keridl - Mathbox for Jeff Madsen

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Theorem keridl 36491

Description: The kernel of a ring homomorphism is an ideal. (Contributed by Jeff Madsen, 3-Jan-2011.)

**Hypotheses**
Ref	Expression
keridl.1	⊢ 𝐺 = (1^st ‘𝑆)
keridl.2	⊢ 𝑍 = (GId‘𝐺)

**Assertion**
Ref	Expression
keridl	⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (^◡𝐹 “ {𝑍}) ∈ (Idl‘𝑅))

Proof of Theorem keridl Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnvimass 6033	. . 3 ⊢ (^◡𝐹 “ {𝑍}) ⊆ dom 𝐹
2		eqid 2736	. . . 4 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
3		eqid 2736	. . . 4 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
4		keridl.1	. . . 4 ⊢ 𝐺 = (1^st ‘𝑆)
5		eqid 2736	. . . 4 ⊢ ran 𝐺 = ran 𝐺
6	2, 3, 4, 5	rngohomf 36425	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹:ran (1^st ‘𝑅)⟶ran 𝐺)
7	1, 6	fssdm 6688	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (^◡𝐹 “ {𝑍}) ⊆ ran (1^st ‘𝑅))
8		eqid 2736	. . . . 5 ⊢ (GId‘(1^st ‘𝑅)) = (GId‘(1^st ‘𝑅))
9	2, 3, 8	rngo0cl 36378	. . . 4 ⊢ (𝑅 ∈ RingOps → (GId‘(1^st ‘𝑅)) ∈ ran (1^st ‘𝑅))
10	9	3ad2ant1 1133	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (GId‘(1^st ‘𝑅)) ∈ ran (1^st ‘𝑅))
11		keridl.2	. . . . 5 ⊢ 𝑍 = (GId‘𝐺)
12	2, 8, 4, 11	rngohom0 36431	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘(GId‘(1^st ‘𝑅))) = 𝑍)
13		fvex 6855	. . . . 5 ⊢ (𝐹‘(GId‘(1^st ‘𝑅))) ∈ V
14	13	elsn 4601	. . . 4 ⊢ ((𝐹‘(GId‘(1^st ‘𝑅))) ∈ {𝑍} ↔ (𝐹‘(GId‘(1^st ‘𝑅))) = 𝑍)
15	12, 14	sylibr 233	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘(GId‘(1^st ‘𝑅))) ∈ {𝑍})
16		ffn 6668	. . . 4 ⊢ (𝐹:ran (1^st ‘𝑅)⟶ran 𝐺 → 𝐹 Fn ran (1^st ‘𝑅))
17		elpreima 7008	. . . 4 ⊢ (𝐹 Fn ran (1^st ‘𝑅) → ((GId‘(1^st ‘𝑅)) ∈ (^◡𝐹 “ {𝑍}) ↔ ((GId‘(1^st ‘𝑅)) ∈ ran (1^st ‘𝑅) ∧ (𝐹‘(GId‘(1^st ‘𝑅))) ∈ {𝑍})))
18	6, 16, 17	3syl 18	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((GId‘(1^st ‘𝑅)) ∈ (^◡𝐹 “ {𝑍}) ↔ ((GId‘(1^st ‘𝑅)) ∈ ran (1^st ‘𝑅) ∧ (𝐹‘(GId‘(1^st ‘𝑅))) ∈ {𝑍})))
19	10, 15, 18	mpbir2and 711	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (GId‘(1^st ‘𝑅)) ∈ (^◡𝐹 “ {𝑍}))
20		an4 654	. . . . . . . 8 ⊢ (((𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) ∈ {𝑍}) ∧ (𝑦 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑦) ∈ {𝑍})) ↔ ((𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) ∧ ((𝐹‘𝑥) ∈ {𝑍} ∧ (𝐹‘𝑦) ∈ {𝑍})))
21	2, 3, 4	rngohomadd 36428	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) = ((𝐹‘𝑥)𝐺(𝐹‘𝑦)))
22	21	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) ∧ ((𝐹‘𝑥) = 𝑍 ∧ (𝐹‘𝑦) = 𝑍)) → (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) = ((𝐹‘𝑥)𝐺(𝐹‘𝑦)))
23		oveq12 7366	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑥) = 𝑍 ∧ (𝐹‘𝑦) = 𝑍) → ((𝐹‘𝑥)𝐺(𝐹‘𝑦)) = (𝑍𝐺𝑍))
24	23	adantl 482	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) ∧ ((𝐹‘𝑥) = 𝑍 ∧ (𝐹‘𝑦) = 𝑍)) → ((𝐹‘𝑥)𝐺(𝐹‘𝑦)) = (𝑍𝐺𝑍))
25	4	rngogrpo 36369	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∈ RingOps → 𝐺 ∈ GrpOp)
26	5, 11	grpoidcl 29456	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ GrpOp → 𝑍 ∈ ran 𝐺)
27	5, 11	grpolid 29458	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑍 ∈ ran 𝐺) → (𝑍𝐺𝑍) = 𝑍)
28	25, 26, 27	syl2anc2 585	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍)
29	28	3ad2ant2 1134	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝑍𝐺𝑍) = 𝑍)
30	29	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) ∧ ((𝐹‘𝑥) = 𝑍 ∧ (𝐹‘𝑦) = 𝑍)) → (𝑍𝐺𝑍) = 𝑍)
31	22, 24, 30	3eqtrd 2780	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) ∧ ((𝐹‘𝑥) = 𝑍 ∧ (𝐹‘𝑦) = 𝑍)) → (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) = 𝑍)
32	31	ex 413	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (((𝐹‘𝑥) = 𝑍 ∧ (𝐹‘𝑦) = 𝑍) → (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) = 𝑍))
