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Mirrors > Home > ILE Home > Th. List > iscrngd | GIF version |
Description: Properties that determine a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.) |
Ref | Expression |
---|---|
isringd.b | ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) |
isringd.p | ⊢ (𝜑 → + = (+g‘𝑅)) |
isringd.t | ⊢ (𝜑 → · = (.r‘𝑅)) |
isringd.g | ⊢ (𝜑 → 𝑅 ∈ Grp) |
isringd.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵) |
isringd.a | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) |
isringd.d | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))) |
isringd.e | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) |
isringd.u | ⊢ (𝜑 → 1 ∈ 𝐵) |
isringd.i | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 1 · 𝑥) = 𝑥) |
isringd.h | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 · 1 ) = 𝑥) |
iscrngd.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) = (𝑦 · 𝑥)) |
Ref | Expression |
---|---|
iscrngd | ⊢ (𝜑 → 𝑅 ∈ CRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isringd.b | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) | |
2 | isringd.p | . . 3 ⊢ (𝜑 → + = (+g‘𝑅)) | |
3 | isringd.t | . . 3 ⊢ (𝜑 → · = (.r‘𝑅)) | |
4 | isringd.g | . . 3 ⊢ (𝜑 → 𝑅 ∈ Grp) | |
5 | isringd.c | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵) | |
6 | isringd.a | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) | |
7 | isringd.d | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))) | |
8 | isringd.e | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) | |
9 | isringd.u | . . 3 ⊢ (𝜑 → 1 ∈ 𝐵) | |
10 | isringd.i | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 1 · 𝑥) = 𝑥) | |
11 | isringd.h | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 · 1 ) = 𝑥) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | isringd 13173 | . 2 ⊢ (𝜑 → 𝑅 ∈ Ring) |
13 | eqid 2177 | . . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
14 | eqid 2177 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
15 | 13, 14 | mgpbasg 13089 | . . . . 5 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘(mulGrp‘𝑅))) |
16 | 12, 15 | syl 14 | . . . 4 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(mulGrp‘𝑅))) |
17 | 1, 16 | eqtrd 2210 | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝑅))) |
18 | eqid 2177 | . . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
19 | 13, 18 | mgpplusgg 13087 | . . . . 5 ⊢ (𝑅 ∈ Ring → (.r‘𝑅) = (+g‘(mulGrp‘𝑅))) |
20 | 12, 19 | syl 14 | . . . 4 ⊢ (𝜑 → (.r‘𝑅) = (+g‘(mulGrp‘𝑅))) |
21 | 3, 20 | eqtrd 2210 | . . 3 ⊢ (𝜑 → · = (+g‘(mulGrp‘𝑅))) |
22 | 17, 21, 5, 6, 9, 10, 11 | ismndd 12792 | . . 3 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ Mnd) |
23 | iscrngd.c | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) = (𝑦 · 𝑥)) | |
24 | 17, 21, 22, 23 | iscmnd 13054 | . 2 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ CMnd) |
25 | 13 | iscrng 13139 | . 2 ⊢ (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ (mulGrp‘𝑅) ∈ CMnd)) |
26 | 12, 24, 25 | sylanbrc 417 | 1 ⊢ (𝜑 → 𝑅 ∈ CRing) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ‘cfv 5216 (class class class)co 5874 Basecbs 12456 +gcplusg 12530 .rcmulr 12531 Grpcgrp 12831 CMndccmn 13041 mulGrpcmgp 13083 Ringcrg 13132 CRingccrg 13133 |
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 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-pre-ltirr 7922 ax-pre-ltadd 7926 |
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-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 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-iota 5178 df-fun 5218 df-fn 5219 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-ltxr 7995 df-inn 8918 df-2 8976 df-3 8977 df-ndx 12459 df-slot 12460 df-base 12462 df-sets 12463 df-plusg 12543 df-mulr 12544 df-mgm 12729 df-sgrp 12762 df-mnd 12772 df-cmn 13043 df-mgp 13084 df-ring 13134 df-cring 13135 |
This theorem is referenced by: cncrng 13354 |
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