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Mirrors > Home > MPE Home > Th. List > ringidval | Structured version Visualization version GIF version |
Description: The value of the unity element of a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) |
Ref | Expression |
---|---|
ringidval.g | ⊢ 𝐺 = (mulGrp‘𝑅) |
ringidval.u | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
ringidval | ⊢ 1 = (0g‘𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ur 18710 | . . . . 5 ⊢ 1r = (0g ∘ mulGrp) | |
2 | 1 | fveq1i 6333 | . . . 4 ⊢ (1r‘𝑅) = ((0g ∘ mulGrp)‘𝑅) |
3 | fnmgp 18699 | . . . . 5 ⊢ mulGrp Fn V | |
4 | fvco2 6415 | . . . . 5 ⊢ ((mulGrp Fn V ∧ 𝑅 ∈ V) → ((0g ∘ mulGrp)‘𝑅) = (0g‘(mulGrp‘𝑅))) | |
5 | 3, 4 | mpan 670 | . . . 4 ⊢ (𝑅 ∈ V → ((0g ∘ mulGrp)‘𝑅) = (0g‘(mulGrp‘𝑅))) |
6 | 2, 5 | syl5eq 2817 | . . 3 ⊢ (𝑅 ∈ V → (1r‘𝑅) = (0g‘(mulGrp‘𝑅))) |
7 | 0g0 17471 | . . . 4 ⊢ ∅ = (0g‘∅) | |
8 | fvprc 6326 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (1r‘𝑅) = ∅) | |
9 | fvprc 6326 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (mulGrp‘𝑅) = ∅) | |
10 | 9 | fveq2d 6336 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (0g‘(mulGrp‘𝑅)) = (0g‘∅)) |
11 | 7, 8, 10 | 3eqtr4a 2831 | . . 3 ⊢ (¬ 𝑅 ∈ V → (1r‘𝑅) = (0g‘(mulGrp‘𝑅))) |
12 | 6, 11 | pm2.61i 176 | . 2 ⊢ (1r‘𝑅) = (0g‘(mulGrp‘𝑅)) |
13 | ringidval.u | . 2 ⊢ 1 = (1r‘𝑅) | |
14 | ringidval.g | . . 3 ⊢ 𝐺 = (mulGrp‘𝑅) | |
15 | 14 | fveq2i 6335 | . 2 ⊢ (0g‘𝐺) = (0g‘(mulGrp‘𝑅)) |
16 | 12, 13, 15 | 3eqtr4i 2803 | 1 ⊢ 1 = (0g‘𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1631 ∈ wcel 2145 Vcvv 3351 ∅c0 4063 ∘ ccom 5253 Fn wfn 6026 ‘cfv 6031 0gc0g 16308 mulGrpcmgp 18697 1rcur 18709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-fv 6039 df-ov 6796 df-slot 16068 df-base 16070 df-0g 16310 df-mgp 18698 df-ur 18710 |
This theorem is referenced by: dfur2 18712 srgidcl 18726 srgidmlem 18728 issrgid 18731 srgpcomp 18740 srg1expzeq1 18747 srgbinom 18753 ringidcl 18776 ringidmlem 18778 isringid 18781 prds1 18822 oppr1 18842 unitsubm 18878 rngidpropd 18903 dfrhm2 18927 isrhm2d 18938 rhm1 18940 subrgsubm 19003 issubrg3 19018 assamulgscmlem1 19563 mplcoe3 19681 mplcoe5 19683 mplbas2 19685 evlslem1 19730 ply1scltm 19866 lply1binomsc 19892 evls1gsummul 19905 evl1gsummul 19939 cnfldexp 19994 expmhm 20030 nn0srg 20031 rge0srg 20032 madetsumid 20485 mat1mhm 20508 scmatmhm 20558 mdet0pr 20616 mdetunilem7 20642 smadiadetlem4 20694 mat2pmatmhm 20758 pm2mpmhm 20845 chfacfscmulgsum 20885 chfacfpmmulgsum 20889 cpmadugsumlemF 20901 efsubm 24518 amgmlem 24937 amgm 24938 wilthlem2 25016 wilthlem3 25017 dchrelbas3 25184 dchrzrh1 25190 dchrmulcl 25195 dchrn0 25196 dchrinvcl 25199 dchrfi 25201 dchrabs 25206 sumdchr2 25216 rpvmasum2 25422 psgnid 30187 iistmd 30288 isdomn3 38308 mon1psubm 38310 deg1mhm 38311 c0rhm 42440 c0rnghm 42441 amgmwlem 43079 amgmlemALT 43080 |
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