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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 18940 | . 2 ⊢ (𝜑 → 𝑅 ∈ Ring) |
13 | eqid 2826 | . . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
14 | eqid 2826 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
15 | 13, 14 | mgpbas 18850 | . . . 4 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅)) |
16 | 1, 15 | syl6eq 2878 | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝑅))) |
17 | eqid 2826 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
18 | 13, 17 | mgpplusg 18848 | . . . 4 ⊢ (.r‘𝑅) = (+g‘(mulGrp‘𝑅)) |
19 | 3, 18 | syl6eq 2878 | . . 3 ⊢ (𝜑 → · = (+g‘(mulGrp‘𝑅))) |
20 | 16, 19, 5, 6, 9, 10, 11 | ismndd 17667 | . . 3 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ Mnd) |
21 | iscrngd.c | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) = (𝑦 · 𝑥)) | |
22 | 16, 19, 20, 21 | iscmnd 18559 | . 2 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ CMnd) |
23 | 13 | iscrng 18909 | . 2 ⊢ (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ (mulGrp‘𝑅) ∈ CMnd)) |
24 | 12, 22, 23 | sylanbrc 580 | 1 ⊢ (𝜑 → 𝑅 ∈ CRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ‘cfv 6124 (class class class)co 6906 Basecbs 16223 +gcplusg 16306 .rcmulr 16307 Grpcgrp 17777 CMndccmn 18547 mulGrpcmgp 18844 Ringcrg 18902 CRingccrg 18903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-2 11415 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-plusg 16319 df-mgm 17596 df-sgrp 17638 df-mnd 17649 df-cmn 18549 df-mgp 18845 df-ring 18904 df-cring 18905 |
This theorem is referenced by: cncrng 20128 |
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