Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 19602 | . . . . 5 ⊢ (𝑥 ∈ CRing → 𝑥 ∈ Ring) | |
6 | 5 | ssriv 3920 | . . . 4 ⊢ CRing ⊆ Ring |
7 | fss 6581 | . . . 4 ⊢ ((𝑅:𝐼⟶CRing ∧ CRing ⊆ Ring) → 𝑅:𝐼⟶Ring) | |
8 | 4, 6, 7 | sylancl 589 | . . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Ring) |
9 | 1, 2, 3, 8 | prdsringd 19658 | . 2 ⊢ (𝜑 → 𝑌 ∈ Ring) |
10 | eqid 2738 | . . . 4 ⊢ (𝑆Xs(mulGrp ∘ 𝑅)) = (𝑆Xs(mulGrp ∘ 𝑅)) | |
11 | fnmgp 19534 | . . . . . . 7 ⊢ mulGrp Fn V | |
12 | ssv 3940 | . . . . . . 7 ⊢ CRing ⊆ V | |
13 | fnssres 6519 | . . . . . . 7 ⊢ ((mulGrp Fn V ∧ CRing ⊆ V) → (mulGrp ↾ CRing) Fn CRing) | |
14 | 11, 12, 13 | mp2an 692 | . . . . . 6 ⊢ (mulGrp ↾ CRing) Fn CRing |
15 | fvres 6755 | . . . . . . . 8 ⊢ (𝑥 ∈ CRing → ((mulGrp ↾ CRing)‘𝑥) = (mulGrp‘𝑥)) | |
16 | eqid 2738 | . . . . . . . . 9 ⊢ (mulGrp‘𝑥) = (mulGrp‘𝑥) | |
17 | 16 | crngmgp 19598 | . . . . . . . 8 ⊢ (𝑥 ∈ CRing → (mulGrp‘𝑥) ∈ CMnd) |
18 | 15, 17 | eqeltrd 2839 | . . . . . . 7 ⊢ (𝑥 ∈ CRing → ((mulGrp ↾ CRing)‘𝑥) ∈ CMnd) |
19 | 18 | rgen 3072 | . . . . . 6 ⊢ ∀𝑥 ∈ CRing ((mulGrp ↾ CRing)‘𝑥) ∈ CMnd |
20 | ffnfv 6954 | . . . . . 6 ⊢ ((mulGrp ↾ CRing):CRing⟶CMnd ↔ ((mulGrp ↾ CRing) Fn CRing ∧ ∀𝑥 ∈ CRing ((mulGrp ↾ CRing)‘𝑥) ∈ CMnd)) | |
21 | 14, 19, 20 | mpbir2an 711 | . . . . 5 ⊢ (mulGrp ↾ CRing):CRing⟶CMnd |
22 | fco2 6591 | . . . . 5 ⊢ (((mulGrp ↾ CRing):CRing⟶CMnd ∧ 𝑅:𝐼⟶CRing) → (mulGrp ∘ 𝑅):𝐼⟶CMnd) | |
23 | 21, 4, 22 | sylancr 590 | . . . 4 ⊢ (𝜑 → (mulGrp ∘ 𝑅):𝐼⟶CMnd) |
24 | 10, 2, 3, 23 | prdscmnd 19274 | . . 3 ⊢ (𝜑 → (𝑆Xs(mulGrp ∘ 𝑅)) ∈ CMnd) |
25 | eqidd 2739 | . . . 4 ⊢ (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(mulGrp‘𝑌))) | |
26 | eqid 2738 | . . . . . 6 ⊢ (mulGrp‘𝑌) = (mulGrp‘𝑌) | |
27 | 4 | ffnd 6565 | . . . . . 6 ⊢ (𝜑 → 𝑅 Fn 𝐼) |
28 | 1, 26, 10, 2, 3, 27 | prdsmgp 19656 | . . . . 5 ⊢ (𝜑 → ((Base‘(mulGrp‘𝑌)) = (Base‘(𝑆Xs(mulGrp ∘ 𝑅))) ∧ (+g‘(mulGrp‘𝑌)) = (+g‘(𝑆Xs(mulGrp ∘ 𝑅))))) |
29 | 28 | simpld 498 | . . . 4 ⊢ (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(𝑆Xs(mulGrp ∘ 𝑅)))) |
30 | 28 | simprd 499 | . . . . 5 ⊢ (𝜑 → (+g‘(mulGrp‘𝑌)) = (+g‘(𝑆Xs(mulGrp ∘ 𝑅)))) |
31 | 30 | oveqdr 7260 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(mulGrp‘𝑌)) ∧ 𝑦 ∈ (Base‘(mulGrp‘𝑌)))) → (𝑥(+g‘(mulGrp‘𝑌))𝑦) = (𝑥(+g‘(𝑆Xs(mulGrp ∘ 𝑅)))𝑦)) |
32 | 25, 29, 31 | cmnpropd 19208 | . . 3 ⊢ (𝜑 → ((mulGrp‘𝑌) ∈ CMnd ↔ (𝑆Xs(mulGrp ∘ 𝑅)) ∈ CMnd)) |
