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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 12244 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 − 𝐵) ∈ ℤ) | |
2 | simpr 488 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℤ) | |
3 | 1, 2 | jca 515 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 − 𝐵) ∈ ℤ ∧ 𝐵 ∈ ℤ)) |
4 | zsubcl 12244 | . . . . 5 ⊢ ((𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) → (𝐶 − 𝐷) ∈ ℤ) | |
5 | simpr 488 | . . . . 5 ⊢ ((𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) → 𝐷 ∈ ℤ) | |
6 | 4, 5 | jca 515 | . . . 4 ⊢ ((𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) → ((𝐶 − 𝐷) ∈ ℤ ∧ 𝐷 ∈ ℤ)) |
7 | zlmodzxz.z | . . . . 5 ⊢ 𝑍 = (ℤring freeLMod {0, 1}) | |
8 | eqid 2738 | . . . . 5 ⊢ (+g‘𝑍) = (+g‘𝑍) | |
9 | 7, 8 | zlmodzxzadd 45398 | . . . 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 599 | . . 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 12206 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ) | |
12 | zcn 12206 | . . . . . . 7 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℂ) | |
13 | npcan 11112 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) + 𝐵) = 𝐴) | |
14 | 11, 12, 13 | syl2an 599 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 − 𝐵) + 𝐵) = 𝐴) |
15 | 14 | adantr 484 | . . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ((𝐴 − 𝐵) + 𝐵) = 𝐴) |
16 | 15 | opeq2d 4806 | . . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 〈0, ((𝐴 − 𝐵) + 𝐵)〉 = 〈0, 𝐴〉) |
17 | zcn 12206 | . . . . . . 7 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ ℂ) | |
18 | zcn 12206 | . . . . . . 7 ⊢ (𝐷 ∈ ℤ → 𝐷 ∈ ℂ) | |
19 | npcan 11112 | . . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) → ((𝐶 − 𝐷) + 𝐷) = 𝐶) | |
20 | 17, 18, 19 | syl2an 599 | . . . . . 6 ⊢ ((𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) → ((𝐶 − 𝐷) + 𝐷) = 𝐶) |
21 | 20 | adantl 485 | . . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ((𝐶 − 𝐷) + 𝐷) = 𝐶) |
22 | 21 | opeq2d 4806 | . . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 〈1, ((𝐶 − 𝐷) + 𝐷)〉 = 〈1, 𝐶〉) |
23 | 16, 22 | preq12d 4672 | . . 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 2778 | . 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 45394 | . . . 4 ⊢ (𝑍 ∈ LMod ∧ ℤring = (Scalar‘𝑍)) |
26 | lmodgrp 19934 | . . . . 5 ⊢ (𝑍 ∈ LMod → 𝑍 ∈ Grp) | |
27 | 26 | adantr 484 | . . . 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 45395 | . . . 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 45395 | . . . 4 ⊢ ((𝐵 ∈ ℤ ∧ 𝐷 ∈ ℤ) → {〈0, 𝐵〉, 〈1, 𝐷〉} ∈ (Base‘𝑍)) |
32 | 2, 5, 31 | syl2an 599 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → {〈0, 𝐵〉, 〈1, 𝐷〉} ∈ (Base‘𝑍)) |
33 | 7 | zlmodzxzel 45395 | . . . 4 ⊢ (((𝐴 − 𝐵) ∈ ℤ ∧ (𝐶 − 𝐷) ∈ ℤ) → {〈0, (𝐴 − 𝐵)〉, 〈1, (𝐶 − 𝐷)〉} ∈ (Base‘𝑍)) |
34 | 1, 4, 33 | syl2an 599 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → {〈0, (𝐴 − 𝐵)〉, 〈1, (𝐶 − 𝐷)〉} ∈ (Base‘𝑍)) |
35 | eqid 2738 | . . . 4 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
36 | zlmodzxzsub.m | . . . 4 ⊢ − = (-g‘𝑍) | |
37 | 35, 8, 36 | grpsubadd 18479 | . . 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 1374 | . 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 260 | 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 209 ∧ wa 399 = wceq 1543 ∈ wcel 2111 {cpr 4558 〈cop 4562 ‘cfv 6398 (class class class)co 7232 ℂcc 10752 0cc0 10754 1c1 10755 + caddc 10757 − cmin 11087 ℤcz 12201 Basecbs 16788 +gcplusg 16830 Scalarcsca 16833 Grpcgrp 18393 -gcsg 18395 LModclmod 19927 ℤringzring 20463 freeLMod cfrlm 20736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5194 ax-sep 5207 ax-nul 5214 ax-pow 5273 ax-pr 5337 ax-un 7542 ax-cnex 10810 ax-resscn 10811 ax-1cn 10812 ax-icn 10813 ax-addcl 10814 ax-addrcl 10815 ax-mulcl 10816 ax-mulrcl 10817 ax-mulcom 10818 ax-addass 10819 ax-mulass 10820 ax-distr 10821 ax-i2m1 10822 ax-1ne0 10823 ax-1rid 10824 ax-rnegex 10825 ax-rrecex 10826 ax-cnre 10827 ax-pre-lttri 10828 ax-pre-lttrn 10829 ax-pre-ltadd 10830 ax-pre-mulgt0 10831 ax-addf 10833 ax-mulf 10834 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3423 df-sbc 3710 df-csb 3827 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4253 df-if 4455 df-pw 4530 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4835 df-iun 4921 df-br 5069 df-opab 5131 df-mpt 5151 df-tr 5177 df-id 5470 df-eprel 5475 df-po 5483 df-so 5484 df-fr 5524 df-we 5526 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-res 5578 df-ima 5579 df-pred 6176 df-ord 6234 df-on 6235 df-lim 6236 df-suc 6237 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-f1 6403 df-fo 6404 df-f1o 6405 df-fv 6406 df-riota 7189 df-ov 7235 df-oprab 7236 df-mpo 7237 df-of 7488 df-om 7664 df-1st 7780 df-2nd 7781 df-supp 7925 df-wrecs 8068 df-recs 8129 df-rdg 8167 df-1o 8223 df-er 8412 df-map 8531 df-ixp 8600 df-en 8648 df-dom 8649 df-sdom 8650 df-fin 8651 df-fsupp 9011 df-sup 9083 df-pnf 10894 df-mnf 10895 df-xr 10896 df-ltxr 10897 df-le 10898 df-sub 11089 df-neg 11090 df-nn 11856 df-2 11918 df-3 11919 df-4 11920 df-5 11921 df-6 11922 df-7 11923 df-8 11924 df-9 11925 df-n0 12116 df-z 12202 df-dec 12319 df-uz 12464 df-fz 13121 df-struct 16728 df-sets 16745 df-slot 16763 df-ndx 16773 df-base 16789 df-ress 16813 df-plusg 16843 df-mulr 16844 df-starv 16845 df-sca 16846 df-vsca 16847 df-ip 16848 df-tset 16849 df-ple 16850 df-ds 16852 df-unif 16853 df-hom 16854 df-cco 16855 df-0g 16974 df-prds 16980 df-pws 16982 df-mgm 18142 df-sgrp 18191 df-mnd 18202 df-grp 18396 df-minusg 18397 df-sbg 18398 df-subg 18568 df-cmn 19200 df-mgp 19533 df-ur 19545 df-ring 19592 df-cring 19593 df-subrg 19826 df-lmod 19929 df-lss 19997 df-sra 20237 df-rgmod 20238 df-cnfld 20392 df-zring 20464 df-dsmm 20722 df-frlm 20737 |
This theorem is referenced by: zlmodzxzequa 45541 |
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