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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 19338 | . 2 ⊢ (𝜑 → 𝑅 ∈ Ring) |
13 | eqid 2824 | . . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
14 | eqid 2824 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
15 | 13, 14 | mgpbas 19248 | . . . 4 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅)) |
16 | 1, 15 | syl6eq 2875 | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝑅))) |
17 | eqid 2824 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
18 | 13, 17 | mgpplusg 19246 | . . . 4 ⊢ (.r‘𝑅) = (+g‘(mulGrp‘𝑅)) |
19 | 3, 18 | syl6eq 2875 | . . 3 ⊢ (𝜑 → · = (+g‘(mulGrp‘𝑅))) |
20 | 16, 19, 5, 6, 9, 10, 11 | ismndd 17936 | . . 3 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ Mnd) |
21 | iscrngd.c | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) = (𝑦 · 𝑥)) | |
22 | 16, 19, 20, 21 | iscmnd 18922 | . 2 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ CMnd) |
23 | 13 | iscrng 19307 | . 2 ⊢ (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ (mulGrp‘𝑅) ∈ CMnd)) |
24 | 12, 22, 23 | sylanbrc 585 | 1 ⊢ (𝜑 → 𝑅 ∈ CRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 +gcplusg 16568 .rcmulr 16569 Grpcgrp 18106 CMndccmn 18909 mulGrpcmgp 19242 Ringcrg 19300 CRingccrg 19301 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-plusg 16581 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-cmn 18911 df-mgp 19243 df-ring 19302 df-cring 19303 |
This theorem is referenced by: cncrng 20569 |
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