Theorem rngisom1 20414

Description: If there is a non-unital ring isomorphism between a unital ring and a non-unital ring, then the function value of the ring unity of the unital ring is a ring unity of the non-unital ring. (Contributed by AV, 27-Feb-2025.)

Hypotheses

Ref	Expression
rngisom1.1	⊢ 1 = (1r‘𝑅)
rngisom1.b	⊢ 𝐵 = (Base‘𝑆)
rngisom1.t	⊢ · = (.r‘𝑆)

Assertion

Ref	Expression
rngisom1	⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → ∀𝑥 ∈ 𝐵 (((𝐹‘ 1 ) · 𝑥) = 𝑥 ∧ (𝑥 · (𝐹‘ 1 )) = 𝑥))

Distinct variable groups:   𝑥,𝐹   𝑥,𝑅   𝑥,𝑆

Allowed substitution hints:   𝐵(𝑥)   · (𝑥)   1 (𝑥)

Proof of Theorem rngisom1

Step	Hyp	Ref	Expression
1		rngimcnv 20404	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑅 RngIso 𝑆) → ◡𝐹 ∈ (𝑆 RngIso 𝑅))
2		rngisom1.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑆)
3		eqid 2737	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
4	2, 3	rngimrnghm 20403	. . . . . . . . 9 ⊢ (◡𝐹 ∈ (𝑆 RngIso 𝑅) → ◡𝐹 ∈ (𝑆 RngHom 𝑅))
5	1, 4	syl 17	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑅 RngIso 𝑆) → ◡𝐹 ∈ (𝑆 RngHom 𝑅))
6	5	3ad2ant3 1136	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → ◡𝐹 ∈ (𝑆 RngHom 𝑅))
7	6	adantr 480	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → ◡𝐹 ∈ (𝑆 RngHom 𝑅))
8		rngisom1.1	. . . . . . . . 9 ⊢ 1 = (1r‘𝑅)
9	8, 2	rngisomfv1 20413	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (𝐹‘ 1 ) ∈ 𝐵)
10	9	3adant2 1132	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (𝐹‘ 1 ) ∈ 𝐵)
11	10	adantr 480	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘ 1 ) ∈ 𝐵)
12		simpr 484	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
13		rngisom1.t	. . . . . . 7 ⊢ · = (.r‘𝑆)
14		eqid 2737	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
15	2, 13, 14	rnghmmul 20397	. . . . . 6 ⊢ ((◡𝐹 ∈ (𝑆 RngHom 𝑅) ∧ (𝐹‘ 1 ) ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (◡𝐹‘((𝐹‘ 1 ) · 𝑥)) = ((◡𝐹‘(𝐹‘ 1 ))(.r‘𝑅)(◡𝐹‘𝑥)))
16	7, 11, 12, 15	syl3anc 1374	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (◡𝐹‘((𝐹‘ 1 ) · 𝑥)) = ((◡𝐹‘(𝐹‘ 1 ))(.r‘𝑅)(◡𝐹‘𝑥)))
17	16	fveq2d 6846	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘((𝐹‘ 1 ) · 𝑥))) = (𝐹‘((◡𝐹‘(𝐹‘ 1 ))(.r‘𝑅)(◡𝐹‘𝑥))))
18	3, 2	rngimf1o 20402	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RngIso 𝑆) → 𝐹:(Base‘𝑅)–1-1-onto→𝐵)
19	18	3ad2ant3 1136	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹:(Base‘𝑅)–1-1-onto→𝐵)
20		simpl2 1194	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → 𝑆 ∈ Rng)
