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Theorem rngqiprngfulem4 21517

Description: Lemma 4 for rngqiprngfu 21520. (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 7424	. . . . 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 20290	. . . . . 6 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
9	7, 8	syl 18	. . . . 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 21515	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝐵)
20		rnggrp 20293	. . . . . . 7 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
21	7, 20	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
22	7, 10, 11, 12, 4, 13, 14	rngqiprng1elbas 21489	. . . . . . 7 ⊢ (𝜑 → 1 ∈ 𝐵)
23	4, 13	rngcl 20299	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 1 ∈ 𝐵 ∧ 𝐸 ∈ 𝐵) → ( 1 · 𝐸) ∈ 𝐵)
24	7, 22, 19, 23	syl3anc 1398	. . . . . 6 ⊢ (𝜑 → ( 1 · 𝐸) ∈ 𝐵)
25	4, 6	grpsubcl 19143	. . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 𝐸 ∈ 𝐵 ∧ ( 1 · 𝐸) ∈ 𝐵) → (𝐸 − ( 1 · 𝐸)) ∈ 𝐵)
26	21, 19, 24, 25	syl3anc 1398	. . . . 5 ⊢ (𝜑 → (𝐸 − ( 1 · 𝐸)) ∈ 𝐵)
27	4, 5, 6, 9, 19, 26, 22	ablsubsub4 19945	. . . 4 ⊢ (𝜑 → ((𝐸 − (𝐸 − ( 1 · 𝐸))) − 1 ) = (𝐸 − ((𝐸 − ( 1 · 𝐸)) + 1 )))
28	4, 6, 9, 19, 24	ablnncan 19947	. . . . 5 ⊢ (𝜑 → (𝐸 − (𝐸 − ( 1 · 𝐸))) = ( 1 · 𝐸))
29	28	oveq1d 7428	. . . 4 ⊢ (𝜑 → ((𝐸 − (𝐸 − ( 1 · 𝐸))) − 1 ) = (( 1 · 𝐸) − 1 ))
30	3, 27, 29	3eqtr2d 2801	. . 3 ⊢ (𝜑 → (𝐸 − 𝑈) = (( 1 · 𝐸) − 1 ))
31		ringrng 20426	. . . . . . . . . 10 ⊢ (𝐽 ∈ Ring → 𝐽 ∈ Rng)
32	12, 31	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ Rng)
33	11, 32	eqeltrrid 2865	. . . . . . . 8 ⊢ (𝜑 → (𝑅 ↾_s 𝐼) ∈ Rng)
34	7, 10, 33	rng2idlnsg 21468	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
35		nsgsubg 19281	. . . . . . 7 ⊢ (𝐼 ∈ (NrmSGrp‘𝑅) → 𝐼 ∈ (SubGrp‘𝑅))
36	34, 35	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (SubGrp‘𝑅))
37	7, 10, 11, 12, 4, 13, 14	rngqiprngghmlem1 21490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 ∈ 𝐵) → ( 1 · 𝐸) ∈ (Base‘𝐽))
38	19, 37	mpdan 700	. . . . . . 7 ⊢ (𝜑 → ( 1 · 𝐸) ∈ (Base‘𝐽))
39		eqid 2760	. . . . . . . 8 ⊢ (Base‘𝐽) = (Base‘𝐽)
40	10, 11, 39	2idlbas 21465	. . . . . . 7 ⊢ (𝜑 → (Base‘𝐽) = 𝐼)
41	38, 40	eleqtrd 2862	. . . . . 6 ⊢ (𝜑 → ( 1 · 𝐸) ∈ 𝐼)
42	39, 14	ringidcl 20406	. . . . . . . 8 ⊢ (𝐽 ∈ Ring → 1 ∈ (Base‘𝐽))
43	12, 42	syl 18	. . . . . . 7 ⊢ (𝜑 → 1 ∈ (Base‘𝐽))
44	43, 40	eleqtrd 2862	. . . . . 6 ⊢ (𝜑 → 1 ∈ 𝐼)
45		eqid 2760	. . . . . . 7 ⊢ (-_g‘𝐽) = (-_g‘𝐽)
46	6, 11, 45	subgsub 19262	. . . . . 6 ⊢ ((𝐼 ∈ (SubGrp‘𝑅) ∧ ( 1 · 𝐸) ∈ 𝐼 ∧ 1 ∈ 𝐼) → (( 1 · 𝐸) − 1 ) = (( 1 · 𝐸)(-_g‘𝐽) 1 ))
47	36, 41, 44, 46	syl3anc 1398	. . . . 5 ⊢ (𝜑 → (( 1 · 𝐸) − 1 ) = (( 1 · 𝐸)(-_g‘𝐽) 1 ))
48	12	ringgrpd 20381	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Grp)
49	39, 45	grpsubcl 19143	. . . . . 6 ⊢ ((𝐽 ∈ Grp ∧ ( 1 · 𝐸) ∈ (Base‘𝐽) ∧ 1 ∈ (Base‘𝐽)) → (( 1 · 𝐸)(-_g‘𝐽) 1 ) ∈ (Base‘𝐽))
50	48, 38, 43, 49	syl3anc 1398	. . . . 5 ⊢ (𝜑 → (( 1 · 𝐸)(-_g‘𝐽) 1 ) ∈ (Base‘𝐽))
51	47, 50	eqeltrd 2860	. . . 4 ⊢ (𝜑 → (( 1 · 𝐸) − 1 ) ∈ (Base‘𝐽))
52	51, 40	eleqtrd 2862	. . 3 ⊢ (𝜑 → (( 1 · 𝐸) − 1 ) ∈ 𝐼)
53	30, 52	eqeltrd 2860	. 2 ⊢ (𝜑 → (𝐸 − 𝑈) ∈ 𝐼)
54	7, 10, 11, 12, 4, 13, 14, 15, 16, 17, 18, 6, 5, 1	rngqiprngfulem3 21516	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐵)
55	4, 6, 15	qusecsub 19962	. . 3 ⊢ (((𝑅 ∈ Abel ∧ 𝐼 ∈ (SubGrp‘𝑅)) ∧ (𝑈 ∈ 𝐵 ∧ 𝐸 ∈ 𝐵)) → ([𝑈] ∼ = [𝐸] ∼ ↔ (𝐸 − 𝑈) ∈ 𝐼))
56	9, 36, 54, 19, 55	syl22anc 852	. 2 ⊢ (𝜑 → ([𝑈] ∼ = [𝐸] ∼ ↔ (𝐸 − 𝑈) ∈ 𝐼))
57	53, 56	mpbird 260	1 ⊢ (𝜑 → [𝑈] ∼ = [𝐸] ∼ )

