Theorem rngisom1 46703

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 RngIsom 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 46690	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑅 RngIsom 𝑆) → ◡𝐹 ∈ (𝑆 RngIsom 𝑅))
2		rngisom1.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑆)
3		eqid 2732	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
4	2, 3	rngimrnghm 46689	. . . . . . . . 9 ⊢ (◡𝐹 ∈ (𝑆 RngIsom 𝑅) → ◡𝐹 ∈ (𝑆 RngHomo 𝑅))
5	1, 4	syl 17	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑅 RngIsom 𝑆) → ◡𝐹 ∈ (𝑆 RngHomo 𝑅))
6	5	3ad2ant3 1135	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) → ◡𝐹 ∈ (𝑆 RngHomo 𝑅))
7	6	adantr 481	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) ∧ 𝑥 ∈ 𝐵) → ◡𝐹 ∈ (𝑆 RngHomo 𝑅))
8		rngisom1.1	. . . . . . . . 9 ⊢ 1 = (1r‘𝑅)
9	8, 2	rngisomfv1 46702	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) → (𝐹‘ 1 ) ∈ 𝐵)
10	9	3adant2 1131	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) → (𝐹‘ 1 ) ∈ 𝐵)
11	10	adantr 481	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘ 1 ) ∈ 𝐵)
12		simpr 485	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
13		rngisom1.t	. . . . . . 7 ⊢ · = (.r‘𝑆)
14		eqid 2732	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
15	2, 13, 14	rnghmmul 46683	. . . . . 6 ⊢ ((◡𝐹 ∈ (𝑆 RngHomo 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 RngIsom 𝑆)) ∧ 𝑥 ∈ 𝐵) → (◡𝐹‘((𝐹‘ 1 ) · 𝑥)) = ((◡𝐹‘(𝐹‘ 1 ))(.r‘𝑅)(◡𝐹‘𝑥)))
17	16	fveq2d 6892	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘((𝐹‘ 1 ) · 𝑥))) = (𝐹‘((◡𝐹‘(𝐹‘ 1 ))(.r‘𝑅)(◡𝐹‘𝑥))))
18	3, 2	rngimf1o 46688	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RngIsom 𝑆) → 𝐹:(Base‘𝑅)–1-1-onto→𝐵)
19	18	3ad2ant3 1135	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) → 𝐹:(Base‘𝑅)–1-1-onto→𝐵)
20		simpl2 1192	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) ∧ 𝑥 ∈ 𝐵) → 𝑆 ∈ Rng)
21	2, 13	rngcl 46649	. . . . . 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 RngIsom 𝑆)) ∧ 𝑥 ∈ 𝐵) → ((𝐹‘ 1 ) · 𝑥) ∈ 𝐵)
23		f1ocnvfv2 7271	. . . . 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 683	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘((𝐹‘ 1 ) · 𝑥))) = ((𝐹‘ 1 ) · 𝑥))
25	3, 8	ringidcl 20076	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
26	25	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) → 1 ∈ (Base‘𝑅))
27	19, 26	jca 512	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) → (𝐹:(Base‘𝑅)–1-1-onto→𝐵 ∧ 1 ∈ (Base‘𝑅)))
28	27	adantr 481	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹:(Base‘𝑅)–1-1-onto→𝐵 ∧ 1 ∈ (Base‘𝑅)))
29		f1ocnvfv1 7270	. . . . . . . . 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 RngIsom 𝑆)) ∧ 𝑥 ∈ 𝐵) → (◡𝐹‘(𝐹‘ 1 )) = 1 )
31	30	oveq1d 7420	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 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 RngIsom 𝑆)) ∧ 𝑥 ∈ 𝐵) → 𝑅 ∈ Ring)
33	2, 3	rngimf1o 46688	. . . . . . . . . . . 12 ⊢ (◡𝐹 ∈ (𝑆 RngIsom 𝑅) → ◡𝐹:𝐵–1-1-onto→(Base‘𝑅))
34		f1of 6830	. . . . . . . . . . . 12 ⊢ (◡𝐹:𝐵–1-1-onto→(Base‘𝑅) → ◡𝐹:𝐵⟶(Base‘𝑅))
35	33, 34	syl 17	. . . . . . . . . . 11 ⊢ (◡𝐹 ∈ (𝑆 RngIsom 𝑅) → ◡𝐹:𝐵⟶(Base‘𝑅))
