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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 20143 | . . . . 5 ⊢ (𝑥 ∈ CRing → 𝑥 ∈ Ring) | |
6 | 5 | ssriv 3986 | . . . 4 ⊢ CRing ⊆ Ring |
7 | fss 6734 | . . . 4 ⊢ ((𝑅:𝐼⟶CRing ∧ CRing ⊆ Ring) → 𝑅:𝐼⟶Ring) | |
8 | 4, 6, 7 | sylancl 585 | . . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Ring) |
9 | 1, 2, 3, 8 | prdsringd 20213 | . 2 ⊢ (𝜑 → 𝑌 ∈ Ring) |
10 | eqid 2731 | . . . 4 ⊢ (𝑆Xs(mulGrp ∘ 𝑅)) = (𝑆Xs(mulGrp ∘ 𝑅)) | |
11 | fnmgp 20034 | . . . . . . 7 ⊢ mulGrp Fn V | |
12 | ssv 4006 | . . . . . . 7 ⊢ CRing ⊆ V | |
13 | fnssres 6673 | . . . . . . 7 ⊢ ((mulGrp Fn V ∧ CRing ⊆ V) → (mulGrp ↾ CRing) Fn CRing) | |
14 | 11, 12, 13 | mp2an 689 | . . . . . 6 ⊢ (mulGrp ↾ CRing) Fn CRing |
15 | fvres 6910 | . . . . . . . 8 ⊢ (𝑥 ∈ CRing → ((mulGrp ↾ CRing)‘𝑥) = (mulGrp‘𝑥)) | |
16 | eqid 2731 | . . . . . . . . 9 ⊢ (mulGrp‘𝑥) = (mulGrp‘𝑥) | |
17 | 16 | crngmgp 20139 | . . . . . . . 8 ⊢ (𝑥 ∈ CRing → (mulGrp‘𝑥) ∈ CMnd) |
18 | 15, 17 | eqeltrd 2832 | . . . . . . 7 ⊢ (𝑥 ∈ CRing → ((mulGrp ↾ CRing)‘𝑥) ∈ CMnd) |
19 | 18 | rgen 3062 | . . . . . 6 ⊢ ∀𝑥 ∈ CRing ((mulGrp ↾ CRing)‘𝑥) ∈ CMnd |
20 | ffnfv 7120 | . . . . . 6 ⊢ ((mulGrp ↾ CRing):CRing⟶CMnd ↔ ((mulGrp ↾ CRing) Fn CRing ∧ ∀𝑥 ∈ CRing ((mulGrp ↾ CRing)‘𝑥) ∈ CMnd)) | |
21 | 14, 19, 20 | mpbir2an 708 | . . . . 5 ⊢ (mulGrp ↾ CRing):CRing⟶CMnd |
22 | fco2 6744 | . . . . 5 ⊢ (((mulGrp ↾ CRing):CRing⟶CMnd ∧ 𝑅:𝐼⟶CRing) → (mulGrp ∘ 𝑅):𝐼⟶CMnd) | |
23 | 21, 4, 22 | sylancr 586 | . . . 4 ⊢ (𝜑 → (mulGrp ∘ 𝑅):𝐼⟶CMnd) |
24 | 10, 2, 3, 23 | prdscmnd 19774 | . . 3 ⊢ (𝜑 → (𝑆Xs(mulGrp ∘ 𝑅)) ∈ CMnd) |
25 | eqidd 2732 | . . . 4 ⊢ (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(mulGrp‘𝑌))) | |
26 | eqid 2731 | . . . . . 6 ⊢ (mulGrp‘𝑌) = (mulGrp‘𝑌) | |
27 | 4 | ffnd 6718 | . . . . . 6 ⊢ (𝜑 → 𝑅 Fn 𝐼) |
28 | 1, 26, 10, 2, 3, 27 | prdsmgp 20049 | . . . . 5 ⊢ (𝜑 → ((Base‘(mulGrp‘𝑌)) = (Base‘(𝑆Xs(mulGrp ∘ 𝑅))) ∧ (+g‘(mulGrp‘𝑌)) = (+g‘(𝑆Xs(mulGrp ∘ 𝑅))))) |
29 | 28 | simpld 494 | . . . 4 ⊢ (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(𝑆Xs(mulGrp ∘ 𝑅)))) |
30 | 28 | simprd 495 | . . . . 5 ⊢ (𝜑 → (+g‘(mulGrp‘𝑌)) = (+g‘(𝑆Xs(mulGrp ∘ 𝑅)))) |
31 | 30 | oveqdr 7440 | . . . 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 19704 | . . 3 ⊢ (𝜑 → ((mulGrp‘𝑌) ∈ CMnd ↔ (𝑆Xs(mulGrp ∘ 𝑅)) ∈ CMnd)) |
33 | 24, 32 | mpbird 257 | . 2 ⊢ (𝜑 → (mulGrp‘𝑌) ∈ CMnd) |
34 | 26 | iscrng 20138 | . 2 ⊢ (𝑌 ∈ CRing ↔ (𝑌 ∈ Ring ∧ (mulGrp‘𝑌) ∈ CMnd)) |
35 | 9, 33, 34 | sylanbrc 582 | 1 ⊢ (𝜑 → 𝑌 ∈ CRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 Vcvv 3473 ⊆ wss 3948 ↾ cres 5678 ∘ ccom 5680 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 +gcplusg 17204 Xscprds 17398 CMndccmn 19693 mulGrpcmgp 20032 Ringcrg 20131 CRingccrg 20132 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-hom 17228 df-cco 17229 df-0g 17394 df-prds 17400 df-mgm 18568 df-sgrp 18647 df-mnd 18663 df-grp 18861 df-minusg 18862 df-cmn 19695 df-abl 19696 df-mgp 20033 df-rng 20051 df-ur 20080 df-ring 20133 df-cring 20134 |
This theorem is referenced by: pwscrng 20218 |
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