Theorem qusrng 14240

Description: The quotient structure of a non-unital ring is a non-unital ring (qusring2 14354 analog). (Contributed by AV, 23-Feb-2025.)

Hypotheses

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

Assertion

Ref	Expression
qusrng	⊢ (𝜑 → 𝑈 ∈ Rng)

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

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

Proof of Theorem qusrng

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

Step	Hyp	Ref	Expression
1		qusrng.u	. . 3 ⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))
2		qusrng.v	. . 3 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
3		eqid 2238	. . 3 ⊢ (𝑢 ∈ 𝑉 ↦ [𝑢] ∼ ) = (𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )
4		qusrng.r	. . . 4 ⊢ (𝜑 → ∼ Er 𝑉)
5		basfn 13394	. . . . . 6 ⊢ Base Fn V
6		qusrng.x	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Rng)
7	6	elexd 2835	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ V)
8		funfvex 5710	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
9	8	funfni 5481	. . . . . 6 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
10	5, 7, 9	sylancr 418	. . . . 5 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
11	2, 10	eqeltrd 2315	. . . 4 ⊢ (𝜑 → 𝑉 ∈ V)
12		erex 6825	. . . 4 ⊢ ( ∼ Er 𝑉 → (𝑉 ∈ V → ∼ ∈ V))
13	4, 11, 12	sylc 62	. . 3 ⊢ (𝜑 → ∼ ∈ V)
14	1, 2, 3, 13, 6	qusval 13627	. 2 ⊢ (𝜑 → 𝑈 = ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ ) “s 𝑅))
15		qusrng.p	. 2 ⊢ + = (+g‘𝑅)
16		qusrng.t	. 2 ⊢ · = (.r‘𝑅)
17	1, 2, 3, 13, 6	quslem 13628	. 2 ⊢ (𝜑 → (𝑢 ∈ 𝑉 ↦ [𝑢] ∼ ):𝑉–onto→(𝑉 / ∼ ))
18	6	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑅 ∈ Rng)
19		simprl 535	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∈ 𝑉)
20	2	eleq2d 2308	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↔ 𝑥 ∈ (Base‘𝑅)))
21	20	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 ∈ 𝑉 ↔ 𝑥 ∈ (Base‘𝑅)))
22	19, 21	mpbid 147	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∈ (Base‘𝑅))
23		simprr 537	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑦 ∈ 𝑉)
24	2	eleq2d 2308	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝑉 ↔ 𝑦 ∈ (Base‘𝑅)))
25	24	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑦 ∈ 𝑉 ↔ 𝑦 ∈ (Base‘𝑅)))
26	23, 25	mpbid 147	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑦 ∈ (Base‘𝑅))
27		eqid 2238	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
28	27, 15	rngacl 14224	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥 + 𝑦) ∈ (Base‘𝑅))
29	18, 22, 26, 28	syl3anc 1278	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ (Base‘𝑅))
30	2	eleq2d 2308	. . . . 5 ⊢ (𝜑 → ((𝑥 + 𝑦) ∈ 𝑉 ↔ (𝑥 + 𝑦) ∈ (Base‘𝑅)))
31	30	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ((𝑥 + 𝑦) ∈ 𝑉 ↔ (𝑥 + 𝑦) ∈ (Base‘𝑅)))
32	29, 31	mpbird 167	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
33		qusrng.e1	. . 3 ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 + 𝑏) ∼ (𝑝 + 𝑞)))
34	4, 11, 3, 32, 33	ercpbl 13635	. 2 ⊢ ((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))))
35	27, 16	rngcl 14226	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥 · 𝑦) ∈ (Base‘𝑅))
36	18, 22, 26, 35	syl3anc 1278	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 · 𝑦) ∈ (Base‘𝑅))
37	2	eleq2d 2308	. . . . 5 ⊢ (𝜑 → ((𝑥 · 𝑦) ∈ 𝑉 ↔ (𝑥 · 𝑦) ∈ (Base‘𝑅)))
38	37	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ((𝑥 · 𝑦) ∈ 𝑉 ↔ (𝑥 · 𝑦) ∈ (Base‘𝑅)))
39	36, 38	mpbird 167	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 · 𝑦) ∈ 𝑉)
40		qusrng.e2	. . 3 ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞)))
41	4, 11, 3, 39, 40	ercpbl 13635	. 2 ⊢ ((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))))
42	14, 2, 15, 16, 17, 34, 41, 6	imasrng 14238	1 ⊢ (𝜑 → 𝑈 ∈ Rng)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  Vcvv 2821   class class class wbr 4128   ↦ cmpt 4190   Fn wfn 5370  ‘cfv 5375  (class class class)co 6079   Er wer 6798  [cec 6799   / cqs 6800  Basecbs 13335  +_gcplusg 13414  ._rcmulr 13415   /_s cqus 13606  Rngcrng 14214

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-pre-ltirr 8285  ax-pre-lttrn 8287  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-tp 3716  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-er 6801  df-ec 6803  df-qs 6807  df-pnf 8356  df-mnf 8357  df-ltxr 8359  df-inn 9288  df-2 9346  df-3 9347  df-ndx 13338  df-slot 13339  df-base 13341  df-sets 13342  df-plusg 13427  df-mulr 13428  df-0g 13595  df-iimas 13607  df-qus 13608  df-mgm 13659  df-sgrp 13700  df-mnd 13713  df-grp 13791  df-minusg 13792  df-cmn 14072  df-abl 14073  df-mgp 14201  df-rng 14215

This theorem is referenced by:  qus2idrng  14845