Theorem rngisom1 20492

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 20482	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑅 RngIso 𝑆) → ◡𝐹 ∈ (𝑆 RngIso 𝑅))
2		rngisom1.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑆)
3		eqid 2740	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
4	2, 3	rngimrnghm 20481	. . . . . . . . 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 1135	. . . . . . 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 20491	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (𝐹‘ 1 ) ∈ 𝐵)
10	9	3adant2 1131	. . . . . . 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 2740	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
15	2, 13, 14	rnghmmul 20475	. . . . . 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 1371	. . . . 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 6924	. . . 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 20480	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RngIso 𝑆) → 𝐹:(Base‘𝑅)–1-1-onto→𝐵)
19	18	3ad2ant3 1135	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹:(Base‘𝑅)–1-1-onto→𝐵)
20		simpl2 1192	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → 𝑆 ∈ Rng)
21	2, 13	rngcl 20191	. . . . . 6 ⊢ ((𝑆 ∈ Rng ∧ (𝐹‘ 1 ) ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝐹‘ 1 ) · 𝑥) ∈ 𝐵)
22	20, 11, 12, 21	syl3anc 1371	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → ((𝐹‘ 1 ) · 𝑥) ∈ 𝐵)
23		f1ocnvfv2 7313	. . . . 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 684	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘((𝐹‘ 1 ) · 𝑥))) = ((𝐹‘ 1 ) · 𝑥))
25	3, 8	ringidcl 20289	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
26	25	3ad2ant1 1133	. . . . . . . . . . 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 7312	. . . . . . . . 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 7463	. . . . . . 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 1191	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → 𝑅 ∈ Ring)
33	2, 3	rngimf1o 20480	. . . . . . . . . . . 12 ⊢ (◡𝐹 ∈ (𝑆 RngIso 𝑅) → ◡𝐹:𝐵–1-1-onto→(Base‘𝑅))
34		f1of 6862	. . . . . . . . . . . 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 1135	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → ◡𝐹:𝐵⟶(Base‘𝑅))
38	37	ffvelcdmda 7118	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (◡𝐹‘𝑥) ∈ (Base‘𝑅))
39	3, 14, 8, 32, 38	ringlidmd 20295	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → ( 1 (.r‘𝑅)(◡𝐹‘𝑥)) = (◡𝐹‘𝑥))
40	31, 39	eqtrd 2780	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → ((◡𝐹‘(𝐹‘ 1 ))(.r‘𝑅)(◡𝐹‘𝑥)) = (◡𝐹‘𝑥))
41	40	fveq2d 6924	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘((◡𝐹‘(𝐹‘ 1 ))(.r‘𝑅)(◡𝐹‘𝑥))) = (𝐹‘(◡𝐹‘𝑥)))
42		f1ocnvfv2 7313	. . . . . 6 ⊢ ((𝐹:(Base‘𝑅)–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
43	19, 42	sylan 579	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
44	41, 43	eqtrd 2780	. . . 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 2788	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → ((𝐹‘ 1 ) · 𝑥) = 𝑥)
46	1	3ad2ant3 1135	. . . . . . . . 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 20475	. . . . . . 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 1371	. . . . . 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 7464	. . . . . 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 20296	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → ((◡𝐹‘𝑥)(.r‘𝑅) 1 ) = (◡𝐹‘𝑥))
53	50, 51, 52	3eqtrd 2784	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (◡𝐹‘(𝑥 · (𝐹‘ 1 ))) = (◡𝐹‘𝑥))
54	53	fveq2d 6924	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘(𝑥 · (𝐹‘ 1 )))) = (𝐹‘(◡𝐹‘𝑥)))
55	2, 13	rngcl 20191	. . . . . 6 ⊢ ((𝑆 ∈ Rng ∧ 𝑥 ∈ 𝐵 ∧ (𝐹‘ 1 ) ∈ 𝐵) → (𝑥 · (𝐹‘ 1 )) ∈ 𝐵)
56	20, 12, 11, 55	syl3anc 1371	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝑥 · (𝐹‘ 1 )) ∈ 𝐵)
57		f1ocnvfv2 7313	. . . . 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 684	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘(𝑥 · (𝐹‘ 1 )))) = (𝑥 · (𝐹‘ 1 )))
59	54, 58, 43	3eqtr3d 2788	. . 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 3152	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 1537   ∈ wcel 2108  ∀wral 3067  ^◡ccnv 5699  ⟶wf 6569  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  ._rcmulr 17312  Rngcrng 20179  1_rcur 20208  Ringcrg 20260   RngHom crnghm 20460   RngIso crngim 20461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-plusg 17324  df-0g 17501  df-mgm 18678  df-mgmhm 18730  df-sgrp 18757  df-mnd 18773  df-grp 18976  df-ghm 19253  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-ring 20262  df-rnghm 20462  df-rngim 20463

This theorem is referenced by:  rngisomring  20493  rngisomring1  20494