33		fvex 6855	. . . . . . . . . . . . 13 ⊢ (𝐹‘𝑥) ∈ V
34	33	elsn 4601	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑥) ∈ {𝑍} ↔ (𝐹‘𝑥) = 𝑍)
35		fvex 6855	. . . . . . . . . . . . 13 ⊢ (𝐹‘𝑦) ∈ V
36	35	elsn 4601	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑦) ∈ {𝑍} ↔ (𝐹‘𝑦) = 𝑍)
37	34, 36	anbi12i 627	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) ∈ {𝑍} ∧ (𝐹‘𝑦) ∈ {𝑍}) ↔ ((𝐹‘𝑥) = 𝑍 ∧ (𝐹‘𝑦) = 𝑍))
38		fvex 6855	. . . . . . . . . . . 12 ⊢ (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) ∈ V
39	38	elsn 4601	. . . . . . . . . . 11 ⊢ ((𝐹‘(𝑥(1^st ‘𝑅)𝑦)) ∈ {𝑍} ↔ (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) = 𝑍)
40	32, 37, 39	3imtr4g 295	. . . . . . . . . 10 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (((𝐹‘𝑥) ∈ {𝑍} ∧ (𝐹‘𝑦) ∈ {𝑍}) → (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) ∈ {𝑍}))
41	40	imdistanda 572	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (((𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) ∧ ((𝐹‘𝑥) ∈ {𝑍} ∧ (𝐹‘𝑦) ∈ {𝑍})) → ((𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) ∧ (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) ∈ {𝑍})))
42	2, 3	rngogcl 36371	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) → (𝑥(1^st ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅))
43	42	3expib 1122	. . . . . . . . . . 11 ⊢ (𝑅 ∈ RingOps → ((𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) → (𝑥(1^st ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅)))
44	43	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) → (𝑥(1^st ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅)))
45	44	anim1d 611	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (((𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) ∧ (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) ∈ {𝑍}) → ((𝑥(1^st ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅) ∧ (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) ∈ {𝑍})))
46	41, 45	syld 47	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (((𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) ∧ ((𝐹‘𝑥) ∈ {𝑍} ∧ (𝐹‘𝑦) ∈ {𝑍})) → ((𝑥(1^st ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅) ∧ (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) ∈ {𝑍})))
47	20, 46	biimtrid 241	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (((𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) ∈ {𝑍}) ∧ (𝑦 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑦) ∈ {𝑍})) → ((𝑥(1^st ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅) ∧ (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) ∈ {𝑍})))
48		elpreima 7008	. . . . . . . . 9 ⊢ (𝐹 Fn ran (1^st ‘𝑅) → (𝑥 ∈ (^◡𝐹 “ {𝑍}) ↔ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) ∈ {𝑍})))
49	6, 16, 48	3syl 18	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝑥 ∈ (^◡𝐹 “ {𝑍}) ↔ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) ∈ {𝑍})))
50		elpreima 7008	. . . . . . . . 9 ⊢ (𝐹 Fn ran (1^st ‘𝑅) → (𝑦 ∈ (^◡𝐹 “ {𝑍}) ↔ (𝑦 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑦) ∈ {𝑍})))
51	6, 16, 50	3syl 18	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝑦 ∈ (^◡𝐹 “ {𝑍}) ↔ (𝑦 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑦) ∈ {𝑍})))
52	49, 51	anbi12d 631	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ (^◡𝐹 “ {𝑍}) ∧ 𝑦 ∈ (^◡𝐹 “ {𝑍})) ↔ ((𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) ∈ {𝑍}) ∧ (𝑦 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑦) ∈ {𝑍}))))
53		elpreima 7008	. . . . . . . 8 ⊢ (𝐹 Fn ran (1^st ‘𝑅) → ((𝑥(1^st ‘𝑅)𝑦) ∈ (^◡𝐹 “ {𝑍}) ↔ ((𝑥(1^st ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅) ∧ (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) ∈ {𝑍})))
