Theorem rhmqusnsg 21195

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 2729	. 2 ⊢ (Base‘𝑄) = (Base‘𝑄)
2		eqid 2729	. 2 ⊢ (1r‘𝑄) = (1r‘𝑄)
3		eqid 2729	. 2 ⊢ (1r‘𝐻) = (1r‘𝐻)
4		eqid 2729	. 2 ⊢ (.r‘𝑄) = (.r‘𝑄)
5		eqid 2729	. 2 ⊢ (.r‘𝐻) = (.r‘𝐻)
6		rhmqusnsg.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ CRing)
7	6	crngringd 20155	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Ring)
8		rhmqusnsg.1	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (LIdeal‘𝐺))
9		eqid 2729	. . . . . . 7 ⊢ (LIdeal‘𝐺) = (LIdeal‘𝐺)
10	9	crng2idl 21191	. . . . . 6 ⊢ (𝐺 ∈ CRing → (LIdeal‘𝐺) = (2Ideal‘𝐺))
11	6, 10	syl 17	. . . . 5 ⊢ (𝜑 → (LIdeal‘𝐺) = (2Ideal‘𝐺))
12	8, 11	eleqtrd 2830	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (2Ideal‘𝐺))
13		rhmqusnsg.q	. . . . 5 ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))
14		eqid 2729	. . . . 5 ⊢ (2Ideal‘𝐺) = (2Ideal‘𝐺)
15		eqid 2729	. . . . 5 ⊢ (1r‘𝐺) = (1r‘𝐺)
16	13, 14, 15	qus1 21184	. . . 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 20386	. . 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 20393	. . . . 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 21158	. . . . 5 ⊢ ((𝐺 ∈ Ring ∧ 𝑁 ∈ (LIdeal‘𝐺)) → 𝑁 ∈ (NrmSGrp‘𝐺))
29	7, 8, 28	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))
30		eqid 2729	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
31	30, 15	ringidcl 20174	. . . . 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 19212	. . 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 6862	. . 3 ⊢ (𝜑 → (𝐽‘[(1r‘𝐺)](𝐺 ~QG 𝑁)) = (𝐽‘(1r‘𝑄)))
36	15, 3	rhm1 20398	. . . 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 2772	. 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 2730	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
42		ovexd 7422	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 ~QG 𝑁) ∈ V)
43	40, 41, 42, 6	qusbas 17508	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝑁)) = (Base‘𝑄))
44		nsgsubg 19090	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
45		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ~QG 𝑁) = (𝐺 ~QG 𝑁)
46	30, 45	eqger 19110	. . . . . . . . . . . . . 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 8749	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝑁)) ⊆ 𝒫 (Base‘𝐺))
49	43, 48	eqsstrrd 3982	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝑄) ⊆ 𝒫 (Base‘𝐺))
50	49	sselda 3946	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ 𝒫 (Base‘𝐺))
51	50	elpwid 4572	. . . . . . . . 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 3947	. . . . . . 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 3946	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ∈ 𝒫 (Base‘𝐺))
56	55	elpwid 4572	. . . . . . . . . 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 3947	. . . . . . 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 2729	. . . . . . . 8 ⊢ (.r‘𝐺) = (.r‘𝐺)
62	30, 61, 5	rhmmul 20395	. . . . . . 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 2831	. . . . . . . . . . 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 8769	. . . . . . . . . . 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 2831	. . . . . . . . . . 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 8769	. . . . . . . . . . 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 7405	. . . . . . . . 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 21194	. . . . . . . . 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 2765	. . . . . . . 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 6862	. . . . . . 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 20385	. . . . . . . . . 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 20169	. . . . . . . 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 19212	. . . . . . 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 2766	. . . . . 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 7405	. . . . . 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 2774	. . . . 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 19213	. . . . 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 3141	. . . 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 19213	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → ∃𝑥 ∈ 𝑟 (𝐽‘𝑟) = (𝐹‘𝑥))
103	97, 102	r19.29a 3141	. . 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 19214	. 2 ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpHom 𝐻))
106	1, 2, 3, 4, 5, 18, 21, 38, 104, 105	isrhm2d 20396	1 ⊢ (𝜑 → 𝐽 ∈ (𝑄 RingHom 𝐻))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447   ⊆ wss 3914  𝒫 cpw 4563  {csn 4589  ∪ cuni 4871   ↦ cmpt 5188  ^◡ccnv 5637   “ cima 5641  ‘cfv 6511  (class class class)co 7387   Er wer 8668  [cec 8669   / cqs 8670  Basecbs 17179  ._rcmulr 17221  0_gc0g 17402   /_s cqus 17468  SubGrpcsubg 19052  NrmSGrpcnsg 19053   ~_QG cqg 19054   GrpHom cghm 19144  1_rcur 20090  Ringcrg 20142  CRingccrg 20143   RingHom crh 20378  LIdealclidl 21116  2Idealc2idl 21159

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-tpos 8205  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-ec 8673  df-qs 8677  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-sup 9393  df-inf 9394  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-0g 17404  df-imas 17471  df-qus 17472  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-mhm 18710  df-grp 18868  df-minusg 18869  df-sbg 18870  df-subg 19055  df-nsg 19056  df-eqg 19057  df-ghm 19145  df-cmn 19712  df-abl 19713  df-mgp 20050  df-rng 20062  df-ur 20091  df-ring 20144  df-cring 20145  df-oppr 20246  df-rhm 20381  df-subrg 20479  df-lmod 20768  df-lss 20838  df-lsp 20878  df-sra 21080  df-rgmod 21081  df-lidl 21118  df-rsp 21119  df-2idl 21160

This theorem is referenced by:  zndvdchrrhm  41960  rhmqusspan  42173