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Theorem rngqiprngfulem4 21324

Description: Lemma 4 for rngqiprngfu 21327. (Contributed by AV, 16-Mar-2025.)

**Hypotheses**
Ref	Expression
rngqiprngfu.r	⊢ (𝜑 → 𝑅 ∈ Rng)
rngqiprngfu.i	⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))
rngqiprngfu.j	⊢ 𝐽 = (𝑅 ↾_s 𝐼)
rngqiprngfu.u	⊢ (𝜑 → 𝐽 ∈ Ring)
rngqiprngfu.b	⊢ 𝐵 = (Base‘𝑅)
rngqiprngfu.t	⊢ · = (._r‘𝑅)
rngqiprngfu.1	⊢ 1 = (1_r‘𝐽)
rngqiprngfu.g	⊢ ∼ = (𝑅 ~_QG 𝐼)
rngqiprngfu.q	⊢ 𝑄 = (𝑅 /_s ∼ )
rngqiprngfu.v	⊢ (𝜑 → 𝑄 ∈ Ring)
rngqiprngfu.e	⊢ (𝜑 → 𝐸 ∈ (1_r‘𝑄))
rngqiprngfu.m	⊢ − = (-_g‘𝑅)
rngqiprngfu.a	⊢ + = (+_g‘𝑅)
rngqiprngfu.n	⊢ 𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 )

