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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 12345 | . . . . . . . 8 ⊢ 3 ∈ ℂ | |
2 | 1 | 2timesi 12402 | . . . . . . 7 ⊢ (2 · 3) = (3 + 3) |
3 | 3p3e6 12416 | . . . . . . 7 ⊢ (3 + 3) = 6 | |
4 | 2, 3 | eqtri 2763 | . . . . . 6 ⊢ (2 · 3) = 6 |
5 | 3t2e6 12430 | . . . . . 6 ⊢ (3 · 2) = 6 | |
6 | 4, 5 | oveq12i 7443 | . . . . 5 ⊢ ((2 · 3) − (3 · 2)) = (6 − 6) |
7 | 6cn 12355 | . . . . . 6 ⊢ 6 ∈ ℂ | |
8 | 7 | subidi 11578 | . . . . 5 ⊢ (6 − 6) = 0 |
9 | 6, 8 | eqtri 2763 | . . . 4 ⊢ ((2 · 3) − (3 · 2)) = 0 |
10 | 9 | opeq2i 4882 | . . 3 ⊢ 〈0, ((2 · 3) − (3 · 2))〉 = 〈0, 0〉 |
11 | 2t6m3t4e0 48193 | . . . 4 ⊢ ((2 · 6) − (3 · 4)) = 0 | |
12 | 11 | opeq2i 4882 | . . 3 ⊢ 〈1, ((2 · 6) − (3 · 4))〉 = 〈1, 0〉 |
13 | 10, 12 | preq12i 4743 | . 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 12647 | . . . . . 6 ⊢ 2 ∈ ℤ | |
17 | 3z 12648 | . . . . . 6 ⊢ 3 ∈ ℤ | |
18 | 6nn 12353 | . . . . . . 7 ⊢ 6 ∈ ℕ | |
19 | 18 | nnzi 12639 | . . . . . 6 ⊢ 6 ∈ ℤ |
20 | zlmodzxzequa.z | . . . . . . 7 ⊢ 𝑍 = (ℤring freeLMod {0, 1}) | |
21 | zlmodzxzequa.t | . . . . . . 7 ⊢ ∙ = ( ·𝑠 ‘𝑍) | |
22 | 20, 21 | zlmodzxzscm 48202 | . . . . . 6 ⊢ ((2 ∈ ℤ ∧ 3 ∈ ℤ ∧ 6 ∈ ℤ) → (2 ∙ {〈0, 3〉, 〈1, 6〉}) = {〈0, (2 · 3)〉, 〈1, (2 · 6)〉}) |
23 | 16, 17, 19, 22 | mp3an 1460 | . . . . 5 ⊢ (2 ∙ {〈0, 3〉, 〈1, 6〉}) = {〈0, (2 · 3)〉, 〈1, (2 · 6)〉} |
24 | 15, 23 | eqtri 2763 | . . . 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 12649 | . . . . . 6 ⊢ 4 ∈ ℤ | |
28 | 20, 21 | zlmodzxzscm 48202 | . . . . . 6 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ ∧ 4 ∈ ℤ) → (3 ∙ {〈0, 2〉, 〈1, 4〉}) = {〈0, (3 · 2)〉, 〈1, (3 · 4)〉}) |
29 | 17, 16, 27, 28 | mp3an 1460 | . . . . 5 ⊢ (3 ∙ {〈0, 2〉, 〈1, 4〉}) = {〈0, (3 · 2)〉, 〈1, (3 · 4)〉} |
30 | 26, 29 | eqtri 2763 | . . . 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 12664 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ 3 ∈ ℤ) → (2 · 3) ∈ ℤ) | |
33 | 16, 17, 32 | mp2an 692 | . . . 4 ⊢ (2 · 3) ∈ ℤ |
34 | zmulcl 12664 | . . . . 5 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ) → (3 · 2) ∈ ℤ) | |
35 | 17, 16, 34 | mp2an 692 | . . . 4 ⊢ (3 · 2) ∈ ℤ |
36 | zmulcl 12664 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ 6 ∈ ℤ) → (2 · 6) ∈ ℤ) | |
37 | 16, 19, 36 | mp2an 692 | . . . 4 ⊢ (2 · 6) ∈ ℤ |
38 | zmulcl 12664 | . . . . 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 48205 | . . . 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 2763 | . 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 2773 | 1 ⊢ ((2 ∙ 𝐴) − (3 ∙ 𝐵)) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 {cpr 4633 〈cop 4637 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 − cmin 11490 2c2 12319 3c3 12320 4c4 12321 6c6 12323 ℤcz 12611 ·𝑠 cvsca 17302 -gcsg 18966 ℤringczring 21475 freeLMod cfrlm 21784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17488 df-prds 17494 df-pws 17496 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-subrng 20563 df-subrg 20587 df-lmod 20877 df-lss 20948 df-sra 21190 df-rgmod 21191 df-cnfld 21383 df-zring 21476 df-dsmm 21770 df-frlm 21785 |
This theorem is referenced by: (None) |
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