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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 18381 | . . . . 5 ⊢ (𝑥 ∈ CRing → 𝑥 ∈ Ring) | |
6 | 5 | ssriv 3572 | . . . 4 ⊢ CRing ⊆ Ring |
7 | fss 5969 | . . . 4 ⊢ ((𝑅:𝐼⟶CRing ∧ CRing ⊆ Ring) → 𝑅:𝐼⟶Ring) | |
8 | 4, 6, 7 | sylancl 693 | . . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Ring) |
9 | 1, 2, 3, 8 | prdsringd 18435 | . 2 ⊢ (𝜑 → 𝑌 ∈ Ring) |
10 | eqid 2610 | . . . 4 ⊢ (𝑆Xs(mulGrp ∘ 𝑅)) = (𝑆Xs(mulGrp ∘ 𝑅)) | |
11 | fnmgp 18314 | . . . . . . 7 ⊢ mulGrp Fn V | |
12 | ssv 3588 | . . . . . . 7 ⊢ CRing ⊆ V | |
13 | fnssres 5918 | . . . . . . 7 ⊢ ((mulGrp Fn V ∧ CRing ⊆ V) → (mulGrp ↾ CRing) Fn CRing) | |
14 | 11, 12, 13 | mp2an 704 | . . . . . 6 ⊢ (mulGrp ↾ CRing) Fn CRing |
15 | fvres 6117 | . . . . . . . 8 ⊢ (𝑥 ∈ CRing → ((mulGrp ↾ CRing)‘𝑥) = (mulGrp‘𝑥)) | |
16 | eqid 2610 | . . . . . . . . 9 ⊢ (mulGrp‘𝑥) = (mulGrp‘𝑥) | |
17 | 16 | crngmgp 18378 | . . . . . . . 8 ⊢ (𝑥 ∈ CRing → (mulGrp‘𝑥) ∈ CMnd) |
18 | 15, 17 | eqeltrd 2688 | . . . . . . 7 ⊢ (𝑥 ∈ CRing → ((mulGrp ↾ CRing)‘𝑥) ∈ CMnd) |
19 | 18 | rgen 2906 | . . . . . 6 ⊢ ∀𝑥 ∈ CRing ((mulGrp ↾ CRing)‘𝑥) ∈ CMnd |
20 | ffnfv 6295 | . . . . . 6 ⊢ ((mulGrp ↾ CRing):CRing⟶CMnd ↔ ((mulGrp ↾ CRing) Fn CRing ∧ ∀𝑥 ∈ CRing ((mulGrp ↾ CRing)‘𝑥) ∈ CMnd)) | |
21 | 14, 19, 20 | mpbir2an 957 | . . . . 5 ⊢ (mulGrp ↾ CRing):CRing⟶CMnd |
22 | fco2 5972 | . . . . 5 ⊢ (((mulGrp ↾ CRing):CRing⟶CMnd ∧ 𝑅:𝐼⟶CRing) → (mulGrp ∘ 𝑅):𝐼⟶CMnd) | |
23 | 21, 4, 22 | sylancr 694 | . . . 4 ⊢ (𝜑 → (mulGrp ∘ 𝑅):𝐼⟶CMnd) |
24 | 10, 2, 3, 23 | prdscmnd 18087 | . . 3 ⊢ (𝜑 → (𝑆Xs(mulGrp ∘ 𝑅)) ∈ CMnd) |
25 | eqidd 2611 | . . . 4 ⊢ (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(mulGrp‘𝑌))) | |
26 | eqid 2610 | . . . . . 6 ⊢ (mulGrp‘𝑌) = (mulGrp‘𝑌) | |
27 | ffn 5958 | . . . . . . 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 18433 | . . . . 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 6573 | . . . 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 18025 | . . 3 ⊢ (𝜑 → ((mulGrp‘𝑌) ∈ CMnd ↔ (𝑆Xs(mulGrp ∘ 𝑅)) ∈ CMnd)) |
34 | 24, 33 | mpbird 246 | . 2 ⊢ (𝜑 → (mulGrp‘𝑌) ∈ CMnd) |
35 | 26 | iscrng 18377 | . 2 ⊢ (𝑌 ∈ CRing ↔ (𝑌 ∈ Ring ∧ (mulGrp‘𝑌) ∈ CMnd)) |
36 | 9, 34, 35 | sylanbrc 695 | 1 ⊢ (𝜑 → 𝑌 ∈ CRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⊆ wss 3540 ↾ cres 5040 ∘ ccom 5042 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 Xscprds 15929 CMndccmn 18016 mulGrpcmgp 18312 Ringcrg 18370 CRingccrg 18371 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-hom 15793 df-cco 15794 df-0g 15925 df-prds 15931 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-cmn 18018 df-mgp 18313 df-ring 18372 df-cring 18373 |
This theorem is referenced by: pwscrng 18440 |
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