21	2, 13	rngcl 20111	. . . . . 6 ⊢ ((𝑆 ∈ Rng ∧ (𝐹‘ 1 ) ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝐹‘ 1 ) · 𝑥) ∈ 𝐵)
22	20, 11, 12, 21	syl3anc 1374	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → ((𝐹‘ 1 ) · 𝑥) ∈ 𝐵)
23		f1ocnvfv2 7233	. . . . 5 ⊢ ((𝐹:(Base‘𝑅)–1-1-onto→𝐵 ∧ ((𝐹‘ 1 ) · 𝑥) ∈ 𝐵) → (𝐹‘(◡𝐹‘((𝐹‘ 1 ) · 𝑥))) = ((𝐹‘ 1 ) · 𝑥))
24	19, 22, 23	syl2an2r 686	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘((𝐹‘ 1 ) · 𝑥))) = ((𝐹‘ 1 ) · 𝑥))
25	3, 8	ringidcl 20212	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
26	25	3ad2ant1 1134	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 1 ∈ (Base‘𝑅))
27	19, 26	jca 511	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (𝐹:(Base‘𝑅)–1-1-onto→𝐵 ∧ 1 ∈ (Base‘𝑅)))
28	27	adantr 480	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹:(Base‘𝑅)–1-1-onto→𝐵 ∧ 1 ∈ (Base‘𝑅)))
29		f1ocnvfv1 7232	. . . . . . . . 9 ⊢ ((𝐹:(Base‘𝑅)–1-1-onto→𝐵 ∧ 1 ∈ (Base‘𝑅)) → (◡𝐹‘(𝐹‘ 1 )) = 1 )
30	28, 29	syl 17	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (◡𝐹‘(𝐹‘ 1 )) = 1 )
31	30	oveq1d 7383	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → ((◡𝐹‘(𝐹‘ 1 ))(.r‘𝑅)(◡𝐹‘𝑥)) = ( 1 (.r‘𝑅)(◡𝐹‘𝑥)))
32		simpl1 1193	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → 𝑅 ∈ Ring)
33	2, 3	rngimf1o 20402	. . . . . . . . . . . 12 ⊢ (◡𝐹 ∈ (𝑆 RngIso 𝑅) → ◡𝐹:𝐵–1-1-onto→(Base‘𝑅))
34		f1of 6782	. . . . . . . . . . . 12 ⊢ (◡𝐹:𝐵–1-1-onto→(Base‘𝑅) → ◡𝐹:𝐵⟶(Base‘𝑅))
35	33, 34	syl 17	. . . . . . . . . . 11 ⊢ (◡𝐹 ∈ (𝑆 RngIso 𝑅) → ◡𝐹:𝐵⟶(Base‘𝑅))
36	1, 35	syl 17	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑅 RngIso 𝑆) → ◡𝐹:𝐵⟶(Base‘𝑅))
37	36	3ad2ant3 1136	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → ◡𝐹:𝐵⟶(Base‘𝑅))
38	37	ffvelcdmda 7038	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (◡𝐹‘𝑥) ∈ (Base‘𝑅))
39	3, 14, 8, 32, 38	ringlidmd 20219	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → ( 1 (.r‘𝑅)(◡𝐹‘𝑥)) = (◡𝐹‘𝑥))
40	31, 39	eqtrd 2772	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → ((◡𝐹‘(𝐹‘ 1 ))(.r‘𝑅)(◡𝐹‘𝑥)) = (◡𝐹‘𝑥))
41	40	fveq2d 6846	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘((◡𝐹‘(𝐹‘ 1 ))(.r‘𝑅)(◡𝐹‘𝑥))) = (𝐹‘(◡𝐹‘𝑥)))
42		f1ocnvfv2 7233	. . . . . 6 ⊢ ((𝐹:(Base‘𝑅)–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
43	19, 42	sylan 581	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
44	41, 43	eqtrd 2772	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘((◡𝐹‘(𝐹‘ 1 ))(.r‘𝑅)(◡𝐹‘𝑥))) = 𝑥)
45	17, 24, 44	3eqtr3d 2780	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → ((𝐹‘ 1 ) · 𝑥) = 𝑥)
