Theorem rngisom1 20426

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 20416	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑅 RngIso 𝑆) → ◡𝐹 ∈ (𝑆 RngIso 𝑅))
2		rngisom1.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑆)
3		eqid 2735	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
4	2, 3	rngimrnghm 20415	. . . . . . . . 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 20425	. . . . . . . 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 2735	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
15	2, 13, 14	rnghmmul 20409	. . . . . 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 6880	. . . 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 20414	. . . . . 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 20124	. . . . . 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 7270	. . . . 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 20225	. . . . . . . . . . . 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 7269	. . . . . . . . 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 7420	. . . . . . 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 20414	. . . . . . . . . . . 12 ⊢ (◡𝐹 ∈ (𝑆 RngIso 𝑅) → ◡𝐹:𝐵–1-1-onto→(Base‘𝑅))
34		f1of 6818	. . . . . . . . . . . 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 7074	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (◡𝐹‘𝑥) ∈ (Base‘𝑅))
39	3, 14, 8, 32, 38	ringlidmd 20232	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → ( 1 (.r‘𝑅)(◡𝐹‘𝑥)) = (◡𝐹‘𝑥))
40	31, 39	eqtrd 2770	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → ((◡𝐹‘(𝐹‘ 1 ))(.r‘𝑅)(◡𝐹‘𝑥)) = (◡𝐹‘𝑥))
41	40	fveq2d 6880	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘((◡𝐹‘(𝐹‘ 1 ))(.r‘𝑅)(◡𝐹‘𝑥))) = (𝐹‘(◡𝐹‘𝑥)))
42		f1ocnvfv2 7270	. . . . . 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 2770	. . . 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 2778	. . 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 20409	. . . . . . 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 7421	. . . . . 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 20233	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → ((◡𝐹‘𝑥)(.r‘𝑅) 1 ) = (◡𝐹‘𝑥))
53	50, 51, 52	3eqtrd 2774	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (◡𝐹‘(𝑥 · (𝐹‘ 1 ))) = (◡𝐹‘𝑥))
54	53	fveq2d 6880	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘(𝑥 · (𝐹‘ 1 )))) = (𝐹‘(◡𝐹‘𝑥)))
55	2, 13	rngcl 20124	. . . . . 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 7270	. . . . 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 2778	. . 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 3132	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 1540   ∈ wcel 2108  ∀wral 3051  ^◡ccnv 5653  ⟶wf 6527  –1-1-onto→wf1o 6530  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  ._rcmulr 17272  Rngcrng 20112  1_rcur 20141  Ringcrg 20193   RngHom crnghm 20394   RngIso crngim 20395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8719  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-plusg 17284  df-0g 17455  df-mgm 18618  df-mgmhm 18670  df-sgrp 18697  df-mnd 18713  df-grp 18919  df-ghm 19196  df-abl 19764  df-mgp 20101  df-rng 20113  df-ur 20142  df-ring 20195  df-rnghm 20396  df-rngim 20397

This theorem is referenced by:  rngisomring  20427  rngisomring1  20428