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Theorem rngqiprngfulem4 21342

Description: Lemma 4 for rngqiprngfu 21345. (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 20173	. . . . . 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 21340	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝐵)
20		rnggrp 20176	. . . . . . 7 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
21	7, 20	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
22	7, 10, 11, 12, 4, 13, 14	rngqiprng1elbas 21314	. . . . . . 7 ⊢ (𝜑 → 1 ∈ 𝐵)
23	4, 13	rngcl 20182	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 1 ∈ 𝐵 ∧ 𝐸 ∈ 𝐵) → ( 1 · 𝐸) ∈ 𝐵)
24	7, 22, 19, 23	syl3anc 1370	. . . . . 6 ⊢ (𝜑 → ( 1 · 𝐸) ∈ 𝐵)
25	4, 6	grpsubcl 19051	. . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 𝐸 ∈ 𝐵 ∧ ( 1 · 𝐸) ∈ 𝐵) → (𝐸 − ( 1 · 𝐸)) ∈ 𝐵)
26	21, 19, 24, 25	syl3anc 1370	. . . . 5 ⊢ (𝜑 → (𝐸 − ( 1 · 𝐸)) ∈ 𝐵)
27	4, 5, 6, 9, 19, 26, 22	ablsubsub4 19851	. . . 4 ⊢ (𝜑 → ((𝐸 − (𝐸 − ( 1 · 𝐸))) − 1 ) = (𝐸 − ((𝐸 − ( 1 · 𝐸)) + 1 )))
28	4, 6, 9, 19, 24	ablnncan 19853	. . . . 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 2781	. . 3 ⊢ (𝜑 → (𝐸 − 𝑈) = (( 1 · 𝐸) − 1 ))
31		ringrng 20299	. . . . . . . . . 10 ⊢ (𝐽 ∈ Ring → 𝐽 ∈ Rng)
32	12, 31	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ Rng)
33	11, 32	eqeltrrid 2844	. . . . . . . 8 ⊢ (𝜑 → (𝑅 ↾_s 𝐼) ∈ Rng)
34	7, 10, 33	rng2idlnsg 21294	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
35		nsgsubg 19189	. . . . . . 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 21315	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 ∈ 𝐵) → ( 1 · 𝐸) ∈ (Base‘𝐽))
38	19, 37	mpdan 687	. . . . . . 7 ⊢ (𝜑 → ( 1 · 𝐸) ∈ (Base‘𝐽))
39		eqid 2735	. . . . . . . 8 ⊢ (Base‘𝐽) = (Base‘𝐽)
40	10, 11, 39	2idlbas 21291	. . . . . . 7 ⊢ (𝜑 → (Base‘𝐽) = 𝐼)
41	38, 40	eleqtrd 2841	. . . . . 6 ⊢ (𝜑 → ( 1 · 𝐸) ∈ 𝐼)
42	39, 14	ringidcl 20280	. . . . . . . 8 ⊢ (𝐽 ∈ Ring → 1 ∈ (Base‘𝐽))
43	12, 42	syl 17	. . . . . . 7 ⊢ (𝜑 → 1 ∈ (Base‘𝐽))
44	43, 40	eleqtrd 2841	. . . . . 6 ⊢ (𝜑 → 1 ∈ 𝐼)
45		eqid 2735	. . . . . . 7 ⊢ (-_g‘𝐽) = (-_g‘𝐽)
46	6, 11, 45	subgsub 19169	. . . . . 6 ⊢ ((𝐼 ∈ (SubGrp‘𝑅) ∧ ( 1 · 𝐸) ∈ 𝐼 ∧ 1 ∈ 𝐼) → (( 1 · 𝐸) − 1 ) = (( 1 · 𝐸)(-_g‘𝐽) 1 ))
47	36, 41, 44, 46	syl3anc 1370	. . . . 5 ⊢ (𝜑 → (( 1 · 𝐸) − 1 ) = (( 1 · 𝐸)(-_g‘𝐽) 1 ))
48	12	ringgrpd 20260	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Grp)
49	39, 45	grpsubcl 19051	. . . . . 6 ⊢ ((𝐽 ∈ Grp ∧ ( 1 · 𝐸) ∈ (Base‘𝐽) ∧ 1 ∈ (Base‘𝐽)) → (( 1 · 𝐸)(-_g‘𝐽) 1 ) ∈ (Base‘𝐽))
50	48, 38, 43, 49	syl3anc 1370	. . . . 5 ⊢ (𝜑 → (( 1 · 𝐸)(-_g‘𝐽) 1 ) ∈ (Base‘𝐽))
51	47, 50	eqeltrd 2839	. . . 4 ⊢ (𝜑 → (( 1 · 𝐸) − 1 ) ∈ (Base‘𝐽))
52	51, 40	eleqtrd 2841	. . 3 ⊢ (𝜑 → (( 1 · 𝐸) − 1 ) ∈ 𝐼)
53	30, 52	eqeltrd 2839	. 2 ⊢ (𝜑 → (𝐸 − 𝑈) ∈ 𝐼)
54	7, 10, 11, 12, 4, 13, 14, 15, 16, 17, 18, 6, 5, 1	rngqiprngfulem3 21341	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐵)
55	4, 6, 15	qusecsub 19868	. . 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 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 [cec 8742 Basecbs 17245 ↾_s cress 17274 +_gcplusg 17298 ._rcmulr 17299 /_s cqus 17552 Grpcgrp 18964 -_gcsg 18966 SubGrpcsubg 19151 NrmSGrpcnsg 19152 ~_QG cqg 19153 Abelcabl 19814 Rngcrng 20170 1_rcur 20199 Ringcrg 20251 2Idealc2idl 21277

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-ec 8746 df-qs 8750 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 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 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-0g 17488 df-imas 17555 df-qus 17556 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-nsg 19155 df-eqg 19156 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-subrng 20563 df-lss 20948 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-2idl 21278

This theorem is referenced by: rngqiprngfu 21345

W3C validator