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Mirrors > Home > MPE Home > Th. List > Mathboxes > zlmodzxzsubm | Structured version Visualization version GIF version |
Description: The subtraction of the ℤ-module ℤ × ℤ expressed as addition. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.) |
Ref | Expression |
---|---|
zlmodzxz.z | ⊢ 𝑍 = (ℤring freeLMod {0, 1}) |
zlmodzxzsub.m | ⊢ − = (-g‘𝑍) |
Ref | Expression |
---|---|
zlmodzxzsubm | ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({〈0, 𝐴〉, 〈1, 𝐶〉} − {〈0, 𝐵〉, 〈1, 𝐷〉}) = ({〈0, 𝐴〉, 〈1, 𝐶〉} (+g‘𝑍)(-1( ·𝑠 ‘𝑍){〈0, 𝐵〉, 〈1, 𝐷〉}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zlmodzxz.z | . . . . . 6 ⊢ 𝑍 = (ℤring freeLMod {0, 1}) | |
2 | 1 | zlmodzxzlmod 44395 | . . . . 5 ⊢ (𝑍 ∈ LMod ∧ ℤring = (Scalar‘𝑍)) |
3 | 2 | simpli 486 | . . . 4 ⊢ 𝑍 ∈ LMod |
4 | 3 | a1i 11 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → 𝑍 ∈ LMod) |
5 | 1 | zlmodzxzel 44396 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) → {〈0, 𝐴〉, 〈1, 𝐶〉} ∈ (Base‘𝑍)) |
6 | 5 | ad2ant2r 745 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → {〈0, 𝐴〉, 〈1, 𝐶〉} ∈ (Base‘𝑍)) |
7 | 1 | zlmodzxzel 44396 | . . . 4 ⊢ ((𝐵 ∈ ℤ ∧ 𝐷 ∈ ℤ) → {〈0, 𝐵〉, 〈1, 𝐷〉} ∈ (Base‘𝑍)) |
8 | 7 | ad2ant2l 744 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → {〈0, 𝐵〉, 〈1, 𝐷〉} ∈ (Base‘𝑍)) |
9 | eqid 2821 | . . . 4 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
10 | eqid 2821 | . . . 4 ⊢ (+g‘𝑍) = (+g‘𝑍) | |
11 | zlmodzxzsub.m | . . . 4 ⊢ − = (-g‘𝑍) | |
12 | 2 | simpri 488 | . . . 4 ⊢ ℤring = (Scalar‘𝑍) |
13 | eqid 2821 | . . . 4 ⊢ ( ·𝑠 ‘𝑍) = ( ·𝑠 ‘𝑍) | |
14 | eqid 2821 | . . . 4 ⊢ (invg‘ℤring) = (invg‘ℤring) | |
15 | zring1 20622 | . . . 4 ⊢ 1 = (1r‘ℤring) | |
16 | 9, 10, 11, 12, 13, 14, 15 | lmodvsubval2 19683 | . . 3 ⊢ ((𝑍 ∈ LMod ∧ {〈0, 𝐴〉, 〈1, 𝐶〉} ∈ (Base‘𝑍) ∧ {〈0, 𝐵〉, 〈1, 𝐷〉} ∈ (Base‘𝑍)) → ({〈0, 𝐴〉, 〈1, 𝐶〉} − {〈0, 𝐵〉, 〈1, 𝐷〉}) = ({〈0, 𝐴〉, 〈1, 𝐶〉} (+g‘𝑍)(((invg‘ℤring)‘1)( ·𝑠 ‘𝑍){〈0, 𝐵〉, 〈1, 𝐷〉}))) |
17 | 4, 6, 8, 16 | syl3anc 1367 | . 2 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({〈0, 𝐴〉, 〈1, 𝐶〉} − {〈0, 𝐵〉, 〈1, 𝐷〉}) = ({〈0, 𝐴〉, 〈1, 𝐶〉} (+g‘𝑍)(((invg‘ℤring)‘1)( ·𝑠 ‘𝑍){〈0, 𝐵〉, 〈1, 𝐷〉}))) |
18 | 1z 12006 | . . . . . 6 ⊢ 1 ∈ ℤ | |
19 | zringinvg 20628 | . . . . . 6 ⊢ (1 ∈ ℤ → -1 = ((invg‘ℤring)‘1)) | |
20 | 18, 19 | mp1i 13 | . . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → -1 = ((invg‘ℤring)‘1)) |
21 | 20 | eqcomd 2827 | . . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ((invg‘ℤring)‘1) = -1) |
22 | 21 | oveq1d 7165 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → (((invg‘ℤring)‘1)( ·𝑠 ‘𝑍){〈0, 𝐵〉, 〈1, 𝐷〉}) = (-1( ·𝑠 ‘𝑍){〈0, 𝐵〉, 〈1, 𝐷〉})) |
23 | 22 | oveq2d 7166 | . 2 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({〈0, 𝐴〉, 〈1, 𝐶〉} (+g‘𝑍)(((invg‘ℤring)‘1)( ·𝑠 ‘𝑍){〈0, 𝐵〉, 〈1, 𝐷〉})) = ({〈0, 𝐴〉, 〈1, 𝐶〉} (+g‘𝑍)(-1( ·𝑠 ‘𝑍){〈0, 𝐵〉, 〈1, 𝐷〉}))) |
24 | 17, 23 | eqtrd 2856 | 1 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({〈0, 𝐴〉, 〈1, 𝐶〉} − {〈0, 𝐵〉, 〈1, 𝐷〉}) = ({〈0, 𝐴〉, 〈1, 𝐶〉} (+g‘𝑍)(-1( ·𝑠 ‘𝑍){〈0, 𝐵〉, 〈1, 𝐷〉}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {cpr 4563 〈cop 4567 ‘cfv 6350 (class class class)co 7150 0cc0 10531 1c1 10532 -cneg 10865 ℤcz 11975 Basecbs 16477 +gcplusg 16559 Scalarcsca 16562 ·𝑠 cvsca 16563 invgcminusg 18098 -gcsg 18099 LModclmod 19628 ℤringzring 20611 freeLMod cfrlm 20884 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-0g 16709 df-prds 16715 df-pws 16717 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-cmn 18902 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-subrg 19527 df-lmod 19630 df-lss 19698 df-sra 19938 df-rgmod 19939 df-cnfld 20540 df-zring 20612 df-dsmm 20870 df-frlm 20885 |
This theorem is referenced by: (None) |
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