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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 44409 | . . . . 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 44410 | . . . 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 44410 | . . . 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 2823 | . . . 4 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
10 | eqid 2823 | . . . 4 ⊢ (+g‘𝑍) = (+g‘𝑍) | |
11 | zlmodzxzsub.m | . . . 4 ⊢ − = (-g‘𝑍) | |
12 | 2 | simpri 488 | . . . 4 ⊢ ℤring = (Scalar‘𝑍) |
13 | eqid 2823 | . . . 4 ⊢ ( ·𝑠 ‘𝑍) = ( ·𝑠 ‘𝑍) | |
14 | eqid 2823 | . . . 4 ⊢ (invg‘ℤring) = (invg‘ℤring) | |
15 | zring1 20630 | . . . 4 ⊢ 1 = (1r‘ℤring) | |
16 | 9, 10, 11, 12, 13, 14, 15 | lmodvsubval2 19691 | . . 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 12015 | . . . . . 6 ⊢ 1 ∈ ℤ | |
19 | zringinvg 20636 | . . . . . 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 2829 | . . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ((invg‘ℤring)‘1) = -1) |
22 | 21 | oveq1d 7173 | . . 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 7174 | . 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 2858 | 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 1537 ∈ wcel 2114 {cpr 4571 〈cop 4575 ‘cfv 6357 (class class class)co 7158 0cc0 10539 1c1 10540 -cneg 10873 ℤcz 11984 Basecbs 16485 +gcplusg 16567 Scalarcsca 16570 ·𝑠 cvsca 16571 invgcminusg 18106 -gcsg 18107 LModclmod 19636 ℤringzring 20619 freeLMod cfrlm 20892 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-0g 16717 df-prds 16723 df-pws 16725 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-cmn 18910 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-subrg 19535 df-lmod 19638 df-lss 19706 df-sra 19946 df-rgmod 19947 df-cnfld 20548 df-zring 20620 df-dsmm 20878 df-frlm 20893 |
This theorem is referenced by: (None) |
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