Theorem rngisomring1 20390

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 2733	. . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅)
2		eqid 2733	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2733	. . . 4 ⊢ (.r‘𝑆) = (.r‘𝑆)
4	1, 2, 3	rngisom1 20388	. . 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 2734	. . . 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 2734	. . . 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 2733	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
8	7, 2	rngimf1o 20376	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑅 RngIso 𝑆) → 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆))
9		f1of 6770	. . . . . . . 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 20187	. . . . . . 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 7026	. . . . 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 7362	. . . . . . . . . 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 2749	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑥) = 𝑥 ↔ ((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑦) = 𝑦))
19		oveq1 7361	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥(.r‘𝑆)(𝐹‘(1r‘𝑅))) = (𝑦(.r‘𝑆)(𝐹‘(1r‘𝑅))))
20	19, 17	eqeq12d 2749	. . . . . . . . 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 3570	. . . . . . 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 20107	. . 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 2739	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 2113  ∀wral 3048  ⟶wf 6484  –1-1-onto→wf1o 6487  ‘cfv 6488  (class class class)co 7354  Basecbs 17124  ._rcmulr 17166  Rngcrng 20074  1_rcur 20103  Ringcrg 20155   RngIso crngim 20357

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 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676  ax-cnex 11071  ax-resscn 11072  ax-1cn 11073  ax-icn 11074  ax-addcl 11075  ax-addrcl 11076  ax-mulcl 11077  ax-mulrcl 11078  ax-mulcom 11079  ax-addass 11080  ax-mulass 11081  ax-distr 11082  ax-i2m1 11083  ax-1ne0 11084  ax-1rid 11085  ax-rnegex 11086  ax-rrecex 11087  ax-cnre 11088  ax-pre-lttri 11089  ax-pre-lttrn 11090  ax-pre-ltadd 11091  ax-pre-mulgt0 11092

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7311  df-ov 7357  df-oprab 7358  df-mpo 7359  df-om 7805  df-1st 7929  df-2nd 7930  df-frecs 8219  df-wrecs 8250  df-recs 8299  df-rdg 8337  df-er 8630  df-map 8760  df-en 8878  df-dom 8879  df-sdom 8880  df-pnf 11157  df-mnf 11158  df-xr 11159  df-ltxr 11160  df-le 11161  df-sub 11355  df-neg 11356  df-nn 12135  df-2 12197  df-sets 17079  df-slot 17097  df-ndx 17109  df-base 17125  df-plusg 17178  df-0g 17349  df-mgm 18552  df-mgmhm 18604  df-sgrp 18631  df-mnd 18647  df-grp 18853  df-ghm 19129  df-abl 19699  df-mgp 20063  df-rng 20075  df-ur 20104  df-ring 20157  df-rnghm 20358  df-rngim 20359

This theorem is referenced by:  rngqiprngu  21259