Theorem rngisom1 20400

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 20390	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑅 RngIso 𝑆) → ◡𝐹 ∈ (𝑆 RngIso 𝑅))
2		rngisom1.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑆)
3		eqid 2734	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
4	2, 3	rngimrnghm 20389	. . . . . . . . 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 20399	. . . . . . . 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 2734	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
15	2, 13, 14	rnghmmul 20383	. . . . . 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 1373	. . . . 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 6836	. . . 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 20388	. . . . . 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 1193	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → 𝑆 ∈ Rng)
21	2, 13	rngcl 20097	. . . . . 6 ⊢ ((𝑆 ∈ Rng ∧ (𝐹‘ 1 ) ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝐹‘ 1 ) · 𝑥) ∈ 𝐵)
22	20, 11, 12, 21	syl3anc 1373	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → ((𝐹‘ 1 ) · 𝑥) ∈ 𝐵)
23		f1ocnvfv2 7221	. . . . 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 685	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘((𝐹‘ 1 ) · 𝑥))) = ((𝐹‘ 1 ) · 𝑥))
25	3, 8	ringidcl 20198	. . . . . . . . . . . 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 7220	. . . . . . . . 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 7371	. . . . . . 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 1192	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → 𝑅 ∈ Ring)
33	2, 3	rngimf1o 20388	. . . . . . . . . . . 12 ⊢ (◡𝐹 ∈ (𝑆 RngIso 𝑅) → ◡𝐹:𝐵–1-1-onto→(Base‘𝑅))
34		f1of 6772	. . . . . . . . . . . 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 7027	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (◡𝐹‘𝑥) ∈ (Base‘𝑅))
39	3, 14, 8, 32, 38	ringlidmd 20205	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → ( 1 (.r‘𝑅)(◡𝐹‘𝑥)) = (◡𝐹‘𝑥))
40	31, 39	eqtrd 2769	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → ((◡𝐹‘(𝐹‘ 1 ))(.r‘𝑅)(◡𝐹‘𝑥)) = (◡𝐹‘𝑥))
41	40	fveq2d 6836	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘((◡𝐹‘(𝐹‘ 1 ))(.r‘𝑅)(◡𝐹‘𝑥))) = (𝐹‘(◡𝐹‘𝑥)))
42		f1ocnvfv2 7221	. . . . . 6 ⊢ ((𝐹:(Base‘𝑅)–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
43	19, 42	sylan 580	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
44	41, 43	eqtrd 2769	. . . 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 2777	. . 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 20383	. . . . . . 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 1373	. . . . . 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 7372	. . . . . 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 20206	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → ((◡𝐹‘𝑥)(.r‘𝑅) 1 ) = (◡𝐹‘𝑥))
53	50, 51, 52	3eqtrd 2773	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (◡𝐹‘(𝑥 · (𝐹‘ 1 ))) = (◡𝐹‘𝑥))
54	53	fveq2d 6836	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘(𝑥 · (𝐹‘ 1 )))) = (𝐹‘(◡𝐹‘𝑥)))
55	2, 13	rngcl 20097	. . . . . 6 ⊢ ((𝑆 ∈ Rng ∧ 𝑥 ∈ 𝐵 ∧ (𝐹‘ 1 ) ∈ 𝐵) → (𝑥 · (𝐹‘ 1 )) ∈ 𝐵)
56	20, 12, 11, 55	syl3anc 1373	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝑥 · (𝐹‘ 1 )) ∈ 𝐵)
57		f1ocnvfv2 7221	. . . . 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 685	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘(𝑥 · (𝐹‘ 1 )))) = (𝑥 · (𝐹‘ 1 )))
59	54, 58, 43	3eqtr3d 2777	. . 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 3126	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 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3049  ^◡ccnv 5621  ⟶wf 6486  –1-1-onto→wf1o 6489  ‘cfv 6490  (class class class)co 7356  Basecbs 17134  ._rcmulr 17176  Rngcrng 20085  1_rcur 20114  Ringcrg 20166   RngHom crnghm 20368   RngIso crngim 20369

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 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

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 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8633  df-map 8763  df-en 8882  df-dom 8883  df-sdom 8884  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-2 12206  df-sets 17089  df-slot 17107  df-ndx 17119  df-base 17135  df-plusg 17188  df-0g 17359  df-mgm 18563  df-mgmhm 18615  df-sgrp 18642  df-mnd 18658  df-grp 18864  df-ghm 19140  df-abl 19710  df-mgp 20074  df-rng 20086  df-ur 20115  df-ring 20168  df-rnghm 20370  df-rngim 20371

This theorem is referenced by:  rngisomring  20401  rngisomring1  20402