54	6, 16, 53	3syl 18	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥(1^st ‘𝑅)𝑦) ∈ (^◡𝐹 “ {𝑍}) ↔ ((𝑥(1^st ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅) ∧ (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) ∈ {𝑍})))
55	47, 52, 54	3imtr4d 293	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ (^◡𝐹 “ {𝑍}) ∧ 𝑦 ∈ (^◡𝐹 “ {𝑍})) → (𝑥(1^st ‘𝑅)𝑦) ∈ (^◡𝐹 “ {𝑍})))
56	55	impl 456	. . . . 5 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ (^◡𝐹 “ {𝑍})) ∧ 𝑦 ∈ (^◡𝐹 “ {𝑍})) → (𝑥(1^st ‘𝑅)𝑦) ∈ (^◡𝐹 “ {𝑍}))
57	56	ralrimiva 3143	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ (^◡𝐹 “ {𝑍})) → ∀𝑦 ∈ (^◡𝐹 “ {𝑍})(𝑥(1^st ‘𝑅)𝑦) ∈ (^◡𝐹 “ {𝑍}))
58	34	anbi2i 623	. . . . . . 7 ⊢ ((𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) ∈ {𝑍}) ↔ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍))
59		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (2^nd ‘𝑅) = (2^nd ‘𝑅)
60	2, 59, 3	rngocl 36360	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1^st ‘𝑅) ∧ 𝑥 ∈ ran (1^st ‘𝑅)) → (𝑧(2^nd ‘𝑅)𝑥) ∈ ran (1^st ‘𝑅))
61	60	3expb 1120	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ RingOps ∧ (𝑧 ∈ ran (1^st ‘𝑅) ∧ 𝑥 ∈ ran (1^st ‘𝑅))) → (𝑧(2^nd ‘𝑅)𝑥) ∈ ran (1^st ‘𝑅))
62	61	3ad2antl1 1185	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑧 ∈ ran (1^st ‘𝑅) ∧ 𝑥 ∈ ran (1^st ‘𝑅))) → (𝑧(2^nd ‘𝑅)𝑥) ∈ ran (1^st ‘𝑅))
63	62	anass1rs 653	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ ran (1^st ‘𝑅)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝑧(2^nd ‘𝑅)𝑥) ∈ ran (1^st ‘𝑅))
64	63	adantlrr 719	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝑧(2^nd ‘𝑅)𝑥) ∈ ran (1^st ‘𝑅))
65		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (2^nd ‘𝑆) = (2^nd ‘𝑆)
66	2, 3, 59, 65	rngohommul 36429	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑧 ∈ ran (1^st ‘𝑅) ∧ 𝑥 ∈ ran (1^st ‘𝑅))) → (𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) = ((𝐹‘𝑧)(2^nd ‘𝑆)(𝐹‘𝑥)))
67	66	anass1rs 653	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ ran (1^st ‘𝑅)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) = ((𝐹‘𝑧)(2^nd ‘𝑆)(𝐹‘𝑥)))
68	67	adantlrr 719	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) = ((𝐹‘𝑧)(2^nd ‘𝑆)(𝐹‘𝑥)))
69		oveq2 7365	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑥) = 𝑍 → ((𝐹‘𝑧)(2^nd ‘𝑆)(𝐹‘𝑥)) = ((𝐹‘𝑧)(2^nd ‘𝑆)𝑍))
70	69	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍) → ((𝐹‘𝑧)(2^nd ‘𝑆)(𝐹‘𝑥)) = ((𝐹‘𝑧)(2^nd ‘𝑆)𝑍))
71	70	ad2antlr 725	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → ((𝐹‘𝑧)(2^nd ‘𝑆)(𝐹‘𝑥)) = ((𝐹‘𝑧)(2^nd ‘𝑆)𝑍))
72	2, 3, 4, 5	rngohomcl 36426	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝐹‘𝑧) ∈ ran 𝐺)
73	11, 5, 4, 65	rngorz 36382	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ RingOps ∧ (𝐹‘𝑧) ∈ ran 𝐺) → ((𝐹‘𝑧)(2^nd ‘𝑆)𝑍) = 𝑍)
74	73	3ad2antl2 1186	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐹‘𝑧) ∈ ran 𝐺) → ((𝐹‘𝑧)(2^nd ‘𝑆)𝑍) = 𝑍)
75	72, 74	syldan 591	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → ((𝐹‘𝑧)(2^nd ‘𝑆)𝑍) = 𝑍)
76	75	adantlr 713	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → ((𝐹‘𝑧)(2^nd ‘𝑆)𝑍) = 𝑍)
77	68, 71, 76	3eqtrd 2780	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) = 𝑍)
78		fvex 6855	. . . . . . . . . . . . 13 ⊢ (𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) ∈ V
79	78	elsn 4601	. . . . . . . . . . . 12 ⊢ ((𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) ∈ {𝑍} ↔ (𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) = 𝑍)
80	77, 79	sylibr 233	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) ∈ {𝑍})