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 = wceq 1570 ∈ wcel 2145 ‘cfv 6533 (class class class)co 7413 [cec 8694 Basecbs 17301 ↾_s cress 17322 +_gcplusg 17342 ._rcmulr 17343 /_s cqus 17591 Grpcgrp 19057 -_gcsg 19059 SubGrpcsubg 19243 NrmSGrpcnsg 19244 ~_QG cqg 19245 Abelcabl 19908 Rngcrng 20287 1_rcur 20320 Ringcrg 20372 2Idealc2idl 21451

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 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-tpos 8224 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-er 8696 df-ec 8698 df-qs 8702 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-inf 9413 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12529 df-z 12616 df-dec 12737 df-uz 12888 df-fz 13562 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-sca 17358 df-vsca 17359 df-ip 17360 df-tset 17361 df-ple 17362 df-ds 17364 df-0g 17526 df-imas 17594 df-qus 17595 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-grp 19060 df-minusg 19061 df-sbg 19062 df-subg 19246 df-nsg 19247 df-eqg 19248 df-cmn 19909 df-abl 19910 df-mgp 20274 df-rng 20288 df-ur 20321 df-ring 20374 df-oppr 20478 df-subrng 20708 df-lss 21116 df-sra 21357 df-rgmod 21358 df-lidl 21395 df-2idl 21452

This theorem is used by: rngqiprngfu 21520

W3C validator