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Theorem rngqiprngfulem4 21224

Description: Lemma 4 for rngqiprngfu 21227. (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 7398	. . . . 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 20064	. . . . . 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 21222	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝐵)
20		rnggrp 20067	. . . . . . 7 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
21	7, 20	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
22	7, 10, 11, 12, 4, 13, 14	rngqiprng1elbas 21196	. . . . . . 7 ⊢ (𝜑 → 1 ∈ 𝐵)
23	4, 13	rngcl 20073	. . . . . . 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 18952	. . . . . 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 19748	. . . 4 ⊢ (𝜑 → ((𝐸 − (𝐸 − ( 1 · 𝐸))) − 1 ) = (𝐸 − ((𝐸 − ( 1 · 𝐸)) + 1 )))
28	4, 6, 9, 19, 24	ablnncan 19750	. . . . 5 ⊢ (𝜑 → (𝐸 − (𝐸 − ( 1 · 𝐸))) = ( 1 · 𝐸))
29	28	oveq1d 7402	. . . 4 ⊢ (𝜑 → ((𝐸 − (𝐸 − ( 1 · 𝐸))) − 1 ) = (( 1 · 𝐸) − 1 ))
30	3, 27, 29	3eqtr2d 2770	. . 3 ⊢ (𝜑 → (𝐸 − 𝑈) = (( 1 · 𝐸) − 1 ))
31		ringrng 20194	. . . . . . . . . 10 ⊢ (𝐽 ∈ Ring → 𝐽 ∈ Rng)
32	12, 31	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ Rng)
33	11, 32	eqeltrrid 2833	. . . . . . . 8 ⊢ (𝜑 → (𝑅 ↾_s 𝐼) ∈ Rng)
34	7, 10, 33	rng2idlnsg 21176	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
35		nsgsubg 19090	. . . . . . 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 21197	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 ∈ 𝐵) → ( 1 · 𝐸) ∈ (Base‘𝐽))
38	19, 37	mpdan 687	. . . . . . 7 ⊢ (𝜑 → ( 1 · 𝐸) ∈ (Base‘𝐽))
39		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝐽) = (Base‘𝐽)
40	10, 11, 39	2idlbas 21173	. . . . . . 7 ⊢ (𝜑 → (Base‘𝐽) = 𝐼)
41	38, 40	eleqtrd 2830	. . . . . 6 ⊢ (𝜑 → ( 1 · 𝐸) ∈ 𝐼)
42	39, 14	ringidcl 20174	. . . . . . . 8 ⊢ (𝐽 ∈ Ring → 1 ∈ (Base‘𝐽))
43	12, 42	syl 17	. . . . . . 7 ⊢ (𝜑 → 1 ∈ (Base‘𝐽))
44	43, 40	eleqtrd 2830	. . . . . 6 ⊢ (𝜑 → 1 ∈ 𝐼)
45		eqid 2729	. . . . . . 7 ⊢ (-_g‘𝐽) = (-_g‘𝐽)
46	6, 11, 45	subgsub 19070	. . . . . 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 20151	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Grp)
49	39, 45	grpsubcl 18952	. . . . . 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 2828	. . . 4 ⊢ (𝜑 → (( 1 · 𝐸) − 1 ) ∈ (Base‘𝐽))
52	51, 40	eleqtrd 2830	. . 3 ⊢ (𝜑 → (( 1 · 𝐸) − 1 ) ∈ 𝐼)
53	30, 52	eqeltrd 2828	. 2 ⊢ (𝜑 → (𝐸 − 𝑈) ∈ 𝐼)
54	7, 10, 11, 12, 4, 13, 14, 15, 16, 17, 18, 6, 5, 1	rngqiprngfulem3 21223	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐵)
55	4, 6, 15	qusecsub 19765	. . 3 ⊢ (((𝑅 ∈ Abel ∧ 𝐼 ∈ (SubGrp‘𝑅)) ∧ (𝑈 ∈ 𝐵 ∧ 𝐸 ∈ 𝐵)) → ([𝑈] ∼ = [𝐸] ∼ ↔ (𝐸 − 𝑈) ∈ 𝐼))
56	9, 36, 54, 19, 55	syl22anc 838	. 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 2109 ‘cfv 6511 (class class class)co 7387 [cec 8669 Basecbs 17179 ↾_s cress 17200 +_gcplusg 17220 ._rcmulr 17221 /_s cqus 17468 Grpcgrp 18865 -_gcsg 18867 SubGrpcsubg 19052 NrmSGrpcnsg 19053 ~_QG cqg 19054 Abelcabl 19711 Rngcrng 20061 1_rcur 20090 Ringcrg 20142 2Idealc2idl 21159

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-ec 8673 df-qs 8677 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-fz 13469 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-0g 17404 df-imas 17471 df-qus 17472 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-grp 18868 df-minusg 18869 df-sbg 18870 df-subg 19055 df-nsg 19056 df-eqg 19057 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-oppr 20246 df-subrng 20455 df-lss 20838 df-sra 21080 df-rgmod 21081 df-lidl 21118 df-2idl 21160

This theorem is referenced by: rngqiprngfu 21227

W3C validator