⊢ − = (-g‘𝑍) | |
41 | 20, 40 | zlmodzxzsub 44428 | . . . 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 691 | . . 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 2844 | . 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 2854 | 1 ⊢ ((2 ∙ 𝐴) − (3 ∙ 𝐵)) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 {cpr 4569 〈cop 4573 ‘cfv 6355 (class class class)co 7156 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 − cmin 10870 2c2 11693 3c3 11694 4c4 11695 6c6 11697 ℤcz 11982 ·𝑠 cvsca 16569 -gcsg 18105 ℤringzring 20617 freeLMod cfrlm 20890 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-0g 16715 df-prds 16721 df-pws 16723 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-cmn 18908 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-subrg 19533 df-lmod 19636 df-lss 19704 df-sra 19944 df-rgmod 19945 df-cnfld 20546 df-zring 20618 df-dsmm 20876 df-frlm 20891 |
This theorem is referenced by: (None) |
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