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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 12290 | . . . . . . . 8 ⊢ 3 ∈ ℂ | |
2 | 1 | 2timesi 12347 | . . . . . . 7 ⊢ (2 · 3) = (3 + 3) |
3 | 3p3e6 12361 | . . . . . . 7 ⊢ (3 + 3) = 6 | |
4 | 2, 3 | eqtri 2761 | . . . . . 6 ⊢ (2 · 3) = 6 |
5 | 3t2e6 12375 | . . . . . 6 ⊢ (3 · 2) = 6 | |
6 | 4, 5 | oveq12i 7418 | . . . . 5 ⊢ ((2 · 3) − (3 · 2)) = (6 − 6) |
7 | 6cn 12300 | . . . . . 6 ⊢ 6 ∈ ℂ | |
8 | 7 | subidi 11528 | . . . . 5 ⊢ (6 − 6) = 0 |
9 | 6, 8 | eqtri 2761 | . . . 4 ⊢ ((2 · 3) − (3 · 2)) = 0 |
10 | 9 | opeq2i 4877 | . . 3 ⊢ 〈0, ((2 · 3) − (3 · 2))〉 = 〈0, 0〉 |
11 | 2t6m3t4e0 46978 | . . . 4 ⊢ ((2 · 6) − (3 · 4)) = 0 | |
12 | 11 | opeq2i 4877 | . . 3 ⊢ 〈1, ((2 · 6) − (3 · 4))〉 = 〈1, 0〉 |
13 | 10, 12 | preq12i 4742 | . 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 7417 | . . . . 5 ⊢ (2 ∙ 𝐴) = (2 ∙ {〈0, 3〉, 〈1, 6〉}) |
16 | 2z 12591 | . . . . . 6 ⊢ 2 ∈ ℤ | |
17 | 3z 12592 | . . . . . 6 ⊢ 3 ∈ ℤ | |
18 | 6nn 12298 | . . . . . . 7 ⊢ 6 ∈ ℕ | |
19 | 18 | nnzi 12583 | . . . . . 6 ⊢ 6 ∈ ℤ |
20 | zlmodzxzequa.z | . . . . . . 7 ⊢ 𝑍 = (ℤring freeLMod {0, 1}) | |
21 | zlmodzxzequa.t | . . . . . . 7 ⊢ ∙ = ( ·𝑠 ‘𝑍) | |
22 | 20, 21 | zlmodzxzscm 46987 | . . . . . 6 ⊢ ((2 ∈ ℤ ∧ 3 ∈ ℤ ∧ 6 ∈ ℤ) → (2 ∙ {〈0, 3〉, 〈1, 6〉}) = {〈0, (2 · 3)〉, 〈1, (2 · 6)〉}) |
23 | 16, 17, 19, 22 | mp3an 1462 | . . . . 5 ⊢ (2 ∙ {〈0, 3〉, 〈1, 6〉}) = {〈0, (2 · 3)〉, 〈1, (2 · 6)〉} |
24 | 15, 23 | eqtri 2761 | . . . 4 ⊢ (2 ∙ 𝐴) = {〈0, (2 · 3)〉, 〈1, (2 · 6)〉} |
25 | zlmodzxzequa.b | . . . . . 6 ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} | |
26 | 25 | oveq2i 7417 | . . . . 5 ⊢ (3 ∙ 𝐵) = (3 ∙ {〈0, 2〉, 〈1, 4〉}) |
27 | 4z 12593 | . . . . . 6 ⊢ 4 ∈ ℤ | |
28 | 20, 21 | zlmodzxzscm 46987 | . . . . . 6 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ ∧ 4 ∈ ℤ) → (3 ∙ {〈0, 2〉, 〈1, 4〉}) = {〈0, (3 · 2)〉, 〈1, (3 · 4)〉}) |
29 | 17, 16, 27, 28 | mp3an 1462 | . . . . 5 ⊢ (3 ∙ {〈0, 2〉, 〈1, 4〉}) = {〈0, (3 · 2)〉, 〈1, (3 · 4)〉} |
30 | 26, 29 | eqtri 2761 | . . . 4 ⊢ (3 ∙ 𝐵) = {〈0, (3 · 2)〉, 〈1, (3 · 4)〉} |
31 | 24, 30 | oveq12i 7418 | . . 3 ⊢ ((2 ∙ 𝐴) − (3 ∙ 𝐵)) = ({〈0, (2 · 3)〉, 〈1, (2 · 6)〉} − {〈0, (3 · 2)〉, 〈1, (3 · 4)〉}) |
32 | zmulcl 12608 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ 3 ∈ ℤ) → (2 · 3) ∈ ℤ) | |
33 | 16, 17, 32 | mp2an 691 | . . . 4 ⊢ (2 · 3) ∈ ℤ |
34 | zmulcl 12608 | . . . . 5 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ) → (3 · 2) ∈ ℤ) | |
35 | 17, 16, 34 | mp2an 691 | . . . 4 ⊢ (3 · 2) ∈ ℤ |
36 | zmulcl 12608 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ 6 ∈ ℤ) → (2 · 6) ∈ ℤ) | |
37 | 16, 19, 36 | mp2an 691 | . . . 4 ⊢ (2 · 6) ∈ ℤ |
38 | zmulcl 12608 | . . . . 5 ⊢ ((3 ∈ ℤ ∧ 4 ∈ ℤ) → (3 · 4) ∈ ℤ) | |
39 | 17, 27, 38 | mp2an 691 | . . . 4 ⊢ (3 · 4) ∈ ℤ |
40 | zlmodzxzequa.m | . . . . 5 ⊢ − = (-g‘𝑍) | |
41 | 20, 40 | zlmodzxzsub 46990 | . . . 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 692 | . . 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 2761 | . 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 2771 | 1 ⊢ ((2 ∙ 𝐴) − (3 ∙ 𝐵)) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 {cpr 4630 〈cop 4634 ‘cfv 6541 (class class class)co 7406 0cc0 11107 1c1 11108 + caddc 11110 · cmul 11112 − cmin 11441 2c2 12264 3c3 12265 4c4 12266 6c6 12268 ℤcz 12555 ·𝑠 cvsca 17198 -gcsg 18818 ℤringczring 21010 freeLMod cfrlm 21293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-addf 11186 ax-mulf 11187 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-om 7853 df-1st 7972 df-2nd 7973 df-supp 8144 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-map 8819 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-sup 9434 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-starv 17209 df-sca 17210 df-vsca 17211 df-ip 17212 df-tset 17213 df-ple 17214 df-ds 17216 df-unif 17217 df-hom 17218 df-cco 17219 df-0g 17384 df-prds 17390 df-pws 17392 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-grp 18819 df-minusg 18820 df-sbg 18821 df-subg 18998 df-cmn 19645 df-mgp 19983 df-ur 20000 df-ring 20052 df-cring 20053 df-subrg 20354 df-lmod 20466 df-lss 20536 df-sra 20778 df-rgmod 20779 df-cnfld 20938 df-zring 21011 df-dsmm 21279 df-frlm 21294 |
This theorem is referenced by: (None) |
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