keridl - Mathbox for Jeff Madsen

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Theorem keridl 38051

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 RingOpsHom 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 6028	. . 3 ⊢ (^◡𝐹 “ {𝑍}) ⊆ dom 𝐹
2		eqid 2730	. . . 4 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
3		eqid 2730	. . . 4 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
4		keridl.1	. . . 4 ⊢ 𝐺 = (1^st ‘𝑆)
5		eqid 2730	. . . 4 ⊢ ran 𝐺 = ran 𝐺
6	2, 3, 4, 5	rngohomf 37985	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹:ran (1^st ‘𝑅)⟶ran 𝐺)
7	1, 6	fssdm 6666	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (^◡𝐹 “ {𝑍}) ⊆ ran (1^st ‘𝑅))
8		eqid 2730	. . . . 5 ⊢ (GId‘(1^st ‘𝑅)) = (GId‘(1^st ‘𝑅))
9	2, 3, 8	rngo0cl 37938	. . . 4 ⊢ (𝑅 ∈ RingOps → (GId‘(1^st ‘𝑅)) ∈ ran (1^st ‘𝑅))
10	9	3ad2ant1 1133	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (GId‘(1^st ‘𝑅)) ∈ ran (1^st ‘𝑅))
11		keridl.2	. . . . 5 ⊢ 𝑍 = (GId‘𝐺)
12	2, 8, 4, 11	rngohom0 37991	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹‘(GId‘(1^st ‘𝑅))) = 𝑍)
13		fvex 6830	. . . . 5 ⊢ (𝐹‘(GId‘(1^st ‘𝑅))) ∈ V
14	13	elsn 4589	. . . 4 ⊢ ((𝐹‘(GId‘(1^st ‘𝑅))) ∈ {𝑍} ↔ (𝐹‘(GId‘(1^st ‘𝑅))) = 𝑍)
15	12, 14	sylibr 234	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹‘(GId‘(1^st ‘𝑅))) ∈ {𝑍})
16		ffn 6647	. . . 4 ⊢ (𝐹:ran (1^st ‘𝑅)⟶ran 𝐺 → 𝐹 Fn ran (1^st ‘𝑅))
17		elpreima 6986	. . . 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 RingOpsHom 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 713	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (GId‘(1^st ‘𝑅)) ∈ (^◡𝐹 “ {𝑍}))
20		an4 656	. . . . . . . 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 37988	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 480	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 7350	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑥) = 𝑍 ∧ (𝐹‘𝑦) = 𝑍) → ((𝐹‘𝑥)𝐺(𝐹‘𝑦)) = (𝑍𝐺𝑍))
24	23	adantl 481	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 37929	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∈ RingOps → 𝐺 ∈ GrpOp)
26	5, 11	grpoidcl 30484	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ GrpOp → 𝑍 ∈ ran 𝐺)
27	5, 11	grpolid 30486	. . . . . . . . . . . . . . . 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 RingOpsHom 𝑆)) → (𝑍𝐺𝑍) = 𝑍)
30	29	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) ∧ ((𝐹‘𝑥) = 𝑍 ∧ (𝐹‘𝑦) = 𝑍)) → (𝑍𝐺𝑍) = 𝑍)
31	22, 24, 30	3eqtrd 2769	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 412	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 6830	. . . . . . . . . . . . 13 ⊢ (𝐹‘𝑥) ∈ V
34	33	elsn 4589	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑥) ∈ {𝑍} ↔ (𝐹‘𝑥) = 𝑍)
35		fvex 6830	. . . . . . . . . . . . 13 ⊢ (𝐹‘𝑦) ∈ V
36	35	elsn 4589	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑦) ∈ {𝑍} ↔ (𝐹‘𝑦) = 𝑍)
37	34, 36	anbi12i 628	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) ∈ {𝑍} ∧ (𝐹‘𝑦) ∈ {𝑍}) ↔ ((𝐹‘𝑥) = 𝑍 ∧ (𝐹‘𝑦) = 𝑍))
38		fvex 6830	. . . . . . . . . . . 12 ⊢ (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) ∈ V
39	38	elsn 4589	. . . . . . . . . . 11 ⊢ ((𝐹‘(𝑥(1^st ‘𝑅)𝑦)) ∈ {𝑍} ↔ (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) = 𝑍)
40	32, 37, 39	3imtr4g 296	. . . . . . . . . 10 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 571	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 37931	. . . . . . . . . . . 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 RingOpsHom 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 RingOpsHom 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 RingOpsHom 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 242	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 6986	. . . . . . . . 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 RingOpsHom 𝑆)) → (𝑥 ∈ (^◡𝐹 “ {𝑍}) ↔ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) ∈ {𝑍})))
50		elpreima 6986	. . . . . . . . 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 RingOpsHom 𝑆)) → (𝑦 ∈ (^◡𝐹 “ {𝑍}) ↔ (𝑦 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑦) ∈ {𝑍})))
52	49, 51	anbi12d 632	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 6986	. . . . . . . 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 RingOpsHom 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 294	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → ((𝑥 ∈ (^◡𝐹 “ {𝑍}) ∧ 𝑦 ∈ (^◡𝐹 “ {𝑍})) → (𝑥(1^st ‘𝑅)𝑦) ∈ (^◡𝐹 “ {𝑍})))
56	55	impl 455	. . . . 5 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝑥 ∈ (^◡𝐹 “ {𝑍})) ∧ 𝑦 ∈ (^◡𝐹 “ {𝑍})) → (𝑥(1^st ‘𝑅)𝑦) ∈ (^◡𝐹 “ {𝑍}))
57	56	ralrimiva 3122	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 2730	. . . . . . . . . . . . . . . 16 ⊢ (2^nd ‘𝑅) = (2^nd ‘𝑅)
60	2, 59, 3	rngocl 37920	. . . . . . . . . . . . . . 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 1186	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑧 ∈ ran (1^st ‘𝑅) ∧ 𝑥 ∈ ran (1^st ‘𝑅))) → (𝑧(2^nd ‘𝑅)𝑥) ∈ ran (1^st ‘𝑅))
63	62	anass1rs 655	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝑥 ∈ ran (1^st ‘𝑅)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝑧(2^nd ‘𝑅)𝑥) ∈ ran (1^st ‘𝑅))
64	63	adantlrr 721	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 2730	. . . . . . . . . . . . . . . 16 ⊢ (2^nd ‘𝑆) = (2^nd ‘𝑆)
