Theorem rhmqusnsg 21202

Description: The mapping 𝐽 induced by a ring homomorphism 𝐹 from a subring 𝑁 of the quotient group 𝑄 over 𝐹's kernel 𝐾 is a ring homomorphism. (Contributed by Thierry Arnoux, 13-May-2025.)

Hypotheses

Ref	Expression
rhmqusnsg.0	⊢ 0 = (0g‘𝐻)
rhmqusnsg.f	⊢ (𝜑 → 𝐹 ∈ (𝐺 RingHom 𝐻))
rhmqusnsg.k	⊢ 𝐾 = (◡𝐹 “ { 0 })
rhmqusnsg.q	⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))
rhmqusnsg.j	⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))
rhmqusnsg.g	⊢ (𝜑 → 𝐺 ∈ CRing)
rhmqusnsg.n	⊢ (𝜑 → 𝑁 ⊆ 𝐾)
rhmqusnsg.1	⊢ (𝜑 → 𝑁 ∈ (LIdeal‘𝐺))

Assertion

Ref	Expression
rhmqusnsg	⊢ (𝜑 → 𝐽 ∈ (𝑄 RingHom 𝐻))

Distinct variable groups:   𝐹,𝑞   𝐺,𝑞   𝐻,𝑞   𝐽,𝑞   𝐾,𝑞   𝑁,𝑞   𝑄,𝑞   𝜑,𝑞

Allowed substitution hint:   0 (𝑞)