36	1, 35	syl 17	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑅 RngIsom 𝑆) → ◡𝐹:𝐵⟶(Base‘𝑅))
37	36	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) → ◡𝐹:𝐵⟶(Base‘𝑅))
38	37	ffvelcdmda 7083	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) ∧ 𝑥 ∈ 𝐵) → (◡𝐹‘𝑥) ∈ (Base‘𝑅))
39	3, 14, 8, 32, 38	ringlidmd 20082	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) ∧ 𝑥 ∈ 𝐵) → ( 1 (.r‘𝑅)(◡𝐹‘𝑥)) = (◡𝐹‘𝑥))
40	31, 39	eqtrd 2772	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) ∧ 𝑥 ∈ 𝐵) → ((◡𝐹‘(𝐹‘ 1 ))(.r‘𝑅)(◡𝐹‘𝑥)) = (◡𝐹‘𝑥))
41	40	fveq2d 6892	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘((◡𝐹‘(𝐹‘ 1 ))(.r‘𝑅)(◡𝐹‘𝑥))) = (𝐹‘(◡𝐹‘𝑥)))
42		f1ocnvfv2 7271	. . . . . 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 RngIsom 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
44	41, 43	eqtrd 2772	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 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 RngIsom 𝑆)) ∧ 𝑥 ∈ 𝐵) → ((𝐹‘ 1 ) · 𝑥) = 𝑥)
46	1	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) → ◡𝐹 ∈ (𝑆 RngIsom 𝑅))
47	46, 4	syl 17	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) → ◡𝐹 ∈ (𝑆 RngHomo 𝑅))
48	47	adantr 481	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) ∧ 𝑥 ∈ 𝐵) → ◡𝐹 ∈ (𝑆 RngHomo 𝑅))
49	2, 13, 14	rnghmmul 46683	. . . . . . 7 ⊢ ((◡𝐹 ∈ (𝑆 RngHomo 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 RngIsom 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 RngIsom 𝑆)) ∧ 𝑥 ∈ 𝐵) → ((◡𝐹‘𝑥)(.r‘𝑅)(◡𝐹‘(𝐹‘ 1 ))) = ((◡𝐹‘𝑥)(.r‘𝑅) 1 ))
52	3, 14, 8, 32, 38	ringridmd 20083	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) ∧ 𝑥 ∈ 𝐵) → ((◡𝐹‘𝑥)(.r‘𝑅) 1 ) = (◡𝐹‘𝑥))
53	50, 51, 52	3eqtrd 2776	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) ∧ 𝑥 ∈ 𝐵) → (◡𝐹‘(𝑥 · (𝐹‘ 1 ))) = (◡𝐹‘𝑥))
54	53	fveq2d 6892	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘(𝑥 · (𝐹‘ 1 )))) = (𝐹‘(◡𝐹‘𝑥)))
55	2, 13	rngcl 46649	. . . . . 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 RngIsom 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝑥 · (𝐹‘ 1 )) ∈ 𝐵)
57		f1ocnvfv2 7271	. . . . 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 683	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘(𝑥 · (𝐹‘ 1 )))) = (𝑥 · (𝐹‘ 1 )))
59	54, 58, 43	3eqtr3d 2780	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) ∧ 𝑥 ∈ 𝐵) → (𝑥 · (𝐹‘ 1 )) = 𝑥)
60	45, 59	jca 512	. 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) ∧ 𝑥 ∈ 𝐵) → (((𝐹‘ 1 ) · 𝑥) = 𝑥 ∧ (𝑥 · (𝐹‘ 1 )) = 𝑥))
61	60	ralrimiva 3146	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIsom 𝑆)) → ∀𝑥 ∈ 𝐵 (((𝐹‘ 1 ) · 𝑥) = 𝑥 ∧ (𝑥 · (𝐹‘ 1 )) = 𝑥))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ^◡ccnv 5674  ⟶wf 6536  –1-1-onto→wf1o 6539  ‘cfv 6540  (class class class)co 7405  Basecbs 17140  ._rcmulr 17194  1_rcur 19998  Ringcrg 20049  Rngcrng 46634   RngHomo crngh 46668   RngIsom crngs 46669

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-plusg 17206  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-ghm 19084  df-abl 19645  df-mgp 19982  df-ur 19999  df-ring 20051  df-mgmhm 46535  df-rng 46635  df-rnghomo 46670  df-rngisom 46671

This theorem is referenced by:  rngisomring  46704