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Theorem qusrng 20156

Description: The quotient structure of a non-unital ring is a non-unital ring (qusring2 20309 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 2741	. . 3 ⊢ (𝑢 ∈ 𝑉 ↦ [𝑢] ∼ ) = (𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )
4		qusrng.r	. . . 4 ⊢ (𝜑 → ∼ Er 𝑉)
5		fvex 6844	. . . . 5 ⊢ (Base‘𝑅) ∈ V
6	2, 5	eqeltrdi 2849	. . . 4 ⊢ (𝜑 → 𝑉 ∈ V)
7		erex 8662	. . . 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 17501	. 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 17502	. 2 ⊢ (𝜑 → (𝑢 ∈ 𝑉 ↦ [𝑢] ∼ ):𝑉–onto→(𝑉 / ∼ ))
14	9	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑅 ∈ Rng)
15		simprl 777	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∈ 𝑉)
16	2	eleq2d 2827	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↔ 𝑥 ∈ (Base‘𝑅)))
17	16	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 ∈ 𝑉 ↔ 𝑥 ∈ (Base‘𝑅)))
18	15, 17	mpbid 234	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑥 ∈ (Base‘𝑅))
19		simprr 779	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑦 ∈ 𝑉)
20	2	eleq2d 2827	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝑉 ↔ 𝑦 ∈ (Base‘𝑅)))
21	20	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑦 ∈ 𝑉 ↔ 𝑦 ∈ (Base‘𝑅)))
22	19, 21	mpbid 234	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝑦 ∈ (Base‘𝑅))
23		eqid 2741	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
24	23, 11	rngacl 20138	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥 + 𝑦) ∈ (Base‘𝑅))
25	14, 18, 22, 24	syl3anc 1380	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ (Base‘𝑅))
26	2	eleq2d 2827	. . . . 5 ⊢ (𝜑 → ((𝑥 + 𝑦) ∈ 𝑉 ↔ (𝑥 + 𝑦) ∈ (Base‘𝑅)))
27	26	adantr 482	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ((𝑥 + 𝑦) ∈ 𝑉 ↔ (𝑥 + 𝑦) ∈ (Base‘𝑅)))
28	25, 27	mpbird 259	. . 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 17508	. 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 20140	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥 · 𝑦) ∈ (Base‘𝑅))
32	14, 18, 22, 31	syl3anc 1380	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 · 𝑦) ∈ (Base‘𝑅))
33	2	eleq2d 2827	. . . . 5 ⊢ (𝜑 → ((𝑥 · 𝑦) ∈ 𝑉 ↔ (𝑥 · 𝑦) ∈ (Base‘𝑅)))
34	33	adantr 482	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ((𝑥 · 𝑦) ∈ 𝑉 ↔ (𝑥 · 𝑦) ∈ (Base‘𝑅)))
35	32, 34	mpbird 259	. . 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 17508	. 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 20153	1 ⊢ (𝜑 → 𝑈 ∈ Rng)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 397 = wceq 1548 ∈ wcel 2121 Vcvv 3433 class class class wbr 5075 ↦ cmpt 5156 ‘cfv 6489 (class class class)co 7360 Er wer 8634 [cec 8635 / cqs 8636 Basecbs 17174 +_gcplusg 17215 ._rcmulr 17216 /_s cqus 17464 Rngcrng 20128

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-ec 8639 df-qs 8643 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-inf 9350 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-fz 13457 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-plusg 17228 df-mulr 17229 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-0g 17399 df-imas 17467 df-qus 17468 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18907 df-minusg 18908 df-cmn 19752 df-abl 19753 df-mgp 20117 df-rng 20129

This theorem is referenced by: qus2idrng 21270

W3C validator