Theorem rngisomring1 20468

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 2737	. . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅)
2		eqid 2737	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2737	. . . 4 ⊢ (.r‘𝑆) = (.r‘𝑆)
4	1, 2, 3	rngisom1 20466	. . 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 2738	. . . 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 2738	. . . 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 2737	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
8	7, 2	rngimf1o 20454	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑅 RngIso 𝑆) → 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆))
9		f1of 6848	. . . . . . . 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 1136	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
12	7, 1	ringidcl 20262	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
13	12	3ad2ant1 1134	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (1r‘𝑅) ∈ (Base‘𝑅))
14	11, 13	ffvelcdmd 7105	. . . . 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 7439	. . . . . . . . . 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 2753	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑥) = 𝑥 ↔ ((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑦) = 𝑦))
19		oveq1 7438	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥(.r‘𝑆)(𝐹‘(1r‘𝑅))) = (𝑦(.r‘𝑆)(𝐹‘(1r‘𝑅))))
20	19, 17	eqeq12d 2753	. . . . . . . . 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 3619	. . . . . . 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 20182	. . 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 2743	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (1r‘𝑆) = (𝐹‘(1r‘𝑅)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ⟶wf 6557  –1-1-onto→wf1o 6560  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  ._rcmulr 17298  Rngcrng 20149  1_rcur 20178  Ringcrg 20230   RngIso crngim 20435

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-plusg 17310  df-0g 17486  df-mgm 18653  df-mgmhm 18705  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-ghm 19231  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-rnghm 20436  df-rngim 20437

This theorem is referenced by:  rngqiprngu  21328