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Mirrors > Home > MPE Home > Th. List > iscrngd | Structured version Visualization version 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 18408 | . 2 ⊢ (𝜑 → 𝑅 ∈ Ring) |
13 | eqid 2610 | . . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
14 | eqid 2610 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
15 | 13, 14 | mgpbas 18318 | . . . 4 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅)) |
16 | 1, 15 | syl6eq 2660 | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝑅))) |
17 | eqid 2610 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
18 | 13, 17 | mgpplusg 18316 | . . . 4 ⊢ (.r‘𝑅) = (+g‘(mulGrp‘𝑅)) |
19 | 3, 18 | syl6eq 2660 | . . 3 ⊢ (𝜑 → · = (+g‘(mulGrp‘𝑅))) |
20 | 16, 19, 5, 6, 9, 10, 11 | ismndd 17136 | . . 3 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ Mnd) |
21 | iscrngd.c | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) = (𝑦 · 𝑥)) | |
22 | 16, 19, 20, 21 | iscmnd 18028 | . 2 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ CMnd) |
23 | 13 | iscrng 18377 | . 2 ⊢ (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ (mulGrp‘𝑅) ∈ CMnd)) |
24 | 12, 22, 23 | sylanbrc 695 | 1 ⊢ (𝜑 → 𝑅 ∈ CRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 .rcmulr 15769 Grpcgrp 17245 CMndccmn 18016 mulGrpcmgp 18312 Ringcrg 18370 CRingccrg 18371 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-cmn 18018 df-mgp 18313 df-ring 18372 df-cring 18373 |
This theorem is referenced by: cncrng 19586 |
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