qusrng - Metamath Proof Explorer

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Theorem qusrng 20321

Description: The quotient structure of a non-unital ring is a non-unital ring (qusring2 20481 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 2762	. . 3 ⊢ (𝑢 ∈ 𝑉 ↦ [𝑢] ∼ ) = (𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )
4		qusrng.r	. . . 4 ⊢ (𝜑 → ∼ Er 𝑉)
5		fvex 6895	. . . . 5 ⊢ (Base‘𝑅) ∈ V
6	2, 5	eqeltrdi 2870	. . . 4 ⊢ (𝜑 → 𝑉 ∈ V)
7		erex 8725	. . . 4 ⊢ ( ∼ Er 𝑉 → (𝑉 ∈ V → ∼ ∈ V))
8	4, 6, 7	sylc 66	. . 3 ⊢ (𝜑 → ∼ ∈ V)
9		qusrng.x	. . 3 ⊢ (𝜑 → 𝑅 ∈ Rng)
10	1, 2, 3, 8, 9	qusval 17634	. 2 ⊢ (𝜑 → 𝑈 = ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ ) “_s 𝑅))
11		qusrng.p	. 2 ⊢ + = (+_g‘𝑅)
12		qusrng.t	. 2 ⊢ · = (._r‘𝑅)
13	1, 2, 3, 8, 9	quslem 17635	. 2 ⊢ (𝜑 → (𝑢 ∈ 𝑉 ↦ [𝑢] ∼ ):𝑉–onto→(𝑉 / ∼ ))
14	9	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑅 ∈ Rng)
15		simprl 783	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∈ 𝑉)
16	2	eleq2d 2848	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↔ 𝑥 ∈ (Base‘𝑅)))
17	16	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 ∈ 𝑉 ↔ 𝑥 ∈ (Base‘𝑅)))
18	15, 17	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∈ (Base‘𝑅))
19		simprr 785	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑦 ∈ 𝑉)
20	2	eleq2d 2848	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝑉 ↔ 𝑦 ∈ (Base‘𝑅)))
21	20	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑦 ∈ 𝑉 ↔ 𝑦 ∈ (Base‘𝑅)))
22	19, 21	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑦 ∈ (Base‘𝑅))
23		eqid 2762	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
24	23, 11	rngacl 20303	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥 + 𝑦) ∈ (Base‘𝑅))
25	14, 18, 22, 24	syl3anc 1398	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ (Base‘𝑅))
26	2	eleq2d 2848	. . . . 5 ⊢ (𝜑 → ((𝑥 + 𝑦) ∈ 𝑉 ↔ (𝑥 + 𝑦) ∈ (Base‘𝑅)))
27	26	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ((𝑥 + 𝑦) ∈ 𝑉 ↔ (𝑥 + 𝑦) ∈ (Base‘𝑅)))
28	25, 27	mpbird 260	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
29		qusrng.e1	. . 3 ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 + 𝑏) ∼ (𝑝 + 𝑞)))
30	4, 6, 3, 28, 29	ercpbl 17641	. 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))))
31	23, 12	rngcl 20305	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥 · 𝑦) ∈ (Base‘𝑅))
32	14, 18, 22, 31	syl3anc 1398	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 · 𝑦) ∈ (Base‘𝑅))
33	2	eleq2d 2848	. . . . 5 ⊢ (𝜑 → ((𝑥 · 𝑦) ∈ 𝑉 ↔ (𝑥 · 𝑦) ∈ (Base‘𝑅)))
34	33	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ((𝑥 · 𝑦) ∈ 𝑉 ↔ (𝑥 · 𝑦) ∈ (Base‘𝑅)))
35	32, 34	mpbird 260	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 · 𝑦) ∈ 𝑉)
36		qusrng.e2	. . 3 ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞)))
37	4, 6, 3, 35, 36	ercpbl 17641	. 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))))
38	10, 2, 11, 12, 13, 30, 37, 9	imasrng 20318	1 ⊢ (𝜑 → 𝑈 ∈ Rng)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 Vcvv 3453 class class class wbr 5107 ↦ cmpt 5190 ‘cfv 6537 (class class class)co 7417 Er wer 8697 [cec 8698 / cqs 8699 Basecbs 17307 +_gcplusg 17348 ._rcmulr 17349 /_s cqus 17597 Rngcrng 20293

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-ec 8702 df-qs 8706 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-sup 9416 df-inf 9417 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-nn 12262 df-2 12331 df-3 12332 df-4 12333 df-5 12334 df-6 12335 df-7 12336 df-8 12337 df-9 12338 df-n0 12533 df-z 12620 df-dec 12741 df-uz 12892 df-fz 13566 df-struct 17245 df-sets 17262 df-slot 17280 df-ndx 17292 df-base 17308 df-plusg 17361 df-mulr 17362 df-sca 17364 df-vsca 17365 df-ip 17366 df-tset 17367 df-ple 17368 df-ds 17370 df-0g 17532 df-imas 17600 df-qus 17601 df-mgm 18736 df-sgrp 18827 df-mnd 18843 df-grp 19066 df-minusg 19067 df-cmn 19915 df-abl 19916 df-mgp 20280 df-rng 20294

This theorem is used by: qus2idrng 21481

W3C validator