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Theorem rngqiprngfulem4 21314

Description: Lemma 4 for rngqiprngfu 21317. (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 7374	. . . . 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 20134	. . . . . 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 21312	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝐵)
20		rnggrp 20137	. . . . . . 7 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
21	7, 20	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
22	7, 10, 11, 12, 4, 13, 14	rngqiprng1elbas 21286	. . . . . . 7 ⊢ (𝜑 → 1 ∈ 𝐵)
23	4, 13	rngcl 20143	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 1 ∈ 𝐵 ∧ 𝐸 ∈ 𝐵) → ( 1 · 𝐸) ∈ 𝐵)
24	7, 22, 19, 23	syl3anc 1379	. . . . . 6 ⊢ (𝜑 → ( 1 · 𝐸) ∈ 𝐵)
25	4, 6	grpsubcl 18994	. . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 𝐸 ∈ 𝐵 ∧ ( 1 · 𝐸) ∈ 𝐵) → (𝐸 − ( 1 · 𝐸)) ∈ 𝐵)
26	21, 19, 24, 25	syl3anc 1379	. . . . 5 ⊢ (𝜑 → (𝐸 − ( 1 · 𝐸)) ∈ 𝐵)
27	4, 5, 6, 9, 19, 26, 22	ablsubsub4 19791	. . . 4 ⊢ (𝜑 → ((𝐸 − (𝐸 − ( 1 · 𝐸))) − 1 ) = (𝐸 − ((𝐸 − ( 1 · 𝐸)) + 1 )))
28	4, 6, 9, 19, 24	ablnncan 19793	. . . . 5 ⊢ (𝜑 → (𝐸 − (𝐸 − ( 1 · 𝐸))) = ( 1 · 𝐸))
29	28	oveq1d 7378	. . . 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 20264	. . . . . . . . . 10 ⊢ (𝐽 ∈ Ring → 𝐽 ∈ Rng)
32	12, 31	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ Rng)
33	11, 32	eqeltrrid 2845	. . . . . . . 8 ⊢ (𝜑 → (𝑅 ↾_s 𝐼) ∈ Rng)
34	7, 10, 33	rng2idlnsg 21266	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
35		nsgsubg 19131	. . . . . . 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 21287	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 ∈ 𝐵) → ( 1 · 𝐸) ∈ (Base‘𝐽))
38	19, 37	mpdan 693	. . . . . . 7 ⊢ (𝜑 → ( 1 · 𝐸) ∈ (Base‘𝐽))
39		eqid 2740	. . . . . . . 8 ⊢ (Base‘𝐽) = (Base‘𝐽)
40	10, 11, 39	2idlbas 21263	. . . . . . 7 ⊢ (𝜑 → (Base‘𝐽) = 𝐼)
41	38, 40	eleqtrd 2842	. . . . . 6 ⊢ (𝜑 → ( 1 · 𝐸) ∈ 𝐼)
42	39, 14	ringidcl 20244	. . . . . . . 8 ⊢ (𝐽 ∈ Ring → 1 ∈ (Base‘𝐽))
43	12, 42	syl 17	. . . . . . 7 ⊢ (𝜑 → 1 ∈ (Base‘𝐽))
44	43, 40	eleqtrd 2842	. . . . . 6 ⊢ (𝜑 → 1 ∈ 𝐼)
45		eqid 2740	. . . . . . 7 ⊢ (-_g‘𝐽) = (-_g‘𝐽)
46	6, 11, 45	subgsub 19112	. . . . . 6 ⊢ ((𝐼 ∈ (SubGrp‘𝑅) ∧ ( 1 · 𝐸) ∈ 𝐼 ∧ 1 ∈ 𝐼) → (( 1 · 𝐸) − 1 ) = (( 1 · 𝐸)(-_g‘𝐽) 1 ))
47	36, 41, 44, 46	syl3anc 1379	. . . . 5 ⊢ (𝜑 → (( 1 · 𝐸) − 1 ) = (( 1 · 𝐸)(-_g‘𝐽) 1 ))
48	12	ringgrpd 20221	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Grp)
49	39, 45	grpsubcl 18994	. . . . . 6 ⊢ ((𝐽 ∈ Grp ∧ ( 1 · 𝐸) ∈ (Base‘𝐽) ∧ 1 ∈ (Base‘𝐽)) → (( 1 · 𝐸)(-_g‘𝐽) 1 ) ∈ (Base‘𝐽))
50	48, 38, 43, 49	syl3anc 1379	. . . . 5 ⊢ (𝜑 → (( 1 · 𝐸)(-_g‘𝐽) 1 ) ∈ (Base‘𝐽))
51	47, 50	eqeltrd 2840	. . . 4 ⊢ (𝜑 → (( 1 · 𝐸) − 1 ) ∈ (Base‘𝐽))
52	51, 40	eleqtrd 2842	. . 3 ⊢ (𝜑 → (( 1 · 𝐸) − 1 ) ∈ 𝐼)
53	30, 52	eqeltrd 2840	. 2 ⊢ (𝜑 → (𝐸 − 𝑈) ∈ 𝐼)
54	7, 10, 11, 12, 4, 13, 14, 15, 16, 17, 18, 6, 5, 1	rngqiprngfulem3 21313	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐵)
55	4, 6, 15	qusecsub 19808	. . 3 ⊢ (((𝑅 ∈ Abel ∧ 𝐼 ∈ (SubGrp‘𝑅)) ∧ (𝑈 ∈ 𝐵 ∧ 𝐸 ∈ 𝐵)) → ([𝑈] ∼ = [𝐸] ∼ ↔ (𝐸 − 𝑈) ∈ 𝐼))
56	9, 36, 54, 19, 55	syl22anc 844	. 2 ⊢ (𝜑 → ([𝑈] ∼ = [𝐸] ∼ ↔ (𝐸 − 𝑈) ∈ 𝐼))
57	53, 56	mpbird 258	1 ⊢ (𝜑 → [𝑈] ∼ = [𝐸] ∼ )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 = wceq 1547 ∈ wcel 2119 ‘cfv 6492 (class class class)co 7363 [cec 8638 Basecbs 17177 ↾_s cress 17198 +_gcplusg 17218 ._rcmulr 17219 /_s cqus 17467 Grpcgrp 18907 -_gcsg 18909 SubGrpcsubg 19094 NrmSGrpcnsg 19095 ~_QG cqg 19096 Abelcabl 19754 Rngcrng 20131 1_rcur 20160 Ringcrg 20212 2Idealc2idl 21249

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-tpos 8173 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-ress 17199 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-sbg 18912 df-subg 19097 df-nsg 19098 df-eqg 19099 df-cmn 19755 df-abl 19756 df-mgp 20120 df-rng 20132 df-ur 20161 df-ring 20214 df-oppr 20315 df-subrng 20525 df-lss 20929 df-sra 21170 df-rgmod 21171 df-lidl 21208 df-2idl 21250

This theorem is referenced by: rngqiprngfu 21317

W3C validator