**Assertion**
Ref	Expression
rngqiprngfulem4	⊢ (𝜑 → [𝑈] ∼ = [𝐸] ∼ )

**Proof of Theorem rngqiprngfulem4**
Step	Hyp	Ref	Expression
1		rngqiprngfu.n	. . . . . 6 ⊢ 𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 )
2	1	oveq2i 7442	. . . . 5 ⊢ (𝐸 − 𝑈) = (𝐸 − ((𝐸 − ( 1 · 𝐸)) + 1 ))
3	2	a1i 11	. . . 4 ⊢ (𝜑 → (𝐸 − 𝑈) = (𝐸 − ((𝐸 − ( 1 · 𝐸)) + 1 )))
4		rngqiprngfu.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
5		rngqiprngfu.a	. . . . 5 ⊢ + = (+_g‘𝑅)
6		rngqiprngfu.m	. . . . 5 ⊢ − = (-_g‘𝑅)
7		rngqiprngfu.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Rng)
8		rngabl 20152	. . . . . 6 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
9	7, 8	syl 17	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Abel)
10		rngqiprngfu.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))
11		rngqiprngfu.j	. . . . . 6 ⊢ 𝐽 = (𝑅 ↾_s 𝐼)
12		rngqiprngfu.u	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Ring)
13		rngqiprngfu.t	. . . . . 6 ⊢ · = (._r‘𝑅)
14		rngqiprngfu.1	. . . . . 6 ⊢ 1 = (1_r‘𝐽)
15		rngqiprngfu.g	. . . . . 6 ⊢ ∼ = (𝑅 ~_QG 𝐼)
16		rngqiprngfu.q	. . . . . 6 ⊢ 𝑄 = (𝑅 /_s ∼ )
17		rngqiprngfu.v	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ Ring)
18		rngqiprngfu.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (1_r‘𝑄))
19	7, 10, 11, 12, 4, 13, 14, 15, 16, 17, 18	rngqiprngfulem2 21322	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝐵)
20		rnggrp 20155	. . . . . . 7 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
21	7, 20	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
22	7, 10, 11, 12, 4, 13, 14	rngqiprng1elbas 21296	. . . . . . 7 ⊢ (𝜑 → 1 ∈ 𝐵)
23	4, 13	rngcl 20161	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 1 ∈ 𝐵 ∧ 𝐸 ∈ 𝐵) → ( 1 · 𝐸) ∈ 𝐵)
24	7, 22, 19, 23	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → ( 1 · 𝐸) ∈ 𝐵)
25	4, 6	grpsubcl 19038	. . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 𝐸 ∈ 𝐵 ∧ ( 1 · 𝐸) ∈ 𝐵) → (𝐸 − ( 1 · 𝐸)) ∈ 𝐵)
26	21, 19, 24, 25	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝐸 − ( 1 · 𝐸)) ∈ 𝐵)
27	4, 5, 6, 9, 19, 26, 22	ablsubsub4 19836	. . . 4 ⊢ (𝜑 → ((𝐸 − (𝐸 − ( 1 · 𝐸))) − 1 ) = (𝐸 − ((𝐸 − ( 1 · 𝐸)) + 1 )))
28	4, 6, 9, 19, 24	ablnncan 19838	. . . . 5 ⊢ (𝜑 → (𝐸 − (𝐸 − ( 1 · 𝐸))) = ( 1 · 𝐸))
29	28	oveq1d 7446	. . . 4 ⊢ (𝜑 → ((𝐸 − (𝐸 − ( 1 · 𝐸))) − 1 ) = (( 1 · 𝐸) − 1 ))
30	3, 27, 29	3eqtr2d 2783	. . 3 ⊢ (𝜑 → (𝐸 − 𝑈) = (( 1 · 𝐸) − 1 ))
31		ringrng 20282	. . . . . . . . . 10 ⊢ (𝐽 ∈ Ring → 𝐽 ∈ Rng)
32	12, 31	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ Rng)
33	11, 32	eqeltrrid 2846	. . . . . . . 8 ⊢ (𝜑 → (𝑅 ↾_s 𝐼) ∈ Rng)
34	7, 10, 33	rng2idlnsg 21276	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
35		nsgsubg 19176	. . . . . . 7 ⊢ (𝐼 ∈ (NrmSGrp‘𝑅) → 𝐼 ∈ (SubGrp‘𝑅))
36	34, 35	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (SubGrp‘𝑅))
37	7, 10, 11, 12, 4, 13, 14	rngqiprngghmlem1 21297	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 ∈ 𝐵) → ( 1 · 𝐸) ∈ (Base‘𝐽))
38	19, 37	mpdan 687	. . . . . . 7 ⊢ (𝜑 → ( 1 · 𝐸) ∈ (Base‘𝐽))
39		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝐽) = (Base‘𝐽)
40	10, 11, 39	2idlbas 21273	. . . . . . 7 ⊢ (𝜑 → (Base‘𝐽) = 𝐼)
41	38, 40	eleqtrd 2843	. . . . . 6 ⊢ (𝜑 → ( 1 · 𝐸) ∈ 𝐼)
42	39, 14	ringidcl 20262	. . . . . . . 8 ⊢ (𝐽 ∈ Ring → 1 ∈ (Base‘𝐽))
43	12, 42	syl 17	. . . . . . 7 ⊢ (𝜑 → 1 ∈ (Base‘𝐽))
44	43, 40	eleqtrd 2843	. . . . . 6 ⊢ (𝜑 → 1 ∈ 𝐼)
45		eqid 2737	. . . . . . 7 ⊢ (-_g‘𝐽) = (-_g‘𝐽)
46	6, 11, 45	subgsub 19156	. . . . . 6 ⊢ ((𝐼 ∈ (SubGrp‘𝑅) ∧ ( 1 · 𝐸) ∈ 𝐼 ∧ 1 ∈ 𝐼) → (( 1 · 𝐸) − 1 ) = (( 1 · 𝐸)(-_g‘𝐽) 1 ))
47	36, 41, 44, 46	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (( 1 · 𝐸) − 1 ) = (( 1 · 𝐸)(-_g‘𝐽) 1 ))
48	12	ringgrpd 20239	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Grp)
49	39, 45	grpsubcl 19038	. . . . . 6 ⊢ ((𝐽 ∈ Grp ∧ ( 1 · 𝐸) ∈ (Base‘𝐽) ∧ 1 ∈ (Base‘𝐽)) → (( 1 · 𝐸)(-_g‘𝐽) 1 ) ∈ (Base‘𝐽))
50	48, 38, 43, 49	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (( 1 · 𝐸)(-_g‘𝐽) 1 ) ∈ (Base‘𝐽))
51	47, 50	eqeltrd 2841	. . . 4 ⊢ (𝜑 → (( 1 · 𝐸) − 1 ) ∈ (Base‘𝐽))
52	51, 40	eleqtrd 2843	. . 3 ⊢ (𝜑 → (( 1 · 𝐸) − 1 ) ∈ 𝐼)
53	30, 52	eqeltrd 2841	. 2 ⊢ (𝜑 → (𝐸 − 𝑈) ∈ 𝐼)
54	7, 10, 11, 12, 4, 13, 14, 15, 16, 17, 18, 6, 5, 1	rngqiprngfulem3 21323	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐵)
55	4, 6, 15	qusecsub 19853	. . 3 ⊢ (((𝑅 ∈ Abel ∧ 𝐼 ∈ (SubGrp‘𝑅)) ∧ (𝑈 ∈ 𝐵 ∧ 𝐸 ∈ 𝐵)) → ([𝑈] ∼ = [𝐸] ∼ ↔ (𝐸 − 𝑈) ∈ 𝐼))
56	9, 36, 54, 19, 55	syl22anc 839	. 2 ⊢ (𝜑 → ([𝑈] ∼ = [𝐸] ∼ ↔ (𝐸 − 𝑈) ∈ 𝐼))
57	53, 56	mpbird 257	1 ⊢ (𝜑 → [𝑈] ∼ = [𝐸] ∼ )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 [cec 8743 Basecbs 17247 ↾_s cress 17274 +_gcplusg 17297 ._rcmulr 17298 /_s cqus 17550 Grpcgrp 18951 -_gcsg 18953 SubGrpcsubg 19138 NrmSGrpcnsg 19139 ~_QG cqg 19140 Abelcabl 19799 Rngcrng 20149 1_rcur 20178 Ringcrg 20230 2Idealc2idl 21259

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-ec 8747 df-qs 8751 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-0g 17486 df-imas 17553 df-qus 17554 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-nsg 19142 df-eqg 19143 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-subrng 20546 df-lss 20930 df-sra 21172 df-rgmod 21173 df-lidl 21218 df-2idl 21260

This theorem is referenced by: rngqiprngfu 21327

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