81		elpreima 7008	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn ran (1^st ‘𝑅) → ((𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}) ↔ ((𝑧(2^nd ‘𝑅)𝑥) ∈ ran (1^st ‘𝑅) ∧ (𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) ∈ {𝑍})))
82	6, 16, 81	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}) ↔ ((𝑧(2^nd ‘𝑅)𝑥) ∈ ran (1^st ‘𝑅) ∧ (𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) ∈ {𝑍})))
83	82	ad2antrr 724	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → ((𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}) ↔ ((𝑧(2^nd ‘𝑅)𝑥) ∈ ran (1^st ‘𝑅) ∧ (𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) ∈ {𝑍})))
84	64, 80, 83	mpbir2and 711	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}))
85	2, 59, 3	rngocl 36360	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝑥(2^nd ‘𝑅)𝑧) ∈ ran (1^st ‘𝑅))
86	85	3expb 1120	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ RingOps ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑧 ∈ ran (1^st ‘𝑅))) → (𝑥(2^nd ‘𝑅)𝑧) ∈ ran (1^st ‘𝑅))
87	86	3ad2antl1 1185	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑧 ∈ ran (1^st ‘𝑅))) → (𝑥(2^nd ‘𝑅)𝑧) ∈ ran (1^st ‘𝑅))
88	87	anassrs 468	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ ran (1^st ‘𝑅)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝑥(2^nd ‘𝑅)𝑧) ∈ ran (1^st ‘𝑅))
89	88	adantlrr 719	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝑥(2^nd ‘𝑅)𝑧) ∈ ran (1^st ‘𝑅))
90	2, 3, 59, 65	rngohommul 36429	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑧 ∈ ran (1^st ‘𝑅))) → (𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) = ((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑧)))
91	90	anassrs 468	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ ran (1^st ‘𝑅)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) = ((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑧)))
92	91	adantlrr 719	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) = ((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑧)))
93		oveq1 7364	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑥) = 𝑍 → ((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑧)) = (𝑍(2^nd ‘𝑆)(𝐹‘𝑧)))
94	93	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍) → ((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑧)) = (𝑍(2^nd ‘𝑆)(𝐹‘𝑧)))
95	94	ad2antlr 725	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → ((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑧)) = (𝑍(2^nd ‘𝑆)(𝐹‘𝑧)))
96	11, 5, 4, 65	rngolz 36381	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ RingOps ∧ (𝐹‘𝑧) ∈ ran 𝐺) → (𝑍(2^nd ‘𝑆)(𝐹‘𝑧)) = 𝑍)
97	96	3ad2antl2 1186	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐹‘𝑧) ∈ ran 𝐺) → (𝑍(2^nd ‘𝑆)(𝐹‘𝑧)) = 𝑍)
98	72, 97	syldan 591	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝑍(2^nd ‘𝑆)(𝐹‘𝑧)) = 𝑍)
99	98	adantlr 713	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝑍(2^nd ‘𝑆)(𝐹‘𝑧)) = 𝑍)
100	92, 95, 99	3eqtrd 2780	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) = 𝑍)
101		fvex 6855	. . . . . . . . . . . . 13 ⊢ (𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) ∈ V
102	101	elsn 4601	. . . . . . . . . . . 12 ⊢ ((𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) ∈ {𝑍} ↔ (𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) = 𝑍)
103	100, 102	sylibr 233	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) ∈ {𝑍})
104		elpreima 7008	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn ran (1^st ‘𝑅) → ((𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍}) ↔ ((𝑥(2^nd ‘𝑅)𝑧) ∈ ran (1^st ‘𝑅) ∧ (𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) ∈ {𝑍})))
105	6, 16, 104	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍}) ↔ ((𝑥(2^nd ‘𝑅)𝑧) ∈ ran (1^st ‘𝑅) ∧ (𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) ∈ {𝑍})))
