Theorem qusring2 13565

Description: The quotient structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)

Hypotheses

Ref	Expression
qusring2.u	⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))
qusring2.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
qusring2.p	⊢ + = (+g‘𝑅)
qusring2.t	⊢ · = (.r‘𝑅)
qusring2.o	⊢ 1 = (1r‘𝑅)
qusring2.r	⊢ (𝜑 → ∼ Er 𝑉)
qusring2.e1	⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 + 𝑏) ∼ (𝑝 + 𝑞)))
qusring2.e2	⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞)))
qusring2.x	⊢ (𝜑 → 𝑅 ∈ Ring)

Assertion

Ref	Expression
qusring2	⊢ (𝜑 → (𝑈 ∈ Ring ∧ [ 1 ] ∼ = (1r‘𝑈)))

Distinct variable groups:   𝑞,𝑝, +   1 ,𝑝,𝑞   𝑎,𝑏,𝑝,𝑞,𝑈   𝑉,𝑎,𝑏,𝑝,𝑞   ∼ ,𝑎,𝑏,𝑝,𝑞   𝜑,𝑎,𝑏,𝑝,𝑞   · ,𝑝,𝑞   𝑅,𝑝,𝑞

Allowed substitution hints:   + (𝑎,𝑏)   𝑅(𝑎,𝑏)   · (𝑎,𝑏)   1 (𝑎,𝑏)

Proof of Theorem qusring2

Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qusring2.u	. . . 4 ⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))
2		qusring2.v	. . . 4 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
3		eqid 2193	. . . 4 ⊢ (𝑢 ∈ 𝑉 ↦ [𝑢] ∼ ) = (𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )
4		qusring2.r	. . . . 5 ⊢ (𝜑 → ∼ Er 𝑉)
5		basfn 12679	. . . . . . 7 ⊢ Base Fn V
6		qusring2.x	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
7	6	elexd 2773	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ V)
8		funfvex 5572	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
9	8	funfni 5355	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
10	5, 7, 9	sylancr 414	. . . . . 6 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
11	2, 10	eqeltrd 2270	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ V)
12		erex 6613	. . . . 5 ⊢ ( ∼ Er 𝑉 → (𝑉 ∈ V → ∼ ∈ V))
13	4, 11, 12	sylc 62	. . . 4 ⊢ (𝜑 → ∼ ∈ V)
14	1, 2, 3, 13, 6	qusval 12909	. . 3 ⊢ (𝜑 → 𝑈 = ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ ) “s 𝑅))
15		qusring2.p	. . 3 ⊢ + = (+g‘𝑅)
16		qusring2.t	. . 3 ⊢ · = (.r‘𝑅)
17		qusring2.o	. . 3 ⊢ 1 = (1r‘𝑅)
18	1, 2, 3, 13, 6	quslem 12910	. . 3 ⊢ (𝜑 → (𝑢 ∈ 𝑉 ↦ [𝑢] ∼ ):𝑉–onto→(𝑉 / ∼ ))
19	6	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑅 ∈ Ring)
20		simprl 529	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∈ 𝑉)
21	2	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑉 = (Base‘𝑅))
22	20, 21	eleqtrd 2272	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∈ (Base‘𝑅))
23		simprr 531	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑦 ∈ 𝑉)
24	23, 21	eleqtrd 2272	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑦 ∈ (Base‘𝑅))
25		eqid 2193	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
26	25, 15	ringacl 13529	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥 + 𝑦) ∈ (Base‘𝑅))
27	19, 22, 24, 26	syl3anc 1249	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ (Base‘𝑅))
28	27, 21	eleqtrrd 2273	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
29		qusring2.e1	. . . 4 ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 + 𝑏) ∼ (𝑝 + 𝑞)))
30	4, 11, 3, 28, 29	ercpbl 12917	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → ((((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘𝑎) = ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘𝑝) ∧ ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘𝑏) = ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘𝑞)) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘(𝑎 + 𝑏)) = ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘(𝑝 + 𝑞))))
31	25, 16	ringcl 13512	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥 · 𝑦) ∈ (Base‘𝑅))
32	19, 22, 24, 31	syl3anc 1249	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 · 𝑦) ∈ (Base‘𝑅))
33	32, 21	eleqtrrd 2273	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 · 𝑦) ∈ 𝑉)
34		qusring2.e2	. . . 4 ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞)))
35	4, 11, 3, 33, 34	ercpbl 12917	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → ((((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘𝑎) = ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘𝑝) ∧ ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘𝑏) = ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘𝑞)) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘(𝑎 · 𝑏)) = ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘(𝑝 · 𝑞))))
36	14, 2, 15, 16, 17, 18, 30, 35, 6	imasring 13563	. 2 ⊢ (𝜑 → (𝑈 ∈ Ring ∧ ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 1 ) = (1r‘𝑈)))
37		ringsrg 13546	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
38	25, 17	srgidcl 13475	. . . . . . . 8 ⊢ (𝑅 ∈ SRing → 1 ∈ (Base‘𝑅))
39	6, 37, 38	3syl 17	. . . . . . 7 ⊢ (𝜑 → 1 ∈ (Base‘𝑅))
40	39, 2	eleqtrrd 2273	. . . . . 6 ⊢ (𝜑 → 1 ∈ 𝑉)
41	4, 11, 3, 40	divsfvalg 12915	. . . . 5 ⊢ (𝜑 → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 1 ) = [ 1 ] ∼ )
42	41	eqcomd 2199	. . . 4 ⊢ (𝜑 → [ 1 ] ∼ = ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 1 ))
43	42	eqeq1d 2202	. . 3 ⊢ (𝜑 → ([ 1 ] ∼ = (1r‘𝑈) ↔ ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 1 ) = (1r‘𝑈)))
44	43	anbi2d 464	. 2 ⊢ (𝜑 → ((𝑈 ∈ Ring ∧ [ 1 ] ∼ = (1r‘𝑈)) ↔ (𝑈 ∈ Ring ∧ ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 1 ) = (1r‘𝑈))))
45	36, 44	mpbird 167	1 ⊢ (𝜑 → (𝑈 ∈ Ring ∧ [ 1 ] ∼ = (1r‘𝑈)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  Vcvv 2760   class class class wbr 4030   ↦ cmpt 4091   Fn wfn 5250  ‘cfv 5255  (class class class)co 5919   Er wer 6586  [cec 6587   / cqs 6588  Basecbs 12621  +_gcplusg 12698  ._rcmulr 12699   /_s cqus 12886  1_rcur 13458  SRingcsrg 13462  Ringcrg 13495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-lttrn 7988  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-tp 3627  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-er 6589  df-ec 6591  df-qs 6595  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-3 9044  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-plusg 12711  df-mulr 12712  df-0g 12872  df-iimas 12888  df-qus 12889  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078  df-minusg 13079  df-cmn 13359  df-abl 13360  df-mgp 13420  df-ur 13459  df-srg 13463  df-ring 13497

This theorem is referenced by:  qus1  14025