keridl - Mathbox for Jeff Madsen

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Theorem keridl 37440

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 6079	. . 3 ⊢ (^◡𝐹 “ {𝑍}) ⊆ dom 𝐹
2		eqid 2727	. . . 4 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
3		eqid 2727	. . . 4 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
4		keridl.1	. . . 4 ⊢ 𝐺 = (1^st ‘𝑆)
5		eqid 2727	. . . 4 ⊢ ran 𝐺 = ran 𝐺
6	2, 3, 4, 5	rngohomf 37374	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹:ran (1^st ‘𝑅)⟶ran 𝐺)
7	1, 6	fssdm 6736	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (^◡𝐹 “ {𝑍}) ⊆ ran (1^st ‘𝑅))
8		eqid 2727	. . . . 5 ⊢ (GId‘(1^st ‘𝑅)) = (GId‘(1^st ‘𝑅))
9	2, 3, 8	rngo0cl 37327	. . . 4 ⊢ (𝑅 ∈ RingOps → (GId‘(1^st ‘𝑅)) ∈ ran (1^st ‘𝑅))
10	9	3ad2ant1 1131	. . 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 37380	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹‘(GId‘(1^st ‘𝑅))) = 𝑍)
13		fvex 6904	. . . . 5 ⊢ (𝐹‘(GId‘(1^st ‘𝑅))) ∈ V
14	13	elsn 4639	. . . 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 RingOpsHom 𝑆)) → (𝐹‘(GId‘(1^st ‘𝑅))) ∈ {𝑍})
16		ffn 6716	. . . 4 ⊢ (𝐹:ran (1^st ‘𝑅)⟶ran 𝐺 → 𝐹 Fn ran (1^st ‘𝑅))
17		elpreima 7061	. . . 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 712	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (GId‘(1^st ‘𝑅)) ∈ (^◡𝐹 “ {𝑍}))
20		an4 655	. . . . . . . 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 37377	. . . . . . . . . . . . . 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 7423	. . . . . . . . . . . . . 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 37318	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∈ RingOps → 𝐺 ∈ GrpOp)
26	5, 11	grpoidcl 30311	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ GrpOp → 𝑍 ∈ ran 𝐺)
27	5, 11	grpolid 30313	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑍 ∈ ran 𝐺) → (𝑍𝐺𝑍) = 𝑍)
28	25, 26, 27	syl2anc2 584	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍)
29	28	3ad2ant2 1132	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝑍𝐺𝑍) = 𝑍)
30	29	ad2antrr 725	. . . . . . . . . . . . 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 2771	. . . . . . . . . . . 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 6904	. . . . . . . . . . . . 13 ⊢ (𝐹‘𝑥) ∈ V
34	33	elsn 4639	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑥) ∈ {𝑍} ↔ (𝐹‘𝑥) = 𝑍)
35		fvex 6904	. . . . . . . . . . . . 13 ⊢ (𝐹‘𝑦) ∈ V
36	35	elsn 4639	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑦) ∈ {𝑍} ↔ (𝐹‘𝑦) = 𝑍)
37	34, 36	anbi12i 626	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) ∈ {𝑍} ∧ (𝐹‘𝑦) ∈ {𝑍}) ↔ ((𝐹‘𝑥) = 𝑍 ∧ (𝐹‘𝑦) = 𝑍))
38		fvex 6904	. . . . . . . . . . . 12 ⊢ (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) ∈ V
39	38	elsn 4639	. . . . . . . . . . 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 37320	. . . . . . . . . . . 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 1120	. . . . . . . . . . 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 1131	. . . . . . . . . 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 610	. . . . . . . . 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 241	. . . . . . 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 7061	. . . . . . . . 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 7061	. . . . . . . . 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 630	. . . . . . 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 7061	. . . . . . . 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 3141	. . . 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 622	. . . . . . 7 ⊢ ((𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) ∈ {𝑍}) ↔ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍))
59		eqid 2727	. . . . . . . . . . . . . . . 16 ⊢ (2^nd ‘𝑅) = (2^nd ‘𝑅)
60	2, 59, 3	rngocl 37309	. . . . . . . . . . . . . . 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 1118	. . . . . . . . . . . . . 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 1183	. . . . . . . . . . . . 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 654	. . . . . . . . . . . 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 720	. . . . . . . . . . 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 2727	. . . . . . . . . . . . . . . 16 ⊢ (2^nd ‘𝑆) = (2^nd ‘𝑆)
