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Theorem rngqiprngim 21259

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 21258	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RngHom 𝑃))
14	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	rngqiprngimf1 21255	. . . 4 ⊢ (𝜑 → 𝐹:𝐵–1-1→(𝐶 × 𝐼))
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	rngqiprngimfo 21256	. . . 4 ⊢ (𝜑 → 𝐹:𝐵–onto→(𝐶 × 𝐼))
16		df-f1o 6499	. . . 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 21250	. . . 4 ⊢ (𝜑 → (Base‘𝑃) = (𝐶 × 𝐼))
19	18	f1oeq3d 6771	. . 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 7392	. . 3 ⊢ 𝑃 ∈ V
22		eqid 2736	. . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃)
23	5, 22	isrngim2 20389	. . 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 1541 ∈ wcel 2113 Vcvv 3440 ⟨cop 4586 ↦ cmpt 5179 × cxp 5622 –1-1→wf1 6489 –onto→wfo 6490 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7358 [cec 8633 Basecbs 17136 ↾_s cress 17157 ._rcmulr 17178 /_s cqus 17426 ×_s cxps 17427 ~_QG cqg 19052 Rngcrng 20087 1_rcur 20116 Ringcrg 20168 RngHom crnghm 20370 RngIso crngim 20371 2Idealc2idl 21204

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 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 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-tpos 8168 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-ec 8637 df-qs 8641 df-map 8765 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-inf 9346 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-fz 13424 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-hom 17201 df-cco 17202 df-0g 17361 df-prds 17367 df-imas 17429 df-qus 17430 df-xps 17431 df-mgm 18565 df-mgmhm 18617 df-sgrp 18644 df-mnd 18660 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19053 df-nsg 19054 df-eqg 19055 df-ghm 19142 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-oppr 20273 df-dvdsr 20293 df-unit 20294 df-invr 20324 df-rnghm 20372 df-rngim 20373 df-subrng 20479 df-lss 20883 df-sra 21125 df-rgmod 21126 df-lidl 21163 df-2idl 21205

This theorem is referenced by: rngringbdlem2 21262 rngqiprngu 21273

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