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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > zlmodzxzsub | Structured version Visualization version GIF version |
Description: The subtraction of the ℤ-module ℤ × ℤ. (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.) |
Ref | Expression |
---|---|
zlmodzxz.z | ⊢ 𝑍 = (ℤring freeLMod {0, 1}) |
zlmodzxzsub.m | ⊢ − = (-g‘𝑍) |
Ref | Expression |
---|---|
zlmodzxzsub | ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({〈0, 𝐴〉, 〈1, 𝐶〉} − {〈0, 𝐵〉, 〈1, 𝐷〉}) = {〈0, (𝐴 − 𝐵)〉, 〈1, (𝐶 − 𝐷)〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zsubcl 12656 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 − 𝐵) ∈ ℤ) | |
2 | simpr 484 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℤ) | |
3 | 1, 2 | jca 511 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 − 𝐵) ∈ ℤ ∧ 𝐵 ∈ ℤ)) |
4 | zsubcl 12656 | . . . . 5 ⊢ ((𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) → (𝐶 − 𝐷) ∈ ℤ) | |
5 | simpr 484 | . . . . 5 ⊢ ((𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) → 𝐷 ∈ ℤ) | |
6 | 4, 5 | jca 511 | . . . 4 ⊢ ((𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) → ((𝐶 − 𝐷) ∈ ℤ ∧ 𝐷 ∈ ℤ)) |
7 | zlmodzxz.z | . . . . 5 ⊢ 𝑍 = (ℤring freeLMod {0, 1}) | |
8 | eqid 2734 | . . . . 5 ⊢ (+g‘𝑍) = (+g‘𝑍) | |
9 | 7, 8 | zlmodzxzadd 48202 | . . . 4 ⊢ ((((𝐴 − 𝐵) ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐶 − 𝐷) ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({〈0, (𝐴 − 𝐵)〉, 〈1, (𝐶 − 𝐷)〉} (+g‘𝑍){〈0, 𝐵〉, 〈1, 𝐷〉}) = {〈0, ((𝐴 − 𝐵) + 𝐵)〉, 〈1, ((𝐶 − 𝐷) + 𝐷)〉}) |
10 | 3, 6, 9 | syl2an 596 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({〈0, (𝐴 − 𝐵)〉, 〈1, (𝐶 − 𝐷)〉} (+g‘𝑍){〈0, 𝐵〉, 〈1, 𝐷〉}) = {〈0, ((𝐴 − 𝐵) + 𝐵)〉, 〈1, ((𝐶 − 𝐷) + 𝐷)〉}) |
11 | zcn 12615 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ) | |
12 | zcn 12615 | . . . . . . 7 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℂ) | |
13 | npcan 11514 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) + 𝐵) = 𝐴) | |
14 | 11, 12, 13 | syl2an 596 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 − 𝐵) + 𝐵) = 𝐴) |
15 | 14 | adantr 480 | . . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ((𝐴 − 𝐵) + 𝐵) = 𝐴) |
16 | 15 | opeq2d 4884 | . . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 〈0, ((𝐴 − 𝐵) + 𝐵)〉 = 〈0, 𝐴〉) |
17 | zcn 12615 | . . . . . . 7 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ ℂ) | |
18 | zcn 12615 | . . . . . . 7 ⊢ (𝐷 ∈ ℤ → 𝐷 ∈ ℂ) | |
19 | npcan 11514 | . . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) → ((𝐶 − 𝐷) + 𝐷) = 𝐶) | |
20 | 17, 18, 19 | syl2an 596 | . . . . . 6 ⊢ ((𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) → ((𝐶 − 𝐷) + 𝐷) = 𝐶) |
21 | 20 | adantl 481 | . . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ((𝐶 − 𝐷) + 𝐷) = 𝐶) |
22 | 21 | opeq2d 4884 | . . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 〈1, ((𝐶 − 𝐷) + 𝐷)〉 = 〈1, 𝐶〉) |
