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Mirrors > Home > ILE Home > Th. List > ringidmlem | GIF version |
Description: Lemma for ringlidm 13029 and ringridm 13030. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) |
Ref | Expression |
---|---|
rngidm.b | ⊢ 𝐵 = (Base‘𝑅) |
rngidm.t | ⊢ · = (.r‘𝑅) |
rngidm.u | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
ringidmlem | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (( 1 · 𝑋) = 𝑋 ∧ (𝑋 · 1 ) = 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2177 | . . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
2 | 1 | ringmgp 13008 | . . 3 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
3 | simpr 110 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
4 | rngidm.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
5 | 1, 4 | mgpbasg 12960 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝐵 = (Base‘(mulGrp‘𝑅))) |
6 | 5 | adantr 276 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝐵 = (Base‘(mulGrp‘𝑅))) |
7 | 3, 6 | eleqtrd 2256 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ (Base‘(mulGrp‘𝑅))) |
8 | eqid 2177 | . . . 4 ⊢ (Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅)) | |
9 | eqid 2177 | . . . 4 ⊢ (+g‘(mulGrp‘𝑅)) = (+g‘(mulGrp‘𝑅)) | |
10 | eqid 2177 | . . . 4 ⊢ (0g‘(mulGrp‘𝑅)) = (0g‘(mulGrp‘𝑅)) | |
11 | 8, 9, 10 | mndlrid 12724 | . . 3 ⊢ (((mulGrp‘𝑅) ∈ Mnd ∧ 𝑋 ∈ (Base‘(mulGrp‘𝑅))) → (((0g‘(mulGrp‘𝑅))(+g‘(mulGrp‘𝑅))𝑋) = 𝑋 ∧ (𝑋(+g‘(mulGrp‘𝑅))(0g‘(mulGrp‘𝑅))) = 𝑋)) |
12 | 2, 7, 11 | syl2an2r 595 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (((0g‘(mulGrp‘𝑅))(+g‘(mulGrp‘𝑅))𝑋) = 𝑋 ∧ (𝑋(+g‘(mulGrp‘𝑅))(0g‘(mulGrp‘𝑅))) = 𝑋)) |
13 | rngidm.t | . . . . . . 7 ⊢ · = (.r‘𝑅) | |
14 | 1, 13 | mgpplusgg 12958 | . . . . . 6 ⊢ (𝑅 ∈ Ring → · = (+g‘(mulGrp‘𝑅))) |
15 | 14 | adantr 276 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → · = (+g‘(mulGrp‘𝑅))) |
16 | rngidm.u | . . . . . . 7 ⊢ 1 = (1r‘𝑅) | |
17 | 1, 16 | ringidvalg 12967 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 1 = (0g‘(mulGrp‘𝑅))) |
18 | 17 | adantr 276 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 1 = (0g‘(mulGrp‘𝑅))) |
19 | eqidd 2178 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 = 𝑋) | |
20 | 15, 18, 19 | oveq123d 5890 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 1 · 𝑋) = ((0g‘(mulGrp‘𝑅))(+g‘(mulGrp‘𝑅))𝑋)) |
21 | 20 | eqeq1d 2186 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (( 1 · 𝑋) = 𝑋 ↔ ((0g‘(mulGrp‘𝑅))(+g‘(mulGrp‘𝑅))𝑋) = 𝑋)) |
22 | 15, 19, 18 | oveq123d 5890 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 1 ) = (𝑋(+g‘(mulGrp‘𝑅))(0g‘(mulGrp‘𝑅)))) |
23 | 22 | eqeq1d 2186 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((𝑋 · 1 ) = 𝑋 ↔ (𝑋(+g‘(mulGrp‘𝑅))(0g‘(mulGrp‘𝑅))) = 𝑋)) |
24 | 21, 23 | anbi12d 473 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((( 1 · 𝑋) = 𝑋 ∧ (𝑋 · 1 ) = 𝑋) ↔ (((0g‘(mulGrp‘𝑅))(+g‘(mulGrp‘𝑅))𝑋) = 𝑋 ∧ (𝑋(+g‘(mulGrp‘𝑅))(0g‘(mulGrp‘𝑅))) = 𝑋))) |
25 | 12, 24 | mpbird 167 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (( 1 · 𝑋) = 𝑋 ∧ (𝑋 · 1 ) = 𝑋)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ‘cfv 5212 (class class class)co 5869 Basecbs 12442 +gcplusg 12515 .rcmulr 12516 0gc0g 12650 Mndcmnd 12706 mulGrpcmgp 12954 1rcur 12965 Ringcrg 13002 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7890 ax-resscn 7891 ax-1cn 7892 ax-1re 7893 ax-icn 7894 ax-addcl 7895 ax-addrcl 7896 ax-mulcl 7897 ax-addcom 7899 ax-addass 7901 ax-i2m1 7904 ax-0lt1 7905 ax-0id 7907 ax-rnegex 7908 ax-pre-ltirr 7911 ax-pre-ltadd 7915 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-pnf 7981 df-mnf 7982 df-ltxr 7984 df-inn 8906 df-2 8964 df-3 8965 df-ndx 12445 df-slot 12446 df-base 12448 df-sets 12449 df-plusg 12528 df-mulr 12529 df-0g 12652 df-mgm 12664 df-sgrp 12697 df-mnd 12707 df-mgp 12955 df-ur 12966 df-ring 13004 |
This theorem is referenced by: ringlidm 13029 ringridm 13030 ringid 13032 |
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