106	105	ad2antrr 724	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → ((𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍}) ↔ ((𝑥(2^nd ‘𝑅)𝑧) ∈ ran (1^st ‘𝑅) ∧ (𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) ∈ {𝑍})))
107	89, 103, 106	mpbir2and 711	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍}))
108	84, 107	jca 512	. . . . . . . . 9 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → ((𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}) ∧ (𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍})))
109	108	ralrimiva 3143	. . . . . . . 8 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) → ∀𝑧 ∈ ran (1^st ‘𝑅)((𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}) ∧ (𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍})))
110	109	ex 413	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍) → ∀𝑧 ∈ ran (1^st ‘𝑅)((𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}) ∧ (𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍}))))
111	58, 110	biimtrid 241	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) ∈ {𝑍}) → ∀𝑧 ∈ ran (1^st ‘𝑅)((𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}) ∧ (𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍}))))
112	49, 111	sylbid 239	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝑥 ∈ (^◡𝐹 “ {𝑍}) → ∀𝑧 ∈ ran (1^st ‘𝑅)((𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}) ∧ (𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍}))))
113	112	imp 407	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ (^◡𝐹 “ {𝑍})) → ∀𝑧 ∈ ran (1^st ‘𝑅)((𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}) ∧ (𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍})))
114	57, 113	jca 512	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ (^◡𝐹 “ {𝑍})) → (∀𝑦 ∈ (^◡𝐹 “ {𝑍})(𝑥(1^st ‘𝑅)𝑦) ∈ (^◡𝐹 “ {𝑍}) ∧ ∀𝑧 ∈ ran (1^st ‘𝑅)((𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}) ∧ (𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍}))))
115	114	ralrimiva 3143	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ∀𝑥 ∈ (^◡𝐹 “ {𝑍})(∀𝑦 ∈ (^◡𝐹 “ {𝑍})(𝑥(1^st ‘𝑅)𝑦) ∈ (^◡𝐹 “ {𝑍}) ∧ ∀𝑧 ∈ ran (1^st ‘𝑅)((𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}) ∧ (𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍}))))
116	2, 59, 3, 8	isidl 36473	. . 3 ⊢ (𝑅 ∈ RingOps → ((^◡𝐹 “ {𝑍}) ∈ (Idl‘𝑅) ↔ ((^◡𝐹 “ {𝑍}) ⊆ ran (1^st ‘𝑅) ∧ (GId‘(1^st ‘𝑅)) ∈ (^◡𝐹 “ {𝑍}) ∧ ∀𝑥 ∈ (^◡𝐹 “ {𝑍})(∀𝑦 ∈ (^◡𝐹 “ {𝑍})(𝑥(1^st ‘𝑅)𝑦) ∈ (^◡𝐹 “ {𝑍}) ∧ ∀𝑧 ∈ ran (1^st ‘𝑅)((𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}) ∧ (𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍}))))))
117	116	3ad2ant1 1133	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((^◡𝐹 “ {𝑍}) ∈ (Idl‘𝑅) ↔ ((^◡𝐹 “ {𝑍}) ⊆ ran (1^st ‘𝑅) ∧ (GId‘(1^st ‘𝑅)) ∈ (^◡𝐹 “ {𝑍}) ∧ ∀𝑥 ∈ (^◡𝐹 “ {𝑍})(∀𝑦 ∈ (^◡𝐹 “ {𝑍})(𝑥(1^st ‘𝑅)𝑦) ∈ (^◡𝐹 “ {𝑍}) ∧ ∀𝑧 ∈ ran (1^st ‘𝑅)((𝑧(2^nd ‘𝑅)𝑥) ∈ (^◡𝐹 “ {𝑍}) ∧ (𝑥(2^nd ‘𝑅)𝑧) ∈ (^◡𝐹 “ {𝑍}))))))
118	7, 19, 115, 117	mpbir3and 1342	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (^◡𝐹 “ {𝑍}) ∈ (Idl‘𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ⊆ wss 3910 {csn 4586 ^◡ccnv 5632 ran crn 5634 “ cima 5636 Fn wfn 6491 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 1^st c1st 7919 2^nd c2nd 7920 GrpOpcgr 29431 GIdcgi 29432 RingOpscrngo 36353 RngHom crnghom 36419 Idlcidl 36466

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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-1st 7921 df-2nd 7922 df-map 8767 df-grpo 29435 df-gid 29436 df-ginv 29437 df-ablo 29487 df-ghomOLD 36343 df-rngo 36354 df-rngohom 36422 df-idl 36469

This theorem is referenced by: (None)

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