46	1	3ad2ant3 1136	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → ◡𝐹 ∈ (𝑆 RngIso 𝑅))
47	46, 4	syl 17	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → ◡𝐹 ∈ (𝑆 RngHom 𝑅))
48	47	adantr 480	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → ◡𝐹 ∈ (𝑆 RngHom 𝑅))
49	2, 13, 14	rnghmmul 20397	. . . . . . 7 ⊢ ((◡𝐹 ∈ (𝑆 RngHom 𝑅) ∧ 𝑥 ∈ 𝐵 ∧ (𝐹‘ 1 ) ∈ 𝐵) → (◡𝐹‘(𝑥 · (𝐹‘ 1 ))) = ((◡𝐹‘𝑥)(.r‘𝑅)(◡𝐹‘(𝐹‘ 1 ))))
50	48, 12, 11, 49	syl3anc 1374	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (◡𝐹‘(𝑥 · (𝐹‘ 1 ))) = ((◡𝐹‘𝑥)(.r‘𝑅)(◡𝐹‘(𝐹‘ 1 ))))
51	30	oveq2d 7384	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → ((◡𝐹‘𝑥)(.r‘𝑅)(◡𝐹‘(𝐹‘ 1 ))) = ((◡𝐹‘𝑥)(.r‘𝑅) 1 ))
52	3, 14, 8, 32, 38	ringridmd 20220	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → ((◡𝐹‘𝑥)(.r‘𝑅) 1 ) = (◡𝐹‘𝑥))
53	50, 51, 52	3eqtrd 2776	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (◡𝐹‘(𝑥 · (𝐹‘ 1 ))) = (◡𝐹‘𝑥))
54	53	fveq2d 6846	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘(𝑥 · (𝐹‘ 1 )))) = (𝐹‘(◡𝐹‘𝑥)))
55	2, 13	rngcl 20111	. . . . . 6 ⊢ ((𝑆 ∈ Rng ∧ 𝑥 ∈ 𝐵 ∧ (𝐹‘ 1 ) ∈ 𝐵) → (𝑥 · (𝐹‘ 1 )) ∈ 𝐵)
56	20, 12, 11, 55	syl3anc 1374	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝑥 · (𝐹‘ 1 )) ∈ 𝐵)
57		f1ocnvfv2 7233	. . . . 5 ⊢ ((𝐹:(Base‘𝑅)–1-1-onto→𝐵 ∧ (𝑥 · (𝐹‘ 1 )) ∈ 𝐵) → (𝐹‘(◡𝐹‘(𝑥 · (𝐹‘ 1 )))) = (𝑥 · (𝐹‘ 1 )))
58	19, 56, 57	syl2an2r 686	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘(𝑥 · (𝐹‘ 1 )))) = (𝑥 · (𝐹‘ 1 )))
59	54, 58, 43	3eqtr3d 2780	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝑥 · (𝐹‘ 1 )) = 𝑥)
60	45, 59	jca 511	. 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (((𝐹‘ 1 ) · 𝑥) = 𝑥 ∧ (𝑥 · (𝐹‘ 1 )) = 𝑥))
61	60	ralrimiva 3130	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → ∀𝑥 ∈ 𝐵 (((𝐹‘ 1 ) · 𝑥) = 𝑥 ∧ (𝑥 · (𝐹‘ 1 )) = 𝑥))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ^◡ccnv 5631  ⟶wf 6496  –1-1-onto→wf1o 6499  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  ._rcmulr 17190  Rngcrng 20099  1_rcur 20128  Ringcrg 20180   RngHom crnghm 20382   RngIso crngim 20383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-er 8645  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-plusg 17202  df-0g 17373  df-mgm 18577  df-mgmhm 18629  df-sgrp 18656  df-mnd 18672  df-grp 18878  df-ghm 19154  df-abl 19724  df-mgp 20088  df-rng 20100  df-ur 20129  df-ring 20182  df-rnghm 20384  df-rngim 20385

This theorem is referenced by:  rngisomring  20415  rngisomring1  20416