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Mirrors > Home > MPE Home > Th. List > zringinvg | Structured version Visualization version GIF version |
Description: The additive inverse of an element of the ring of integers. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.) |
Ref | Expression |
---|---|
zringinvg | ⊢ (𝐴 ∈ ℤ → -𝐴 = ((invg‘ℤring)‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 12322 | . . . 4 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ) | |
2 | 1 | negidd 11320 | . . 3 ⊢ (𝐴 ∈ ℤ → (𝐴 + -𝐴) = 0) |
3 | zringgrp 20673 | . . . 4 ⊢ ℤring ∈ Grp | |
4 | id 22 | . . . 4 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℤ) | |
5 | znegcl 12353 | . . . 4 ⊢ (𝐴 ∈ ℤ → -𝐴 ∈ ℤ) | |
6 | zringbas 20674 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
7 | zringplusg 20675 | . . . . 5 ⊢ + = (+g‘ℤring) | |
8 | zring0 20678 | . . . . 5 ⊢ 0 = (0g‘ℤring) | |
9 | eqid 2738 | . . . . 5 ⊢ (invg‘ℤring) = (invg‘ℤring) | |
10 | 6, 7, 8, 9 | grpinvid1 18628 | . . . 4 ⊢ ((ℤring ∈ Grp ∧ 𝐴 ∈ ℤ ∧ -𝐴 ∈ ℤ) → (((invg‘ℤring)‘𝐴) = -𝐴 ↔ (𝐴 + -𝐴) = 0)) |
11 | 3, 4, 5, 10 | mp3an2i 1465 | . . 3 ⊢ (𝐴 ∈ ℤ → (((invg‘ℤring)‘𝐴) = -𝐴 ↔ (𝐴 + -𝐴) = 0)) |
12 | 2, 11 | mpbird 256 | . 2 ⊢ (𝐴 ∈ ℤ → ((invg‘ℤring)‘𝐴) = -𝐴) |
13 | 12 | eqcomd 2744 | 1 ⊢ (𝐴 ∈ ℤ → -𝐴 = ((invg‘ℤring)‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2106 ‘cfv 6435 (class class class)co 7277 0cc0 10869 + caddc 10872 -cneg 11204 ℤcz 12317 Grpcgrp 18575 invgcminusg 18576 ℤringczring 20668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 ax-addf 10948 ax-mulf 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-iun 4928 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-1o 8295 df-er 8496 df-en 8732 df-dom 8733 df-sdom 8734 df-fin 8735 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-9 12041 df-n0 12232 df-z 12318 df-dec 12436 df-uz 12581 df-fz 13238 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-0g 17150 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-grp 18578 df-minusg 18579 df-subg 18750 df-cmn 19386 df-mgp 19719 df-ur 19736 df-ring 19783 df-cring 19784 df-subrg 20020 df-cnfld 20596 df-zring 20669 |
This theorem is referenced by: zrhpsgnodpm 20795 zlmodzxzsubm 45662 |
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