Proof of Theorem rhmqusnsg

Dummy variables 𝑟 𝑥 𝑦 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2730	. 2 ⊢ (Base‘𝑄) = (Base‘𝑄)
2		eqid 2730	. 2 ⊢ (1r‘𝑄) = (1r‘𝑄)
3		eqid 2730	. 2 ⊢ (1r‘𝐻) = (1r‘𝐻)
4		eqid 2730	. 2 ⊢ (.r‘𝑄) = (.r‘𝑄)
5		eqid 2730	. 2 ⊢ (.r‘𝐻) = (.r‘𝐻)
6		rhmqusnsg.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ CRing)
7	6	crngringd 20162	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Ring)
8		rhmqusnsg.1	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (LIdeal‘𝐺))
9		eqid 2730	. . . . . . 7 ⊢ (LIdeal‘𝐺) = (LIdeal‘𝐺)
10	9	crng2idl 21198	. . . . . 6 ⊢ (𝐺 ∈ CRing → (LIdeal‘𝐺) = (2Ideal‘𝐺))
11	6, 10	syl 17	. . . . 5 ⊢ (𝜑 → (LIdeal‘𝐺) = (2Ideal‘𝐺))
12	8, 11	eleqtrd 2831	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (2Ideal‘𝐺))
13		rhmqusnsg.q	. . . . 5 ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))
14		eqid 2730	. . . . 5 ⊢ (2Ideal‘𝐺) = (2Ideal‘𝐺)
15		eqid 2730	. . . . 5 ⊢ (1r‘𝐺) = (1r‘𝐺)
16	13, 14, 15	qus1 21191	. . . 4 ⊢ ((𝐺 ∈ Ring ∧ 𝑁 ∈ (2Ideal‘𝐺)) → (𝑄 ∈ Ring ∧ [(1r‘𝐺)](𝐺 ~QG 𝑁) = (1r‘𝑄)))
17	7, 12, 16	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑄 ∈ Ring ∧ [(1r‘𝐺)](𝐺 ~QG 𝑁) = (1r‘𝑄)))
18	17	simpld 494	. 2 ⊢ (𝜑 → 𝑄 ∈ Ring)
19		rhmqusnsg.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐺 RingHom 𝐻))
20		rhmrcl2 20393	. . 3 ⊢ (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐻 ∈ Ring)
21	19, 20	syl 17	. 2 ⊢ (𝜑 → 𝐻 ∈ Ring)
22		rhmqusnsg.0	. . . 4 ⊢ 0 = (0g‘𝐻)
23		rhmghm 20400	. . . . 5 ⊢ (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
24	19, 23	syl 17	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))
25		rhmqusnsg.k	. . . 4 ⊢ 𝐾 = (◡𝐹 “ { 0 })
26		rhmqusnsg.j	. . . 4 ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))
27		rhmqusnsg.n	. . . 4 ⊢ (𝜑 → 𝑁 ⊆ 𝐾)
28		lidlnsg 21165	. . . . 5 ⊢ ((𝐺 ∈ Ring ∧ 𝑁 ∈ (LIdeal‘𝐺)) → 𝑁 ∈ (NrmSGrp‘𝐺))
29	7, 8, 28	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))
30		eqid 2730	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
31	30, 15	ringidcl 20181	. . . . 5 ⊢ (𝐺 ∈ Ring → (1r‘𝐺) ∈ (Base‘𝐺))
32	7, 31	syl 17	. . . 4 ⊢ (𝜑 → (1r‘𝐺) ∈ (Base‘𝐺))
33	22, 24, 25, 13, 26, 27, 29, 32	ghmqusnsglem1 19219	. . 3 ⊢ (𝜑 → (𝐽‘[(1r‘𝐺)](𝐺 ~QG 𝑁)) = (𝐹‘(1r‘𝐺)))
34	17	simprd 495	. . . 4 ⊢ (𝜑 → [(1r‘𝐺)](𝐺 ~QG 𝑁) = (1r‘𝑄))
35	34	fveq2d 6865	. . 3 ⊢ (𝜑 → (𝐽‘[(1r‘𝐺)](𝐺 ~QG 𝑁)) = (𝐽‘(1r‘𝑄)))
36	15, 3	rhm1 20405	. . . 4 ⊢ (𝐹 ∈ (𝐺 RingHom 𝐻) → (𝐹‘(1r‘𝐺)) = (1r‘𝐻))
37	19, 36	syl 17	. . 3 ⊢ (𝜑 → (𝐹‘(1r‘𝐺)) = (1r‘𝐻))
38	33, 35, 37	3eqtr3d 2773	. 2 ⊢ (𝜑 → (𝐽‘(1r‘𝑄)) = (1r‘𝐻))
39	19	ad6antr 736	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝐹 ∈ (𝐺 RingHom 𝐻))
40	13	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁)))
41		eqidd 2731	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
42		ovexd 7425	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 ~QG 𝑁) ∈ V)
43	40, 41, 42, 6	qusbas 17515	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝑁)) = (Base‘𝑄))
44		nsgsubg 19097	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
45		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ~QG 𝑁) = (𝐺 ~QG 𝑁)
46	30, 45	eqger 19117	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝑁) Er (Base‘𝐺))
47	29, 44, 46	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 ~QG 𝑁) Er (Base‘𝐺))
48	47	qsss 8752	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝑁)) ⊆ 𝒫 (Base‘𝐺))
49	43, 48	eqsstrrd 3985	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝑄) ⊆ 𝒫 (Base‘𝐺))
50	49	sselda 3949	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ 𝒫 (Base‘𝐺))
