Theorem rngisomring1 20387

Description: If there is a non-unital ring isomorphism between a unital ring and a non-unital ring, then the ring unity of the second ring is the function value of the ring unity of the first ring for the isomorphism. (Contributed by AV, 16-Mar-2025.)

Assertion

Ref	Expression
rngisomring1	⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (1r‘𝑆) = (𝐹‘(1r‘𝑅)))

Proof of Theorem rngisomring1

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2731	. . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅)
2		eqid 2731	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2731	. . . 4 ⊢ (.r‘𝑆) = (.r‘𝑆)
4	1, 2, 3	rngisom1 20385	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → ∀𝑥 ∈ (Base‘𝑆)(((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑥))
5		eqidd 2732	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ ∀𝑥 ∈ (Base‘𝑆)(((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑥)) → (Base‘𝑆) = (Base‘𝑆))
6		eqidd 2732	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ ∀𝑥 ∈ (Base‘𝑆)(((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑥)) → (.r‘𝑆) = (.r‘𝑆))
7		eqid 2731	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
8	7, 2	rngimf1o 20373	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑅 RngIso 𝑆) → 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆))
9		f1of 6763	. . . . . . . 8 ⊢ (𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
10	8, 9	syl 17	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RngIso 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
11	10	3ad2ant3 1135	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
12	7, 1	ringidcl 20184	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
13	12	3ad2ant1 1133	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (1r‘𝑅) ∈ (Base‘𝑅))
14	11, 13	ffvelcdmd 7018	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (𝐹‘(1r‘𝑅)) ∈ (Base‘𝑆))
15	14	adantr 480	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ ∀𝑥 ∈ (Base‘𝑆)(((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑥)) → (𝐹‘(1r‘𝑅)) ∈ (Base‘𝑆))
16		oveq2 7354	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑥) = ((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑦))
17		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
18	16, 17	eqeq12d 2747	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑥) = 𝑥 ↔ ((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑦) = 𝑦))
19		oveq1 7353	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥(.r‘𝑆)(𝐹‘(1r‘𝑅))) = (𝑦(.r‘𝑆)(𝐹‘(1r‘𝑅))))
20	19, 17	eqeq12d 2747	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑥(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑥 ↔ (𝑦(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑦))
21	18, 20	anbi12d 632	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑥) ↔ (((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑦)))
22	21	rspccv 3574	. . . . . . 7 ⊢ (∀𝑥 ∈ (Base‘𝑆)(((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑥) → (𝑦 ∈ (Base‘𝑆) → (((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑦)))
23	22	adantl 481	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ ∀𝑥 ∈ (Base‘𝑆)(((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑥)) → (𝑦 ∈ (Base‘𝑆) → (((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑦)))
24		simpl 482	. . . . . 6 ⊢ ((((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑦) → ((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑦) = 𝑦)
25	23, 24	syl6 35	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ ∀𝑥 ∈ (Base‘𝑆)(((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑥)) → (𝑦 ∈ (Base‘𝑆) → ((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑦) = 𝑦))
26	25	imp 406	. . . 4 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ ∀𝑥 ∈ (Base‘𝑆)(((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑥)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑦) = 𝑦)
27		simpr 484	. . . . . 6 ⊢ ((((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑦) → (𝑦(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑦)
28	23, 27	syl6 35	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ ∀𝑥 ∈ (Base‘𝑆)(((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑥)) → (𝑦 ∈ (Base‘𝑆) → (𝑦(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑦))
29	28	imp 406	. . . 4 ⊢ ((((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ ∀𝑥 ∈ (Base‘𝑆)(((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑥)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑦(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑦)
30	5, 6, 15, 26, 29	ringurd 20104	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ ∀𝑥 ∈ (Base‘𝑆)(((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑥)) → (𝐹‘(1r‘𝑅)) = (1r‘𝑆))
31	4, 30	mpdan 687	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (𝐹‘(1r‘𝑅)) = (1r‘𝑆))
32	31	eqcomd 2737	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (1r‘𝑆) = (𝐹‘(1r‘𝑅)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ⟶wf 6477  –1-1-onto→wf1o 6480  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  ._rcmulr 17162  Rngcrng 20071  1_rcur 20100  Ringcrg 20152   RngIso crngim 20354

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-plusg 17174  df-0g 17345  df-mgm 18548  df-mgmhm 18600  df-sgrp 18627  df-mnd 18643  df-grp 18849  df-ghm 19126  df-abl 19696  df-mgp 20060  df-rng 20072  df-ur 20101  df-ring 20154  df-rnghm 20355  df-rngim 20356

This theorem is referenced by:  rngqiprngu  21256