Theorem rhmquskerlem 33390

Description: The mapping 𝐽 induced by a ring homomorphism 𝐹 from the quotient group 𝑄 over 𝐹's kernel 𝐾 is a ring homomorphism. (Contributed by Thierry Arnoux, 22-Mar-2025.)

Hypotheses

Ref	Expression
rhmqusker.1	⊢ 0 = (0g‘𝐻)
rhmqusker.f	⊢ (𝜑 → 𝐹 ∈ (𝐺 RingHom 𝐻))
rhmqusker.k	⊢ 𝐾 = (◡𝐹 “ { 0 })
rhmqusker.q	⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))
rhmquskerlem.j	⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))
rhmquskerlem.2	⊢ (𝜑 → 𝐺 ∈ CRing)

Assertion

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

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

Allowed substitution hint:   0 (𝑞)

Proof of Theorem rhmquskerlem

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

Step	Hyp	Ref	Expression
1		eqid 2731	. 2 ⊢ (Base‘𝑄) = (Base‘𝑄)
2		eqid 2731	. 2 ⊢ (1r‘𝑄) = (1r‘𝑄)
3		eqid 2731	. 2 ⊢ (1r‘𝐻) = (1r‘𝐻)
4		eqid 2731	. 2 ⊢ (.r‘𝑄) = (.r‘𝑄)
5		eqid 2731	. 2 ⊢ (.r‘𝐻) = (.r‘𝐻)
6		rhmqusker.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐺 RingHom 𝐻))
7		rhmrcl1 20394	. . . . 5 ⊢ (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐺 ∈ Ring)
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Ring)
9		rhmqusker.k	. . . . . 6 ⊢ 𝐾 = (◡𝐹 “ { 0 })
10		eqid 2731	. . . . . . . 8 ⊢ (LIdeal‘𝐺) = (LIdeal‘𝐺)
11		rhmqusker.1	. . . . . . . 8 ⊢ 0 = (0g‘𝐻)
12	10, 11	kerlidl 21215	. . . . . . 7 ⊢ (𝐹 ∈ (𝐺 RingHom 𝐻) → (◡𝐹 “ { 0 }) ∈ (LIdeal‘𝐺))
13	6, 12	syl 17	. . . . . 6 ⊢ (𝜑 → (◡𝐹 “ { 0 }) ∈ (LIdeal‘𝐺))
14	9, 13	eqeltrid 2835	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (LIdeal‘𝐺))
15		rhmquskerlem.2	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ CRing)
16	10	crng2idl 21218	. . . . . 6 ⊢ (𝐺 ∈ CRing → (LIdeal‘𝐺) = (2Ideal‘𝐺))
17	15, 16	syl 17	. . . . 5 ⊢ (𝜑 → (LIdeal‘𝐺) = (2Ideal‘𝐺))
18	14, 17	eleqtrd 2833	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (2Ideal‘𝐺))
19		rhmqusker.q	. . . . 5 ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))
20		eqid 2731	. . . . 5 ⊢ (2Ideal‘𝐺) = (2Ideal‘𝐺)
21		eqid 2731	. . . . 5 ⊢ (1r‘𝐺) = (1r‘𝐺)
22	19, 20, 21	qus1 21211	. . . 4 ⊢ ((𝐺 ∈ Ring ∧ 𝐾 ∈ (2Ideal‘𝐺)) → (𝑄 ∈ Ring ∧ [(1r‘𝐺)](𝐺 ~QG 𝐾) = (1r‘𝑄)))
23	8, 18, 22	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑄 ∈ Ring ∧ [(1r‘𝐺)](𝐺 ~QG 𝐾) = (1r‘𝑄)))
24	23	simpld 494	. 2 ⊢ (𝜑 → 𝑄 ∈ Ring)
25		rhmrcl2 20395	. . 3 ⊢ (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐻 ∈ Ring)
26	6, 25	syl 17	. 2 ⊢ (𝜑 → 𝐻 ∈ Ring)
27		rhmghm 20401	. . . . 5 ⊢ (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
28	6, 27	syl 17	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))
29		rhmquskerlem.j	. . . 4 ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))
30		eqid 2731	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
31	30, 21	ringidcl 20183	. . . . 5 ⊢ (𝐺 ∈ Ring → (1r‘𝐺) ∈ (Base‘𝐺))
32	8, 31	syl 17	. . . 4 ⊢ (𝜑 → (1r‘𝐺) ∈ (Base‘𝐺))
33	11, 28, 9, 19, 29, 32	ghmquskerlem1 19195	. . 3 ⊢ (𝜑 → (𝐽‘[(1r‘𝐺)](𝐺 ~QG 𝐾)) = (𝐹‘(1r‘𝐺)))
34	23	simprd 495	. . . 4 ⊢ (𝜑 → [(1r‘𝐺)](𝐺 ~QG 𝐾) = (1r‘𝑄))
35	34	fveq2d 6826	. . 3 ⊢ (𝜑 → (𝐽‘[(1r‘𝐺)](𝐺 ~QG 𝐾)) = (𝐽‘(1r‘𝑄)))
36	21, 3	rhm1 20406	. . . 4 ⊢ (𝐹 ∈ (𝐺 RingHom 𝐻) → (𝐹‘(1r‘𝐺)) = (1r‘𝐻))
37	6, 36	syl 17	. . 3 ⊢ (𝜑 → (𝐹‘(1r‘𝐺)) = (1r‘𝐻))
38	33, 35, 37	3eqtr3d 2774	. 2 ⊢ (𝜑 → (𝐽‘(1r‘𝑄)) = (1r‘𝐻))
39	6	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	19	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾)))
