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Theorem rngqiprngim 21271

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 21270	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RngHom 𝑃))
14	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	rngqiprngimf1 21267	. . . 4 ⊢ (𝜑 → 𝐹:𝐵–1-1→(𝐶 × 𝐼))
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	rngqiprngimfo 21268	. . . 4 ⊢ (𝜑 → 𝐹:𝐵–onto→(𝐶 × 𝐼))
16		df-f1o 6507	. . . 4 ⊢ (𝐹:𝐵–1-1-onto→(𝐶 × 𝐼) ↔ (𝐹:𝐵–1-1→(𝐶 × 𝐼) ∧ 𝐹:𝐵–onto→(𝐶 × 𝐼)))
17	14, 15, 16	sylanbrc 584	. . 3 ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→(𝐶 × 𝐼))
18	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	rngqipbas 21262	. . . 4 ⊢ (𝜑 → (Base‘𝑃) = (𝐶 × 𝐼))
19	18	f1oeq3d 6779	. . 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 7402	. . 3 ⊢ 𝑃 ∈ V
22		eqid 2737	. . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃)
23	5, 22	isrngim2 20401	. . 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 587	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑅 RngIso 𝑃) ↔ (𝐹 ∈ (𝑅 RngHom 𝑃) ∧ 𝐹:𝐵–1-1-onto→(Base‘𝑃))))
25	13, 20, 24	mpbir2and 714	1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RngIso 𝑃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 Vcvv 3442 ⟨cop 4588 ↦ cmpt 5181 × cxp 5630 –1-1→wf1 6497 –onto→wfo 6498 –1-1-onto→wf1o 6499 ‘cfv 6500 (class class class)co 7368 [cec 8643 Basecbs 17148 ↾_s cress 17169 ._rcmulr 17190 /_s cqus 17438 ×_s cxps 17439 ~_QG cqg 19064 Rngcrng 20099 1_rcur 20128 Ringcrg 20180 RngHom crnghm 20382 RngIso crngim 20383 2Idealc2idl 21216

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-tpos 8178 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-ec 8647 df-qs 8651 df-map 8777 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-inf 9358 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-fz 13436 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-hom 17213 df-cco 17214 df-0g 17373 df-prds 17379 df-imas 17441 df-qus 17442 df-xps 17443 df-mgm 18577 df-mgmhm 18629 df-sgrp 18656 df-mnd 18672 df-grp 18878 df-minusg 18879 df-sbg 18880 df-subg 19065 df-nsg 19066 df-eqg 19067 df-ghm 19154 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20285 df-dvdsr 20305 df-unit 20306 df-invr 20336 df-rnghm 20384 df-rngim 20385 df-subrng 20491 df-lss 20895 df-sra 21137 df-rgmod 21138 df-lidl 21175 df-2idl 21217

This theorem is referenced by: rngringbdlem2 21274 rngqiprngu 21285

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