51	50	elpwid 4575	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ⊆ (Base‘𝐺))
52	51	ad5antr 734	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑟 ⊆ (Base‘𝐺))
53		simp-4r 783	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑥 ∈ 𝑟)
54	52, 53	sseldd 3950	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑥 ∈ (Base‘𝐺))
55	49	sselda 3949	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ∈ 𝒫 (Base‘𝐺))
56	55	elpwid 4575	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ⊆ (Base‘𝐺))
57	56	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ⊆ (Base‘𝐺))
58	57	ad4antr 732	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑠 ⊆ (Base‘𝐺))
59		simplr 768	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑦 ∈ 𝑠)
60	58, 59	sseldd 3950	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑦 ∈ (Base‘𝐺))
61		eqid 2730	. . . . . . . 8 ⊢ (.r‘𝐺) = (.r‘𝐺)
62	30, 61, 5	rhmmul 20402	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐺 RingHom 𝐻) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝐹‘(𝑥(.r‘𝐺)𝑦)) = ((𝐹‘𝑥)(.r‘𝐻)(𝐹‘𝑦)))
63	39, 54, 60, 62	syl3anc 1373	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → (𝐹‘(𝑥(.r‘𝐺)𝑦)) = ((𝐹‘𝑥)(.r‘𝐻)(𝐹‘𝑦)))
64	47	ad6antr 736	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → (𝐺 ~QG 𝑁) Er (Base‘𝐺))
65		simp-6r 787	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑟 ∈ (Base‘𝑄))
66	43	ad6antr 736	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → ((Base‘𝐺) / (𝐺 ~QG 𝑁)) = (Base‘𝑄))
67	65, 66	eleqtrrd 2832	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑁)))
68		qsel 8772	. . . . . . . . . . 11 ⊢ (((𝐺 ~QG 𝑁) Er (Base‘𝐺) ∧ 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑁)) ∧ 𝑥 ∈ 𝑟) → 𝑟 = [𝑥](𝐺 ~QG 𝑁))
69	64, 67, 53, 68	syl3anc 1373	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑟 = [𝑥](𝐺 ~QG 𝑁))
70		simp-5r 785	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑠 ∈ (Base‘𝑄))
71	70, 66	eleqtrrd 2832	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑠 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑁)))
72		qsel 8772	. . . . . . . . . . 11 ⊢ (((𝐺 ~QG 𝑁) Er (Base‘𝐺) ∧ 𝑠 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑁)) ∧ 𝑦 ∈ 𝑠) → 𝑠 = [𝑦](𝐺 ~QG 𝑁))
73	64, 71, 59, 72	syl3anc 1373	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑠 = [𝑦](𝐺 ~QG 𝑁))
74	69, 73	oveq12d 7408	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → (𝑟(.r‘𝑄)𝑠) = ([𝑥](𝐺 ~QG 𝑁)(.r‘𝑄)[𝑦](𝐺 ~QG 𝑁)))
75	6	ad6antr 736	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝐺 ∈ CRing)
76	8	ad6antr 736	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑁 ∈ (LIdeal‘𝐺))
77	13, 30, 61, 4, 75, 76, 54, 60	qusmulcrng 21201	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → ([𝑥](𝐺 ~QG 𝑁)(.r‘𝑄)[𝑦](𝐺 ~QG 𝑁)) = [(𝑥(.r‘𝐺)𝑦)](𝐺 ~QG 𝑁))
78	74, 77	eqtr2d 2766	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → [(𝑥(.r‘𝐺)𝑦)](𝐺 ~QG 𝑁) = (𝑟(.r‘𝑄)𝑠))
79	78	fveq2d 6865	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → (𝐽‘[(𝑥(.r‘𝐺)𝑦)](𝐺 ~QG 𝑁)) = (𝐽‘(𝑟(.r‘𝑄)𝑠)))
80	39, 23	syl 17	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
81	27	ad6antr 736	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑁 ⊆ 𝐾)
82	29	ad6antr 736	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑁 ∈ (NrmSGrp‘𝐺))
83		rhmrcl1 20392	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐺 ∈ Ring)
84	39, 83	syl 17	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝐺 ∈ Ring)
85	30, 61, 84, 54, 60	ringcld 20176	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → (𝑥(.r‘𝐺)𝑦) ∈ (Base‘𝐺))