33 | 24, 32 | mpbird 260 | . 2 ⊢ (𝜑 → (mulGrp‘𝑌) ∈ CMnd) |
34 | 26 | iscrng 19597 | . 2 ⊢ (𝑌 ∈ CRing ↔ (𝑌 ∈ Ring ∧ (mulGrp‘𝑌) ∈ CMnd)) |
35 | 9, 33, 34 | sylanbrc 586 | 1 ⊢ (𝜑 → 𝑌 ∈ CRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 ∀wral 3062 Vcvv 3421 ⊆ wss 3881 ↾ cres 5568 ∘ ccom 5570 Fn wfn 6393 ⟶wf 6394 ‘cfv 6398 (class class class)co 7232 Basecbs 16788 +gcplusg 16830 Xscprds 16978 CMndccmn 19198 mulGrpcmgp 19532 Ringcrg 19590 CRingccrg 19591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5194 ax-sep 5207 ax-nul 5214 ax-pow 5273 ax-pr 5337 ax-un 7542 ax-cnex 10810 ax-resscn 10811 ax-1cn 10812 ax-icn 10813 ax-addcl 10814 ax-addrcl 10815 ax-mulcl 10816 ax-mulrcl 10817 ax-mulcom 10818 ax-addass 10819 ax-mulass 10820 ax-distr 10821 ax-i2m1 10822 ax-1ne0 10823 ax-1rid 10824 ax-rnegex 10825 ax-rrecex 10826 ax-cnre 10827 ax-pre-lttri 10828 ax-pre-lttrn 10829 ax-pre-ltadd 10830 ax-pre-mulgt0 10831 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3423 df-sbc 3710 df-csb 3827 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4253 df-if 4455 df-pw 4530 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4835 df-iun 4921 df-br 5069 df-opab 5131 df-mpt 5151 df-tr 5177 df-id 5470 df-eprel 5475 df-po 5483 df-so 5484 df-fr 5524 df-we 5526 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-res 5578 df-ima 5579 df-pred 6176 df-ord 6234 df-on 6235 df-lim 6236 df-suc 6237 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-f1 6403 df-fo 6404 df-f1o 6405 df-fv 6406 df-riota 7189 df-ov 7235 df-oprab 7236 df-mpo 7237 df-om 7664 df-1st 7780 df-2nd 7781 df-wrecs 8068 df-recs 8129 df-rdg 8167 df-1o 8223 df-er 8412 df-map 8531 df-ixp 8600 df-en 8648 df-dom 8649 df-sdom 8650 df-fin 8651 df-sup 9083 df-pnf 10894 df-mnf 10895 df-xr 10896 df-ltxr 10897 df-le 10898 df-sub 11089 df-neg 11090 df-nn 11856 df-2 11918 df-3 11919 df-4 11920 df-5 11921 df-6 11922 df-7 11923 df-8 11924 df-9 11925 df-n0 12116 df-z 12202 df-dec 12319 df-uz 12464 df-fz 13121 df-struct 16728 df-sets 16745 df-slot 16763 df-ndx 16773 df-base 16789 df-plusg 16843 df-mulr 16844 df-sca 16846 df-vsca 16847 df-ip 16848 df-tset 16849 df-ple 16850 df-ds 16852 df-hom 16854 df-cco 16855 df-0g 16974 df-prds 16980 df-mgm 18142 df-sgrp 18191 df-mnd 18202 df-grp 18396 df-minusg 18397 df-cmn 19200 df-mgp 19533 df-ring 19592 df-cring 19593 |
This theorem is referenced by: pwscrng 19663 |
Copyright terms: Public domain | W3C validator |