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Mathbox for Alexander van der Vekens |
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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 12298 | . . . . . . . 8 ⊢ 3 ∈ ℂ | |
2 | 1 | 2timesi 12355 | . . . . . . 7 ⊢ (2 · 3) = (3 + 3) |
3 | 3p3e6 12369 | . . . . . . 7 ⊢ (3 + 3) = 6 | |
4 | 2, 3 | eqtri 2759 | . . . . . 6 ⊢ (2 · 3) = 6 |
5 | 3t2e6 12383 | . . . . . 6 ⊢ (3 · 2) = 6 | |
6 | 4, 5 | oveq12i 7424 | . . . . 5 ⊢ ((2 · 3) − (3 · 2)) = (6 − 6) |
7 | 6cn 12308 | . . . . . 6 ⊢ 6 ∈ ℂ | |
8 | 7 | subidi 11536 | . . . . 5 ⊢ (6 − 6) = 0 |
9 | 6, 8 | eqtri 2759 | . . . 4 ⊢ ((2 · 3) − (3 · 2)) = 0 |
10 | 9 | opeq2i 4877 | . . 3 ⊢ ⟨0, ((2 · 3) − (3 · 2))⟩ = ⟨0, 0⟩ |
11 | 2t6m3t4e0 47113 | . . . 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 7423 | . . . . 5 ⊢ (2 ∙ 𝐴) = (2 ∙ {⟨0, 3⟩, ⟨1, 6⟩}) |
16 | 2z 12599 | . . . . . 6 ⊢ 2 ∈ ℤ | |
17 | 3z 12600 | . . . . . 6 ⊢ 3 ∈ ℤ | |
18 | 6nn 12306 | . . . . . . 7 ⊢ 6 ∈ ℕ | |
19 | 18 | nnzi 12591 | . . . . . 6 ⊢ 6 ∈ ℤ |
20 | zlmodzxzequa.z | . . . . . . 7 ⊢ 𝑍 = (ℤring freeLMod {0, 1}) | |
21 | zlmodzxzequa.t | . . . . . . 7 ⊢ ∙ = ( ·𝑠 ‘𝑍) | |
22 | 20, 21 | zlmodzxzscm 47122 | . . . . . 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 2759 | . . . 4 ⊢ (2 ∙ 𝐴) = {⟨0, (2 · 3)⟩, ⟨1, (2 · 6)⟩} |
25 | zlmodzxzequa.b | . . . . . 6 ⊢ 𝐵 = {⟨0, 2⟩, ⟨1, 4⟩} | |
26 | 25 | oveq2i 7423 | . . . . 5 ⊢ (3 ∙ 𝐵) = (3 ∙ {⟨0, 2⟩, ⟨1, 4⟩}) |
27 | 4z 12601 | . . . . . 6 ⊢ 4 ∈ ℤ | |
28 | 20, 21 | zlmodzxzscm 47122 | . . . . . 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 2759 | . . . 4 ⊢ (3 ∙ 𝐵) = {⟨0, (3 · 2)⟩, ⟨1, (3 · 4)⟩} |
31 | 24, 30 | oveq12i 7424 | . . 3 ⊢ ((2 ∙ 𝐴) − (3 ∙ 𝐵)) = ({⟨0, (2 · 3)⟩, ⟨1, (2 · 6)⟩} − {⟨0, (3 · 2)⟩, ⟨1, (3 · 4)⟩}) |
32 | zmulcl 12616 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ 3 ∈ ℤ) → (2 · 3) ∈ ℤ) | |
33 | 16, 17, 32 | mp2an 689 | . . . 4 ⊢ (2 · 3) ∈ ℤ |
34 | zmulcl 12616 | . . . . 5 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ) → (3 · 2) ∈ ℤ) | |
35 | 17, 16, 34 | mp2an 689 | . . . 4 ⊢ (3 · 2) ∈ ℤ |
36 | zmulcl 12616 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ 6 ∈ ℤ) → (2 · 6) ∈ ℤ) | |
37 | 16, 19, 36 | mp2an 689 | . . . 4 ⊢ (2 · 6) ∈ ℤ |
38 | zmulcl 12616 | . . . . 5 ⊢ ((3 ∈ ℤ ∧ 4 ∈ ℤ) → (3 · 4) ∈ ℤ) | |
39 | 17, 27, 38 | mp2an 689 | . . . 4 ⊢ (3 · 4) ∈ ℤ |
40 | zlmodzxzequa.m | . . . . 5 ⊢ − = (-g‘𝑍) | |
41 | 20, 40 | zlmodzxzsub 47125 | . . . 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 690 | . . 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 2759 | . 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 2769 | 1 ⊢ ((2 ∙ 𝐴) − (3 ∙ 𝐵)) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 {cpr 4630 ⟨cop 4634 ‘cfv 6543 (class class class)co 7412 0cc0 11114 1c1 11115 + caddc 11117 · cmul 11119 − cmin 11449 2c2 12272 3c3 12273 4c4 12274 6c6 12276 ℤcz 12563 ·𝑠 cvsca 17206 -gcsg 18858 ℤringczring 21218 freeLMod cfrlm 21521 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-addf 11193 ax-mulf 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9366 df-sup 9441 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-fz 13490 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-0g 17392 df-prds 17398 df-pws 17400 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-grp 18859 df-minusg 18860 df-sbg 18861 df-subg 19040 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-subrng 20435 df-subrg 20460 df-lmod 20617 df-lss 20688 df-sra 20931 df-rgmod 20932 df-cnfld 21146 df-zring 21219 df-dsmm 21507 df-frlm 21522 |
This theorem is referenced by: (None) |
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