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Mirrors > Home > MPE Home > Th. List > prdscrngd | Structured version Visualization version GIF version |
Description: A product of commutative rings is a commutative ring. Since the resulting ring will have zero divisors in all nontrivial cases, this cannot be strengthened much further. (Contributed by Mario Carneiro, 11-Mar-2015.) |
Ref | Expression |
---|---|
prdscrngd.y | ⊢ 𝑌 = (𝑆Xs𝑅) |
prdscrngd.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
prdscrngd.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
prdscrngd.r | ⊢ (𝜑 → 𝑅:𝐼⟶CRing) |
Ref | Expression |
---|---|
prdscrngd | ⊢ (𝜑 → 𝑌 ∈ CRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prdscrngd.y | . . 3 ⊢ 𝑌 = (𝑆Xs𝑅) | |
2 | prdscrngd.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
3 | prdscrngd.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
4 | prdscrngd.r | . . . 4 ⊢ (𝜑 → 𝑅:𝐼⟶CRing) | |
5 | crngring 18604 | . . . . 5 ⊢ (𝑥 ∈ CRing → 𝑥 ∈ Ring) | |
6 | 5 | ssriv 3640 | . . . 4 ⊢ CRing ⊆ Ring |
7 | fss 6094 | . . . 4 ⊢ ((𝑅:𝐼⟶CRing ∧ CRing ⊆ Ring) → 𝑅:𝐼⟶Ring) | |
8 | 4, 6, 7 | sylancl 695 | . . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Ring) |
9 | 1, 2, 3, 8 | prdsringd 18658 | . 2 ⊢ (𝜑 → 𝑌 ∈ Ring) |
10 | eqid 2651 | . . . 4 ⊢ (𝑆Xs(mulGrp ∘ 𝑅)) = (𝑆Xs(mulGrp ∘ 𝑅)) | |
11 | fnmgp 18537 | . . . . . . 7 ⊢ mulGrp Fn V | |
12 | ssv 3658 | . . . . . . 7 ⊢ CRing ⊆ V | |
13 | fnssres 6042 | . . . . . . 7 ⊢ ((mulGrp Fn V ∧ CRing ⊆ V) → (mulGrp ↾ CRing) Fn CRing) | |
14 | 11, 12, 13 | mp2an 708 | . . . . . 6 ⊢ (mulGrp ↾ CRing) Fn CRing |
15 | fvres 6245 | . . . . . . . 8 ⊢ (𝑥 ∈ CRing → ((mulGrp ↾ CRing)‘𝑥) = (mulGrp‘𝑥)) | |
16 | eqid 2651 | . . . . . . . . 9 ⊢ (mulGrp‘𝑥) = (mulGrp‘𝑥) | |
17 | 16 | crngmgp 18601 | . . . . . . . 8 ⊢ (𝑥 ∈ CRing → (mulGrp‘𝑥) ∈ CMnd) |
18 | 15, 17 | eqeltrd 2730 | . . . . . . 7 ⊢ (𝑥 ∈ CRing → ((mulGrp ↾ CRing)‘𝑥) ∈ CMnd) |
19 | 18 | rgen 2951 | . . . . . 6 ⊢ ∀𝑥 ∈ CRing ((mulGrp ↾ CRing)‘𝑥) ∈ CMnd |
20 | ffnfv 6428 | . . . . . 6 ⊢ ((mulGrp ↾ CRing):CRing⟶CMnd ↔ ((mulGrp ↾ CRing) Fn CRing ∧ ∀𝑥 ∈ CRing ((mulGrp ↾ CRing)‘𝑥) ∈ CMnd)) | |
21 | 14, 19, 20 | mpbir2an 975 | . . . . 5 ⊢ (mulGrp ↾ CRing):CRing⟶CMnd |
22 | fco2 6097 | . . . . 5 ⊢ (((mulGrp ↾ CRing):CRing⟶CMnd ∧ 𝑅:𝐼⟶CRing) → (mulGrp ∘ 𝑅):𝐼⟶CMnd) | |
23 | 21, 4, 22 | sylancr 696 | . . . 4 ⊢ (𝜑 → (mulGrp ∘ 𝑅):𝐼⟶CMnd) |
24 | 10, 2, 3, 23 | prdscmnd 18310 | . . 3 ⊢ (𝜑 → (𝑆Xs(mulGrp ∘ 𝑅)) ∈ CMnd) |
25 | eqidd 2652 | . . . 4 ⊢ (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(mulGrp‘𝑌))) | |
26 | eqid 2651 | . . . . . 6 ⊢ (mulGrp‘𝑌) = (mulGrp‘𝑌) | |
27 | ffn 6083 | . . . . . . 7 ⊢ (𝑅:𝐼⟶CRing → 𝑅 Fn 𝐼) | |
28 | 4, 27 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 Fn 𝐼) |
29 | 1, 26, 10, 2, 3, 28 | prdsmgp 18656 | . . . . 5 ⊢ (𝜑 → ((Base‘(mulGrp‘𝑌)) = (Base‘(𝑆Xs(mulGrp ∘ 𝑅))) ∧ (+g‘(mulGrp‘𝑌)) = (+g‘(𝑆Xs(mulGrp ∘ 𝑅))))) |
30 | 29 | simpld 474 | . . . 4 ⊢ (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(𝑆Xs(mulGrp ∘ 𝑅)))) |
31 | 29 | simprd 478 | . . . . 5 ⊢ (𝜑 → (+g‘(mulGrp‘𝑌)) = (+g‘(𝑆Xs(mulGrp ∘ 𝑅)))) |
32 | 31 | oveqdr 6714 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(mulGrp‘𝑌)) ∧ 𝑦 ∈ (Base‘(mulGrp‘𝑌)))) → (𝑥(+g‘(mulGrp‘𝑌))𝑦) = (𝑥(+g‘(𝑆Xs(mulGrp ∘ 𝑅)))𝑦)) |
33 | 25, 30, 32 | cmnpropd 18248 | . . 3 ⊢ (𝜑 → ((mulGrp‘𝑌) ∈ CMnd ↔ (𝑆Xs(mulGrp ∘ 𝑅)) ∈ CMnd)) |
34 | 24, 33 | mpbird 247 | . 2 ⊢ (𝜑 → (mulGrp‘𝑌) ∈ CMnd) |
35 | 26 | iscrng 18600 | . 2 ⊢ (𝑌 ∈ CRing ↔ (𝑌 ∈ Ring ∧ (mulGrp‘𝑌) ∈ CMnd)) |
36 | 9, 34, 35 | sylanbrc 699 | 1 ⊢ (𝜑 → 𝑌 ∈ CRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∀wral 2941 Vcvv 3231 ⊆ wss 3607 ↾ cres 5145 ∘ ccom 5147 Fn wfn 5921 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 +gcplusg 15988 Xscprds 16153 CMndccmn 18239 mulGrpcmgp 18535 Ringcrg 18593 CRingccrg 18594 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-sup 8389 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-fz 12365 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-plusg 16001 df-mulr 16002 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-hom 16013 df-cco 16014 df-0g 16149 df-prds 16155 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-minusg 17473 df-cmn 18241 df-mgp 18536 df-ring 18595 df-cring 18596 |
This theorem is referenced by: pwscrng 18663 |
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