23 | 16, 22 | preq12d 4745 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → {〈0, ((𝐴 − 𝐵) + 𝐵)〉, 〈1, ((𝐶 − 𝐷) + 𝐷)〉} = {〈0, 𝐴〉, 〈1, 𝐶〉}) |
24 | 10, 23 | eqtrd 2774 | . 2 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({〈0, (𝐴 − 𝐵)〉, 〈1, (𝐶 − 𝐷)〉} (+g‘𝑍){〈0, 𝐵〉, 〈1, 𝐷〉}) = {〈0, 𝐴〉, 〈1, 𝐶〉}) |
25 | 7 | zlmodzxzlmod 48198 | . . . 4 ⊢ (𝑍 ∈ LMod ∧ ℤring = (Scalar‘𝑍)) |
26 | lmodgrp 20881 | . . . . 5 ⊢ (𝑍 ∈ LMod → 𝑍 ∈ Grp) | |
27 | 26 | adantr 480 | . . . 4 ⊢ ((𝑍 ∈ LMod ∧ ℤring = (Scalar‘𝑍)) → 𝑍 ∈ Grp) |
28 | 25, 27 | mp1i 13 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 𝑍 ∈ Grp) |
29 | 7 | zlmodzxzel 48199 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) → {〈0, 𝐴〉, 〈1, 𝐶〉} ∈ (Base‘𝑍)) |
30 | 29 | ad2ant2r 747 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → {〈0, 𝐴〉, 〈1, 𝐶〉} ∈ (Base‘𝑍)) |
31 | 7 | zlmodzxzel 48199 | . . . 4 ⊢ ((𝐵 ∈ ℤ ∧ 𝐷 ∈ ℤ) → {〈0, 𝐵〉, 〈1, 𝐷〉} ∈ (Base‘𝑍)) |
32 | 2, 5, 31 | syl2an 596 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → {〈0, 𝐵〉, 〈1, 𝐷〉} ∈ (Base‘𝑍)) |
33 | 7 | zlmodzxzel 48199 | . . . 4 ⊢ (((𝐴 − 𝐵) ∈ ℤ ∧ (𝐶 − 𝐷) ∈ ℤ) → {〈0, (𝐴 − 𝐵)〉, 〈1, (𝐶 − 𝐷)〉} ∈ (Base‘𝑍)) |
34 | 1, 4, 33 | syl2an 596 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → {〈0, (𝐴 − 𝐵)〉, 〈1, (𝐶 − 𝐷)〉} ∈ (Base‘𝑍)) |
35 | eqid 2734 | . . . 4 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
36 | zlmodzxzsub.m | . . . 4 ⊢ − = (-g‘𝑍) | |
37 | 35, 8, 36 | grpsubadd 19058 | . . 3 ⊢ ((𝑍 ∈ Grp ∧ ({〈0, 𝐴〉, 〈1, 𝐶〉} ∈ (Base‘𝑍) ∧ {〈0, 𝐵〉, 〈1, 𝐷〉} ∈ (Base‘𝑍) ∧ {〈0, (𝐴 − 𝐵)〉, 〈1, (𝐶 − 𝐷)〉} ∈ (Base‘𝑍))) → (({〈0, 𝐴〉, 〈1, 𝐶〉} − {〈0, 𝐵〉, 〈1, 𝐷〉}) = {〈0, (𝐴 − 𝐵)〉, 〈1, (𝐶 − 𝐷)〉} ↔ ({〈0, (𝐴 − 𝐵)〉, 〈1, (𝐶 − 𝐷)〉} (+g‘𝑍){〈0, 𝐵〉, 〈1, 𝐷〉}) = {〈0, 𝐴〉, 〈1, 𝐶〉})) |
38 | 28, 30, 32, 34, 37 | syl13anc 1371 | . 2 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → (({〈0, 𝐴〉, 〈1, 𝐶〉} − {〈0, 𝐵〉, 〈1, 𝐷〉}) = {〈0, (𝐴 − 𝐵)〉, 〈1, (𝐶 − 𝐷)〉} ↔ ({〈0, (𝐴 − 𝐵)〉, 〈1, (𝐶 − 𝐷)〉} (+g‘𝑍){〈0, 𝐵〉, 〈1, 𝐷〉}) = {〈0, 𝐴〉, 〈1, 𝐶〉})) |
39 | 24, 38 | mpbird 257 | 1 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({〈0, 𝐴〉, 〈1, 𝐶〉} − {〈0, 𝐵〉, 〈1, 𝐷〉}) = {〈0, (𝐴 − 𝐵)〉, 〈1, (𝐶 − 𝐷)〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 {cpr 4632 〈cop 4636 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 0cc0 11152 1c1 11153 + caddc 11155 − cmin 11489 ℤcz 12610 Basecbs 17244 +gcplusg 17297 Scalarcsca 17300 Grpcgrp 18963 -gcsg 18965 LModclmod 20874 ℤringczring 21474 freeLMod cfrlm 21783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-addf 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-sup 9479 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17487 df-prds 17493 df-pws 17495 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 df-sbg 18968 df-subg 19153 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-subrng 20562 df-subrg 20586 df-lmod 20876 df-lss 20947 df-sra 21189 df-rgmod 21190 df-cnfld 21382 df-zring 21475 df-dsmm 21769 df-frlm 21784 |
This theorem is referenced by: zlmodzxzequa 48341 |
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