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Mirrors > Home > ILE Home > Th. List > ringidcl | GIF version |
Description: The unity element of a ring belongs to the base set of the ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) |
Ref | Expression |
---|---|
ringidcl.b | ⊢ 𝐵 = (Base‘𝑅) |
ringidcl.u | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
ringidcl | ⊢ (𝑅 ∈ Ring → 1 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2193 | . . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
2 | 1 | ringmgp 13501 | . . 3 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
3 | eqid 2193 | . . . 4 ⊢ (Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅)) | |
4 | eqid 2193 | . . . 4 ⊢ (0g‘(mulGrp‘𝑅)) = (0g‘(mulGrp‘𝑅)) | |
5 | 3, 4 | mndidcl 13014 | . . 3 ⊢ ((mulGrp‘𝑅) ∈ Mnd → (0g‘(mulGrp‘𝑅)) ∈ (Base‘(mulGrp‘𝑅))) |
6 | 2, 5 | syl 14 | . 2 ⊢ (𝑅 ∈ Ring → (0g‘(mulGrp‘𝑅)) ∈ (Base‘(mulGrp‘𝑅))) |
7 | ringidcl.u | . . 3 ⊢ 1 = (1r‘𝑅) | |
8 | 1, 7 | ringidvalg 13460 | . 2 ⊢ (𝑅 ∈ Ring → 1 = (0g‘(mulGrp‘𝑅))) |
9 | ringidcl.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
10 | 1, 9 | mgpbasg 13425 | . 2 ⊢ (𝑅 ∈ Ring → 𝐵 = (Base‘(mulGrp‘𝑅))) |
11 | 6, 8, 10 | 3eltr4d 2277 | 1 ⊢ (𝑅 ∈ Ring → 1 ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2164 ‘cfv 5255 Basecbs 12621 0gc0g 12870 Mndcmnd 13000 mulGrpcmgp 13419 1rcur 13458 Ringcrg 13495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-addcom 7974 ax-addass 7976 ax-i2m1 7979 ax-0lt1 7980 ax-0id 7982 ax-rnegex 7983 ax-pre-ltirr 7986 ax-pre-ltadd 7990 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-br 4031 df-opab 4092 df-mpt 4093 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-pnf 8058 df-mnf 8059 df-ltxr 8061 df-inn 8985 df-2 9043 df-3 9044 df-ndx 12624 df-slot 12625 df-base 12627 df-sets 12628 df-plusg 12711 df-mulr 12712 df-0g 12872 df-mgm 12942 df-sgrp 12988 df-mnd 13001 df-mgp 13420 df-ur 13459 df-ring 13497 |
This theorem is referenced by: ringid 13525 ringo2times 13527 ringcom 13530 ringnegl 13550 ringnegr 13551 ringmneg1 13552 ringmneg2 13553 ringressid 13562 imasring 13563 opprring 13578 dvdsrid 13599 dvdsrneg 13602 1unit 13606 ringinvdv 13644 elrhmunit 13676 isnzr2 13683 subrgid 13722 rrgnz 13767 lmod1cl 13814 lmodvsneg 13830 lmodsubvs 13842 lmodsubdi 13843 lmodsubdir 13844 lmodprop2d 13847 rmodislmod 13850 lssvnegcl 13875 mulgrhm 14108 zrhmulg 14119 |
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