66	2, 3, 59, 65	rngohommul 37378	. . . . . . . . . . . . . . 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 654	. . . . . . . . . . . . . 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 720	. . . . . . . . . . . . 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 7422	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . 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 37375	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝐹‘𝑧) ∈ ran 𝐺)
73	11, 5, 4, 65	rngorz 37331	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ RingOps ∧ (𝐹‘𝑧) ∈ ran 𝐺) → ((𝐹‘𝑧)(2^nd ‘𝑆)𝑍) = 𝑍)
74	73	3ad2antl2 1184	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝐹‘𝑧) ∈ ran 𝐺) → ((𝐹‘𝑧)(2^nd ‘𝑆)𝑍) = 𝑍)
75	72, 74	syldan 590	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → ((𝐹‘𝑧)(2^nd ‘𝑆)𝑍) = 𝑍)
76	75	adantlr 714	. . . . . . . . . . . . 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 2771	. . . . . . . . . . . 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 6904	. . . . . . . . . . . . 13 ⊢ (𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) ∈ V
79	78	elsn 4639	. . . . . . . . . . . 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 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑧(2^nd ‘𝑅)𝑥)) ∈ {𝑍})
81		elpreima 7061	. . . . . . . . . . . . 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 725	. . . . . . . . . . 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 712	. . . . . . . . . 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 37309	. . . . . . . . . . . . . . 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 1118	. . . . . . . . . . . . . 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 1183	. . . . . . . . . . . . 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 720	. . . . . . . . . . 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 37378	. . . . . . . . . . . . . . 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 720	. . . . . . . . . . . . 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 7421	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . 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 37330	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ RingOps ∧ (𝐹‘𝑧) ∈ ran 𝐺) → (𝑍(2^nd ‘𝑆)(𝐹‘𝑧)) = 𝑍)
97	96	3ad2antl2 1184	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝐹‘𝑧) ∈ ran 𝐺) → (𝑍(2^nd ‘𝑆)(𝐹‘𝑧)) = 𝑍)
98	72, 97	syldan 590	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝑍(2^nd ‘𝑆)(𝐹‘𝑧)) = 𝑍)
99	98	adantlr 714	. . . . . . . . . . . . 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 2771	. . . . . . . . . . . 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 6904	. . . . . . . . . . . . 13 ⊢ (𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) ∈ V
102	101	elsn 4639	. . . . . . . . . . . 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 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ (𝐹‘𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑥(2^nd ‘𝑅)𝑧)) ∈ {𝑍})
104		elpreima 7061	. . . . . . . . . . . . 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 725	. . . . . . . . . . 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 712	. . . . . . . . . 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 3141	. . . . . . . 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 241	. . . . . 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 239	. . . . 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 3141	. 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 37422	. . 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 1131	. 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 1340	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (^◡𝐹 “ {𝑍}) ∈ (Idl‘𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ⊆ wss 3944 {csn 4624 ^◡ccnv 5671 ran crn 5673 “ cima 5675 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 1^st c1st 7985 2^nd c2nd 7986 GrpOpcgr 30286 GIdcgi 30287 RingOpscrngo 37302 RingOpsHom crngohom 37368 Idlcidl 37415

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-map 8838 df-grpo 30290 df-gid 30291 df-ginv 30292 df-ablo 30342 df-ghomOLD 37292 df-rngo 37303 df-rngohom 37371 df-idl 37418

This theorem is referenced by: (None)

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