Theorem rngisomring1 20740

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 2763	. . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅)
2		eqid 2763	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2763	. . . 4 ⊢ (.r‘𝑆) = (.r‘𝑆)
4	1, 2, 3	rngisom1 20738	. . 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 2764	. . . 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 2764	. . . 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 2763	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
8	7, 2	rngimf1o 20726	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑅 RngIso 𝑆) → 𝐹:(Base‘𝑅)⟶1-1onto(Base‘𝑆))
9		f1of 6852	. . . . . . . 8 ⊢ (𝐹:(Base‘𝑅)⟶1-1onto(Base‘𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
10	8, 9	syl 18	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RngIso 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
11	10	3ad2ant3 1153	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
12	7, 1	ringidcl 20536	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
13	12	3ad2ant1 1151	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (1r‘𝑅) ∈ (Base‘𝑅))
14	11, 13	ffvelcdmd 7115	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (𝐹‘(1r‘𝑅)) ∈ (Base‘𝑆))
15	14	adantr 486	. . . 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 7457	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑥) = ((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑦))
17		id 23	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
18	16, 17	eqeq12d 2779	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑥) = 𝑥 ↔ ((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑦) = 𝑦))
19		oveq1 7456	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥(.r‘𝑆)(𝐹‘(1r‘𝑅))) = (𝑦(.r‘𝑆)(𝐹‘(1r‘𝑅))))
20	19, 17	eqeq12d 2779	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑥(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑥 ↔ (𝑦(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑦))
21	18, 20	anbi12d 644	. . . . . . . 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 3577	. . . . . . 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 487	. . . . . 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 488	. . . . . 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 36	. . . . 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 412	. . . 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 490	. . . . . 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 36	. . . . 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 412	. . . 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 20453	. . 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 700	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (𝐹‘(1r‘𝑅)) = (1r‘𝑆))
32	31	eqcomd 2769	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (1r‘𝑆) = (𝐹‘(1r‘𝑅)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ⟶wf 6560  ⟶1-1ontowf1o 6563  ‘cfv 6564  (class class class)co 7449  Basecbs 17428  ._rcmulr 17470  Rngcrng 20416  1_rcur 20449  Ringcrg 20501   RngIso crngim 20707

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7781  ax-cnex 11293  ax-resscn 11294  ax-1cn 11295  ax-icn 11296  ax-addcl 11297  ax-addrcl 11298  ax-mulcl 11299  ax-mulrcl 11300  ax-mulcom 11301  ax-addass 11302  ax-mulass 11303  ax-distr 11304  ax-i2m1 11305  ax-1ne0 11306  ax-1rid 11307  ax-rnegex 11308  ax-rrecex 11309  ax-cnre 11310  ax-pre-lttri 11311  ax-pre-lttrn 11312  ax-pre-ltadd 11313  ax-pre-mulgt0 11314

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6328  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-riota 7406  df-ov 7452  df-oprab 7453  df-mpo 7454  df-om 7910  df-1st 8033  df-2nd 8034  df-frecs 8335  df-wrecs 8366  df-recs 8415  df-rdg 8454  df-er 8753  df-map 8885  df-en 9010  df-dom 9011  df-sdom 9012  df-pnf 11382  df-mnf 11383  df-xr 11384  df-ltxr 11385  df-le 11386  df-sub 11580  df-neg 11581  df-nn 12374  df-2 12443  df-sets 17383  df-slot 17401  df-ndx 17413  df-base 17429  df-plusg 17482  df-0g 17653  df-mgm 18857  df-mgmhm 18922  df-sgrp 18949  df-mnd 18965  df-grp 19188  df-ghm 19469  df-abl 20039  df-mgp 20403  df-rng 20417  df-ur 20450  df-ring 20503  df-rnghm 20708  df-rngim 20709

This theorem is used by:  rngqiprngu  21656