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Theorem qusrng 20159

Description: The quotient structure of a non-unital ring is a non-unital ring (qusring2 20312 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 2740	. . 3 ⊢ (𝑢 ∈ 𝑉 ↦ [𝑢] ∼ ) = (𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )
4		qusrng.r	. . . 4 ⊢ (𝜑 → ∼ Er 𝑉)
5		fvex 6847	. . . . 5 ⊢ (Base‘𝑅) ∈ V
6	2, 5	eqeltrdi 2848	. . . 4 ⊢ (𝜑 → 𝑉 ∈ V)
7		erex 8665	. . . 4 ⊢ ( ∼ Er 𝑉 → (𝑉 ∈ V → ∼ ∈ V))
8	4, 6, 7	sylc 65	. . 3 ⊢ (𝜑 → ∼ ∈ V)
9		qusrng.x	. . 3 ⊢ (𝜑 → 𝑅 ∈ Rng)
10	1, 2, 3, 8, 9	qusval 17504	. 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 17505	. 2 ⊢ (𝜑 → (𝑢 ∈ 𝑉 ↦ [𝑢] ∼ ):𝑉–onto→(𝑉 / ∼ ))
14	9	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑅 ∈ Rng)
15		simprl 776	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∈ 𝑉)
16	2	eleq2d 2826	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↔ 𝑥 ∈ (Base‘𝑅)))
17	16	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 ∈ 𝑉 ↔ 𝑥 ∈ (Base‘𝑅)))
18	15, 17	mpbid 233	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∈ (Base‘𝑅))
19		simprr 778	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑦 ∈ 𝑉)
20	2	eleq2d 2826	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝑉 ↔ 𝑦 ∈ (Base‘𝑅)))
21	20	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑦 ∈ 𝑉 ↔ 𝑦 ∈ (Base‘𝑅)))
22	19, 21	mpbid 233	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑦 ∈ (Base‘𝑅))
23		eqid 2740	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
24	23, 11	rngacl 20141	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥 + 𝑦) ∈ (Base‘𝑅))
25	14, 18, 22, 24	syl3anc 1379	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ (Base‘𝑅))
26	2	eleq2d 2826	. . . . 5 ⊢ (𝜑 → ((𝑥 + 𝑦) ∈ 𝑉 ↔ (𝑥 + 𝑦) ∈ (Base‘𝑅)))
27	26	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ((𝑥 + 𝑦) ∈ 𝑉 ↔ (𝑥 + 𝑦) ∈ (Base‘𝑅)))
28	25, 27	mpbird 258	. . 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 17511	. 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 20143	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥 · 𝑦) ∈ (Base‘𝑅))
32	14, 18, 22, 31	syl3anc 1379	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 · 𝑦) ∈ (Base‘𝑅))
33	2	eleq2d 2826	. . . . 5 ⊢ (𝜑 → ((𝑥 · 𝑦) ∈ 𝑉 ↔ (𝑥 · 𝑦) ∈ (Base‘𝑅)))
34	33	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ((𝑥 · 𝑦) ∈ 𝑉 ↔ (𝑥 · 𝑦) ∈ (Base‘𝑅)))
35	32, 34	mpbird 258	. . 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 17511	. 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 20156	1 ⊢ (𝜑 → 𝑈 ∈ Rng)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 Vcvv 3432 class class class wbr 5079 ↦ cmpt 5160 ‘cfv 6492 (class class class)co 7363 Er wer 8637 [cec 8638 / cqs 8639 Basecbs 17177 +_gcplusg 17218 ._rcmulr 17219 /_s cqus 17467 Rngcrng 20131

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-er 8640 df-ec 8642 df-qs 8646 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9352 df-inf 9353 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-fz 13460 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-0g 17402 df-imas 17470 df-qus 17471 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-grp 18910 df-minusg 18911 df-cmn 19755 df-abl 19756 df-mgp 20120 df-rng 20132

This theorem is referenced by: qus2idrng 21273

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