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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 11631 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 − 𝐵) ∈ ℤ) | |
2 | simpr 479 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℤ) | |
3 | 1, 2 | jca 555 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 − 𝐵) ∈ ℤ ∧ 𝐵 ∈ ℤ)) |
4 | zsubcl 11631 | . . . . 5 ⊢ ((𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) → (𝐶 − 𝐷) ∈ ℤ) | |
5 | simpr 479 | . . . . 5 ⊢ ((𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) → 𝐷 ∈ ℤ) | |
6 | 4, 5 | jca 555 | . . . 4 ⊢ ((𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) → ((𝐶 − 𝐷) ∈ ℤ ∧ 𝐷 ∈ ℤ)) |
7 | zlmodzxz.z | . . . . 5 ⊢ 𝑍 = (ℤring freeLMod {0, 1}) | |
8 | eqid 2760 | . . . . 5 ⊢ (+g‘𝑍) = (+g‘𝑍) | |
9 | 7, 8 | zlmodzxzadd 42664 | . . . 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 495 | . . 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 11594 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ) | |
12 | zcn 11594 | . . . . . . 7 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℂ) | |
13 | npcan 10502 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) + 𝐵) = 𝐴) | |
14 | 11, 12, 13 | syl2an 495 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 − 𝐵) + 𝐵) = 𝐴) |
15 | 14 | adantr 472 | . . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ((𝐴 − 𝐵) + 𝐵) = 𝐴) |
16 | 15 | opeq2d 4560 | . . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 〈0, ((𝐴 − 𝐵) + 𝐵)〉 = 〈0, 𝐴〉) |
17 | zcn 11594 | . . . . . . 7 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ ℂ) | |
18 | zcn 11594 | . . . . . . 7 ⊢ (𝐷 ∈ ℤ → 𝐷 ∈ ℂ) | |
19 | npcan 10502 | . . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) → ((𝐶 − 𝐷) + 𝐷) = 𝐶) | |
20 | 17, 18, 19 | syl2an 495 | . . . . . 6 ⊢ ((𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) → ((𝐶 − 𝐷) + 𝐷) = 𝐶) |
21 | 20 | adantl 473 | . . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ((𝐶 − 𝐷) + 𝐷) = 𝐶) |
22 | 21 | opeq2d 4560 | . . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 〈1, ((𝐶 − 𝐷) + 𝐷)〉 = 〈1, 𝐶〉) |
23 | 16, 22 | preq12d 4420 | . . 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 2794 | . 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 42660 | . . . 4 ⊢ (𝑍 ∈ LMod ∧ ℤring = (Scalar‘𝑍)) |
26 | lmodgrp 19092 | . . . . 5 ⊢ (𝑍 ∈ LMod → 𝑍 ∈ Grp) | |
27 | 26 | adantr 472 | . . . 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 42661 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) → {〈0, 𝐴〉, 〈1, 𝐶〉} ∈ (Base‘𝑍)) |
30 | 29 | ad2ant2r 800 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → {〈0, 𝐴〉, 〈1, 𝐶〉} ∈ (Base‘𝑍)) |
31 | 7 | zlmodzxzel 42661 | . . . 4 ⊢ ((𝐵 ∈ ℤ ∧ 𝐷 ∈ ℤ) → {〈0, 𝐵〉, 〈1, 𝐷〉} ∈ (Base‘𝑍)) |
32 | 2, 5, 31 | syl2an 495 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → {〈0, 𝐵〉, 〈1, 𝐷〉} ∈ (Base‘𝑍)) |
33 | 7 | zlmodzxzel 42661 | . . . 4 ⊢ (((𝐴 − 𝐵) ∈ ℤ ∧ (𝐶 − 𝐷) ∈ ℤ) → {〈0, (𝐴 − 𝐵)〉, 〈1, (𝐶 − 𝐷)〉} ∈ (Base‘𝑍)) |
34 | 1, 4, 33 | syl2an 495 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → {〈0, (𝐴 − 𝐵)〉, 〈1, (𝐶 − 𝐷)〉} ∈ (Base‘𝑍)) |
35 | eqid 2760 | . . . 4 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
36 | zlmodzxzsub.m | . . . 4 ⊢ − = (-g‘𝑍) | |
37 | 35, 8, 36 | grpsubadd 17724 | . . 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 1479 | . 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 247 | 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 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 {cpr 4323 〈cop 4327 ‘cfv 6049 (class class class)co 6814 ℂcc 10146 0cc0 10148 1c1 10149 + caddc 10151 − cmin 10478 ℤcz 11589 Basecbs 16079 +gcplusg 16163 Scalarcsca 16166 Grpcgrp 17643 -gcsg 17645 LModclmod 19085 ℤringzring 20040 freeLMod cfrlm 20312 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 ax-addf 10227 ax-mulf 10228 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-of 7063 df-om 7232 df-1st 7334 df-2nd 7335 df-supp 7465 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-oadd 7734 df-er 7913 df-map 8027 df-ixp 8077 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-fsupp 8443 df-sup 8515 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-nn 11233 df-2 11291 df-3 11292 df-4 11293 df-5 11294 df-6 11295 df-7 11296 df-8 11297 df-9 11298 df-n0 11505 df-z 11590 df-dec 11706 df-uz 11900 df-fz 12540 df-struct 16081 df-ndx 16082 df-slot 16083 df-base 16085 df-sets 16086 df-ress 16087 df-plusg 16176 df-mulr 16177 df-starv 16178 df-sca 16179 df-vsca 16180 df-ip 16181 df-tset 16182 df-ple 16183 df-ds 16186 df-unif 16187 df-hom 16188 df-cco 16189 df-0g 16324 df-prds 16330 df-pws 16332 df-mgm 17463 df-sgrp 17505 df-mnd 17516 df-grp 17646 df-minusg 17647 df-sbg 17648 df-subg 17812 df-cmn 18415 df-mgp 18710 df-ur 18722 df-ring 18769 df-cring 18770 df-subrg 19000 df-lmod 19087 df-lss 19155 df-sra 19394 df-rgmod 19395 df-cnfld 19969 df-zring 20041 df-dsmm 20298 df-frlm 20313 |
This theorem is referenced by: zlmodzxzequa 42813 |
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