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Theorem rngqiprngim 21229

Description: 𝐹 is an isomorphism of non-unital rings. (Contributed by AV, 21-Feb-2025.)

**Hypotheses**
Ref	Expression
rng2idlring.r	⊢ (𝜑 → 𝑅 ∈ Rng)
rng2idlring.i	⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))
rng2idlring.j	⊢ 𝐽 = (𝑅 ↾_s 𝐼)
rng2idlring.u	⊢ (𝜑 → 𝐽 ∈ Ring)
rng2idlring.b	⊢ 𝐵 = (Base‘𝑅)
rng2idlring.t	⊢ · = (._r‘𝑅)
rng2idlring.1	⊢ 1 = (1_r‘𝐽)
rngqiprngim.g	⊢ ∼ = (𝑅 ~_QG 𝐼)
rngqiprngim.q	⊢ 𝑄 = (𝑅 /_s ∼ )
rngqiprngim.c	⊢ 𝐶 = (Base‘𝑄)
rngqiprngim.p	⊢ 𝑃 = (𝑄 ×_s 𝐽)
rngqiprngim.f	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)

**Assertion**
Ref	Expression
rngqiprngim	⊢ (𝜑 → 𝐹 ∈ (𝑅 RngIso 𝑃))

Distinct variable groups: 𝑥,𝐶 𝑥,𝐼 𝑥,𝐵 𝜑,𝑥 𝑥, ∼ 𝑥, 1 𝑥, · 𝑥,𝑅 Allowed substitution hints: 𝑃(𝑥) 𝑄(𝑥) 𝐹(𝑥) 𝐽(𝑥)

**Proof of Theorem rngqiprngim**
Step	Hyp	Ref	Expression
1		rng2idlring.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Rng)
2		rng2idlring.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))
3		rng2idlring.j	. . 3 ⊢ 𝐽 = (𝑅 ↾_s 𝐼)
4		rng2idlring.u	. . 3 ⊢ (𝜑 → 𝐽 ∈ Ring)
5		rng2idlring.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
6		rng2idlring.t	. . 3 ⊢ · = (._r‘𝑅)
7		rng2idlring.1	. . 3 ⊢ 1 = (1_r‘𝐽)
8		rngqiprngim.g	. . 3 ⊢ ∼ = (𝑅 ~_QG 𝐼)
9		rngqiprngim.q	. . 3 ⊢ 𝑄 = (𝑅 /_s ∼ )
10		rngqiprngim.c	. . 3 ⊢ 𝐶 = (Base‘𝑄)
11		rngqiprngim.p	. . 3 ⊢ 𝑃 = (𝑄 ×_s 𝐽)
12		rngqiprngim.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	rngqiprngho 21228	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RngHom 𝑃))
14	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	rngqiprngimf1 21225	. . . 4 ⊢ (𝜑 → 𝐹:𝐵–1-1→(𝐶 × 𝐼))
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	rngqiprngimfo 21226	. . . 4 ⊢ (𝜑 → 𝐹:𝐵–onto→(𝐶 × 𝐼))
16		df-f1o 6493	. . . 4 ⊢ (𝐹:𝐵–1-1-onto→(𝐶 × 𝐼) ↔ (𝐹:𝐵–1-1→(𝐶 × 𝐼) ∧ 𝐹:𝐵–onto→(𝐶 × 𝐼)))
17	14, 15, 16	sylanbrc 583	. . 3 ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→(𝐶 × 𝐼))
18	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	rngqipbas 21220	. . . 4 ⊢ (𝜑 → (Base‘𝑃) = (𝐶 × 𝐼))
19	18	f1oeq3d 6765	. . 3 ⊢ (𝜑 → (𝐹:𝐵–1-1-onto→(Base‘𝑃) ↔ 𝐹:𝐵–1-1-onto→(𝐶 × 𝐼)))
20	17, 19	mpbird 257	. 2 ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→(Base‘𝑃))
21	11	ovexi 7387	. . 3 ⊢ 𝑃 ∈ V
22		eqid 2729	. . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃)
23	5, 22	isrngim2 20356	. . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑃 ∈ V) → (𝐹 ∈ (𝑅 RngIso 𝑃) ↔ (𝐹 ∈ (𝑅 RngHom 𝑃) ∧ 𝐹:𝐵–1-1-onto→(Base‘𝑃))))
24	1, 21, 23	sylancl 586	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑅 RngIso 𝑃) ↔ (𝐹 ∈ (𝑅 RngHom 𝑃) ∧ 𝐹:𝐵–1-1-onto→(Base‘𝑃))))
25	13, 20, 24	mpbir2and 713	1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RngIso 𝑃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 Vcvv 3438 ⟨cop 4585 ↦ cmpt 5176 × cxp 5621 –1-1→wf1 6483 –onto→wfo 6484 –1-1-onto→wf1o 6485 ‘cfv 6486 (class class class)co 7353 [cec 8630 Basecbs 17138 ↾_s cress 17159 ._rcmulr 17180 /_s cqus 17427 ×_s cxps 17428 ~_QG cqg 19019 Rngcrng 20055 1_rcur 20084 Ringcrg 20136 RngHom crnghm 20337 RngIso crngim 20338 2Idealc2idl 21174

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 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

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 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8632 df-ec 8634 df-qs 8638 df-map 8762 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-sup 9351 df-inf 9352 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-fz 13429 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-hom 17203 df-cco 17204 df-0g 17363 df-prds 17369 df-imas 17430 df-qus 17431 df-xps 17432 df-mgm 18532 df-mgmhm 18584 df-sgrp 18611 df-mnd 18627 df-grp 18833 df-minusg 18834 df-sbg 18835 df-subg 19020 df-nsg 19021 df-eqg 19022 df-ghm 19110 df-cmn 19679 df-abl 19680 df-mgp 20044 df-rng 20056 df-ur 20085 df-ring 20138 df-oppr 20240 df-dvdsr 20260 df-unit 20261 df-invr 20291 df-rnghm 20339 df-rngim 20340 df-subrng 20449 df-lss 20853 df-sra 21095 df-rgmod 21096 df-lidl 21133 df-2idl 21175

This theorem is referenced by: rngringbdlem2 21232 rngqiprngu 21243

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