66	2, 3, 59, 65	rngohommul 37989	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 655	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 721	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 7349	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑥) = 𝑍 → ((𝐹‘𝑧)(2^nd ‘𝑆)(𝐹‘𝑥)) = ((𝐹‘𝑧)(2^nd ‘𝑆)𝑍))
70	69	adantl 481	. . . . . . . . . . . . . 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 727	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 37986	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝐹‘𝑧) ∈ ran 𝐺)
73	11, 5, 4, 65	rngorz 37942	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ RingOps ∧ (𝐹‘𝑧) ∈ ran 𝐺) → ((𝐹‘𝑧)(2^nd ‘𝑆)𝑍) = 𝑍)
74	73	3ad2antl2 1187	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝐹‘𝑧) ∈ ran 𝐺) → ((𝐹‘𝑧)(2^nd ‘𝑆)𝑍) = 𝑍)
75	72, 74	syldan 591	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → ((𝐹‘𝑧)(2^nd ‘𝑆)𝑍) = 𝑍)
76	75	adantlr 715	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 2769	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) = 𝑍)
78		fvex 6830	. . . . . . . . . . . . 13 ⊢ (𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) ∈ V
79	78	elsn 4589	. . . . . . . . . . . 12 ⊢ ((𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) ∈ {𝑍} ↔ (𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) = 𝑍)
80	77, 79	sylibr 234	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) ∈ {𝑍})
81		elpreima 6986	. . . . . . . . . . . . 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 RingOpsHom 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 726	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 713	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 37920	. . . . . . . . . . . . . . 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 1186	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑧 ∈ ran (1^st ‘𝑅))) → (𝑥(2^nd ‘𝑅)𝑧) ∈ ran (1^st ‘𝑅))
88	87	anassrs 467	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝑥 ∈ ran (1^st ‘𝑅)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝑥(2^nd ‘𝑅)𝑧) ∈ ran (1^st ‘𝑅))
89	88	adantlrr 721	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 37989	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 467	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 721	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 7348	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑥) = 𝑍 → ((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑧)) = (𝑍(2^nd ‘𝑆)(𝐹‘𝑧)))
94	93	adantl 481	. . . . . . . . . . . . . 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 727	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 37941	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ RingOps ∧ (𝐹‘𝑧) ∈ ran 𝐺) → (𝑍(2^nd ‘𝑆)(𝐹‘𝑧)) = 𝑍)
97	96	3ad2antl2 1187	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝐹‘𝑧) ∈ ran 𝐺) → (𝑍(2^nd ‘𝑆)(𝐹‘𝑧)) = 𝑍)
98	72, 97	syldan 591	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝑍(2^nd ‘𝑆)(𝐹‘𝑧)) = 𝑍)
99	98	adantlr 715	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 2769	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) = 𝑍)
101		fvex 6830	. . . . . . . . . . . . 13 ⊢ (𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) ∈ V
102	101	elsn 4589	. . . . . . . . . . . 12 ⊢ ((𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) ∈ {𝑍} ↔ (𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) = 𝑍)
103	100, 102	sylibr 234	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) ∈ {𝑍})
104		elpreima 6986	. . . . . . . . . . . . 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 RingOpsHom 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 726	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 713	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 511	. . . . . . . . 9 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 3122	. . . . . . . 8 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 412	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 242	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 240	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 406	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 511	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 3122	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 38033	. . 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 RingOpsHom 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 1343	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (^◡𝐹 “ {𝑍}) ∈ (Idl‘𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2110 ∀wral 3045 ⊆ wss 3900 {csn 4574 ^◡ccnv 5613 ran crn 5615 “ cima 5617 Fn wfn 6472 ⟶wf 6473 ‘cfv 6477 (class class class)co 7341 1^st c1st 7914 2^nd c2nd 7915 GrpOpcgr 30459 GIdcgi 30460 RingOpscrngo 37913 RingOpsHom crngohom 37979 Idlcidl 38026

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 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4858 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-1st 7916 df-2nd 7917 df-map 8747 df-grpo 30463 df-gid 30464 df-ginv 30465 df-ablo 30515 df-ghomOLD 37903 df-rngo 37914 df-rngohom 37982 df-idl 38029

This theorem is referenced by: (None)

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