Theorem rngisomring1 20503

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 2761	. . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅)
2		eqid 2761	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2761	. . . 4 ⊢ (.r‘𝑆) = (.r‘𝑆)
4	1, 2, 3	rngisom1 20501	. . 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 2762	. . . 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 2762	. . . 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 2761	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
8	7, 2	rngimf1o 20489	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑅 RngIso 𝑆) → 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆))
9		f1of 6800	. . . . . . . 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 1147	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
12	7, 1	ringidcl 20301	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
13	12	3ad2ant1 1145	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (1r‘𝑅) ∈ (Base‘𝑅))
14	11, 13	ffvelcdmd 7060	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (𝐹‘(1r‘𝑅)) ∈ (Base‘𝑆))
15	14	adantr 484	. . . 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 7398	. . . . . . . . . 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 2777	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑥) = 𝑥 ↔ ((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑦) = 𝑦))
19		oveq1 7397	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥(.r‘𝑆)(𝐹‘(1r‘𝑅))) = (𝑦(.r‘𝑆)(𝐹‘(1r‘𝑅))))
20	19, 17	eqeq12d 2777	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑥(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑥 ↔ (𝑦(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑦))
21	18, 20	anbi12d 641	. . . . . . . 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 485	. . . . . 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 486	. . . . . 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 410	. . . 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 488	. . . . . 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 410	. . . 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 20221	. . 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 697	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (𝐹‘(1r‘𝑅)) = (1r‘𝑆))
32	31	eqcomd 2767	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (1r‘𝑆) = (𝐹‘(1r‘𝑅)))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ⟶wf 6511  –1-1-onto→wf1o 6514  ‘cfv 6515  (class class class)co 7390  Basecbs 17235  ._rcmulr 17277  Rngcrng 20188  1_rcur 20217  Ringcrg 20269   RngIso crngim 20470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712  ax-cnex 11122  ax-resscn 11123  ax-1cn 11124  ax-icn 11125  ax-addcl 11126  ax-addrcl 11127  ax-mulcl 11128  ax-mulrcl 11129  ax-mulcom 11130  ax-addass 11131  ax-mulass 11132  ax-distr 11133  ax-i2m1 11134  ax-1ne0 11135  ax-1rid 11136  ax-rnegex 11137  ax-rrecex 11138  ax-cnre 11139  ax-pre-lttri 11140  ax-pre-lttrn 11141  ax-pre-ltadd 11142  ax-pre-mulgt0 11143

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7841  df-1st 7964  df-2nd 7965  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-rdg 8374  df-er 8671  df-map 8803  df-en 8921  df-dom 8922  df-sdom 8923  df-pnf 11211  df-mnf 11212  df-xr 11213  df-ltxr 11214  df-le 11215  df-sub 11409  df-neg 11410  df-nn 12204  df-2 12273  df-sets 17190  df-slot 17208  df-ndx 17220  df-base 17236  df-plusg 17289  df-0g 17460  df-mgm 18664  df-mgmhm 18716  df-sgrp 18743  df-mnd 18759  df-grp 18968  df-ghm 19244  df-abl 19813  df-mgp 20177  df-rng 20189  df-ur 20218  df-ring 20271  df-rnghm 20471  df-rngim 20472

This theorem is referenced by:  rngqiprngu  21375