86	22, 80, 25, 13, 26, 81, 82, 85	ghmqusnsglem1 19219	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → (𝐽‘[(𝑥(.r‘𝐺)𝑦)](𝐺 ~QG 𝑁)) = (𝐹‘(𝑥(.r‘𝐺)𝑦)))
87	79, 86	eqtr3d 2767	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → (𝐽‘(𝑟(.r‘𝑄)𝑠)) = (𝐹‘(𝑥(.r‘𝐺)𝑦)))
88		simpllr 775	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → (𝐽‘𝑟) = (𝐹‘𝑥))
89		simpr 484	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → (𝐽‘𝑠) = (𝐹‘𝑦))
90	88, 89	oveq12d 7408	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → ((𝐽‘𝑟)(.r‘𝐻)(𝐽‘𝑠)) = ((𝐹‘𝑥)(.r‘𝐻)(𝐹‘𝑦)))
91	63, 87, 90	3eqtr4d 2775	. . . . 5 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → (𝐽‘(𝑟(.r‘𝑄)𝑠)) = ((𝐽‘𝑟)(.r‘𝐻)(𝐽‘𝑠)))
92	24	ad4antr 732	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
93	27	ad4antr 732	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑁 ⊆ 𝐾)
94	29	ad4antr 732	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑁 ∈ (NrmSGrp‘𝐺))
95		simpllr 775	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑠 ∈ (Base‘𝑄))
96	22, 92, 25, 13, 26, 93, 94, 95	ghmqusnsglem2 19220	. . . . 5 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → ∃𝑦 ∈ 𝑠 (𝐽‘𝑠) = (𝐹‘𝑦))
97	91, 96	r19.29a 3142	. . . 4 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → (𝐽‘(𝑟(.r‘𝑄)𝑠)) = ((𝐽‘𝑟)(.r‘𝐻)(𝐽‘𝑠)))
98	24	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
99	27	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑁 ⊆ 𝐾)
100	29	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑁 ∈ (NrmSGrp‘𝐺))
101		simplr 768	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑟 ∈ (Base‘𝑄))
102	22, 98, 25, 13, 26, 99, 100, 101	ghmqusnsglem2 19220	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → ∃𝑥 ∈ 𝑟 (𝐽‘𝑟) = (𝐹‘𝑥))
103	97, 102	r19.29a 3142	. . 3 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → (𝐽‘(𝑟(.r‘𝑄)𝑠)) = ((𝐽‘𝑟)(.r‘𝐻)(𝐽‘𝑠)))
104	103	anasss 466	. 2 ⊢ ((𝜑 ∧ (𝑟 ∈ (Base‘𝑄) ∧ 𝑠 ∈ (Base‘𝑄))) → (𝐽‘(𝑟(.r‘𝑄)𝑠)) = ((𝐽‘𝑟)(.r‘𝐻)(𝐽‘𝑠)))
105	22, 24, 25, 13, 26, 27, 29	ghmqusnsg 19221	. 2 ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpHom 𝐻))
106	1, 2, 3, 4, 5, 18, 21, 38, 104, 105	isrhm2d 20403	1 ⊢ (𝜑 → 𝐽 ∈ (𝑄 RingHom 𝐻))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ⊆ wss 3917  𝒫 cpw 4566  {csn 4592  ∪ cuni 4874   ↦ cmpt 5191  ^◡ccnv 5640   “ cima 5644  ‘cfv 6514  (class class class)co 7390   Er wer 8671  [cec 8672   / cqs 8673  Basecbs 17186  ._rcmulr 17228  0_gc0g 17409   /_s cqus 17475  SubGrpcsubg 19059  NrmSGrpcnsg 19060   ~_QG cqg 19061   GrpHom cghm 19151  1_rcur 20097  Ringcrg 20149  CRingccrg 20150   RingHom crh 20385  LIdealclidl 21123  2Idealc2idl 21166

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-tpos 8208  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-ec 8676  df-qs 8680  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-inf 9401  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-fz 13476  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-mulr 17241  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-0g 17411  df-imas 17478  df-qus 17479  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-mhm 18717  df-grp 18875  df-minusg 18876  df-sbg 18877  df-subg 19062  df-nsg 19063  df-eqg 19064  df-ghm 19152  df-cmn 19719  df-abl 19720  df-mgp 20057  df-rng 20069  df-ur 20098  df-ring 20151  df-cring 20152  df-oppr 20253  df-rhm 20388  df-subrg 20486  df-lmod 20775  df-lss 20845  df-lsp 20885  df-sra 21087  df-rgmod 21088  df-lidl 21125  df-rsp 21126  df-2idl 21167

This theorem is referenced by:  zndvdchrrhm  41967  rhmqusspan  42180