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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 18558 | . . . . 5 ⊢ (𝑥 ∈ CRing → 𝑥 ∈ Ring) | |
6 | 5 | ssriv 3607 | . . . 4 ⊢ CRing ⊆ Ring |
7 | fss 6056 | . . . 4 ⊢ ((𝑅:𝐼⟶CRing ∧ CRing ⊆ Ring) → 𝑅:𝐼⟶Ring) | |
8 | 4, 6, 7 | sylancl 694 | . . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Ring) |
9 | 1, 2, 3, 8 | prdsringd 18612 | . 2 ⊢ (𝜑 → 𝑌 ∈ Ring) |
10 | eqid 2622 | . . . 4 ⊢ (𝑆Xs(mulGrp ∘ 𝑅)) = (𝑆Xs(mulGrp ∘ 𝑅)) | |
11 | fnmgp 18491 | . . . . . . 7 ⊢ mulGrp Fn V | |
12 | ssv 3625 | . . . . . . 7 ⊢ CRing ⊆ V | |
13 | fnssres 6004 | . . . . . . 7 ⊢ ((mulGrp Fn V ∧ CRing ⊆ V) → (mulGrp ↾ CRing) Fn CRing) | |
14 | 11, 12, 13 | mp2an 708 | . . . . . 6 ⊢ (mulGrp ↾ CRing) Fn CRing |
15 | fvres 6207 | . . . . . . . 8 ⊢ (𝑥 ∈ CRing → ((mulGrp ↾ CRing)‘𝑥) = (mulGrp‘𝑥)) | |
16 | eqid 2622 | . . . . . . . . 9 ⊢ (mulGrp‘𝑥) = (mulGrp‘𝑥) | |
17 | 16 | crngmgp 18555 | . . . . . . . 8 ⊢ (𝑥 ∈ CRing → (mulGrp‘𝑥) ∈ CMnd) |
18 | 15, 17 | eqeltrd 2701 | . . . . . . 7 ⊢ (𝑥 ∈ CRing → ((mulGrp ↾ CRing)‘𝑥) ∈ CMnd) |
19 | 18 | rgen 2922 | . . . . . 6 ⊢ ∀𝑥 ∈ CRing ((mulGrp ↾ CRing)‘𝑥) ∈ CMnd |
20 | ffnfv 6388 | . . . . . 6 ⊢ ((mulGrp ↾ CRing):CRing⟶CMnd ↔ ((mulGrp ↾ CRing) Fn CRing ∧ ∀𝑥 ∈ CRing ((mulGrp ↾ CRing)‘𝑥) ∈ CMnd)) | |
21 | 14, 19, 20 | mpbir2an 955 | . . . . 5 ⊢ (mulGrp ↾ CRing):CRing⟶CMnd |
22 | fco2 6059 | . . . . 5 ⊢ (((mulGrp ↾ CRing):CRing⟶CMnd ∧ 𝑅:𝐼⟶CRing) → (mulGrp ∘ 𝑅):𝐼⟶CMnd) | |
23 | 21, 4, 22 | sylancr 695 | . . . 4 ⊢ (𝜑 → (mulGrp ∘ 𝑅):𝐼⟶CMnd) |
24 | 10, 2, 3, 23 | prdscmnd 18264 | . . 3 ⊢ (𝜑 → (𝑆Xs(mulGrp ∘ 𝑅)) ∈ CMnd) |
25 | eqidd 2623 | . . . 4 ⊢ (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(mulGrp‘𝑌))) | |
26 | eqid 2622 | . . . . . 6 ⊢ (mulGrp‘𝑌) = (mulGrp‘𝑌) | |
27 | ffn 6045 | . . . . . . 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 18610 | . . . . 5 ⊢ (𝜑 → ((Base‘(mulGrp‘𝑌)) = (Base‘(𝑆Xs(mulGrp ∘ 𝑅))) ∧ (+g‘(mulGrp‘𝑌)) = (+g‘(𝑆Xs(mulGrp ∘ 𝑅))))) |
30 | 29 | simpld 475 | . . . 4 ⊢ (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(𝑆Xs(mulGrp ∘ 𝑅)))) |
31 | 29 | simprd 479 | . . . . 5 ⊢ (𝜑 → (+g‘(mulGrp‘𝑌)) = (+g‘(𝑆Xs(mulGrp ∘ 𝑅)))) |
32 | 31 | oveqdr 6674 | . . . 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 18202 | . . 3 ⊢ (𝜑 → ((mulGrp‘𝑌) ∈ CMnd ↔ (𝑆Xs(mulGrp ∘ 𝑅)) ∈ CMnd)) |
34 | 24, 33 | mpbird 247 | . 2 ⊢ (𝜑 → (mulGrp‘𝑌) ∈ CMnd) |
35 | 26 | iscrng 18554 | . 2 ⊢ (𝑌 ∈ CRing ↔ (𝑌 ∈ Ring ∧ (mulGrp‘𝑌) ∈ CMnd)) |
36 | 9, 34, 35 | sylanbrc 698 | 1 ⊢ (𝜑 → 𝑌 ∈ CRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ⊆ wss 3574 ↾ cres 5116 ∘ ccom 5118 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 Xscprds 16106 CMndccmn 18193 mulGrpcmgp 18489 Ringcrg 18547 CRingccrg 18548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-0g 16102 df-prds 16108 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-cmn 18195 df-mgp 18490 df-ring 18549 df-cring 18550 |
This theorem is referenced by: pwscrng 18617 |
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