41		eqidd 2732	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
42		ovexd 7381	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 ~QG 𝐾) ∈ V)
43	40, 41, 42, 15	qusbas 17449	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) = (Base‘𝑄))
44	11	ghmker 19154	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
45	28, 44	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
46	9, 45	eqeltrid 2835	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ (NrmSGrp‘𝐺))
47		nsgsubg 19070	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (NrmSGrp‘𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
48		eqid 2731	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ~QG 𝐾) = (𝐺 ~QG 𝐾)
49	30, 48	eqger 19090	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
50	46, 47, 49	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
51	50	qsss 8700	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) ⊆ 𝒫 (Base‘𝐺))
52	43, 51	eqsstrrd 3965	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝑄) ⊆ 𝒫 (Base‘𝐺))
53	52	sselda 3929	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ 𝒫 (Base‘𝐺))
54	53	elpwid 4556	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ⊆ (Base‘𝐺))
55	54	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))
56		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)
57	55, 56	sseldd 3930	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑥 ∈ (Base‘𝐺))
58	52	sselda 3929	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ∈ 𝒫 (Base‘𝐺))
59	58	elpwid 4556	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ⊆ (Base‘𝐺))
60	59	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ⊆ (Base‘𝐺))
61	60	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))
62		simplr 768	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑦 ∈ 𝑠)
63	61, 62	sseldd 3930	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝑦 ∈ (Base‘𝐺))
64		eqid 2731	. . . . . . . 8 ⊢ (.r‘𝐺) = (.r‘𝐺)
65	30, 64, 5	rhmmul 20403	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐺 RingHom 𝐻) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝐹‘(𝑥(.r‘𝐺)𝑦)) = ((𝐹‘𝑥)(.r‘𝐻)(𝐹‘𝑦)))
66	39, 57, 63, 65	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)))
67	50	ad6antr 736	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
68		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))
69	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 𝐾)) = (Base‘𝑄))
70	68, 69	eleqtrrd 2834	. . . . . . . . . . 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 𝐾)))
71		qsel 8720	. . . . . . . . . . 11 ⊢ (((𝐺 ~QG 𝐾) Er (Base‘𝐺) ∧ 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)) ∧ 𝑥 ∈ 𝑟) → 𝑟 = [𝑥](𝐺 ~QG 𝐾))
72	67, 70, 56, 71	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 𝐾))
73		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))
74	73, 69	eleqtrrd 2834	. . . . . . . . . . 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 𝐾)))
75		qsel 8720	. . . . . . . . . . 11 ⊢ (((𝐺 ~QG 𝐾) Er (Base‘𝐺) ∧ 𝑠 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)) ∧ 𝑦 ∈ 𝑠) → 𝑠 = [𝑦](𝐺 ~QG 𝐾))
76	67, 74, 62, 75	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 𝐾))
77	72, 76	oveq12d 7364	. . . . . . . . 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 𝐾)(.r‘𝑄)[𝑦](𝐺 ~QG 𝐾)))
78	15	ad6antr 736	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝐺 ∈ CRing)
79	14	ad6antr 736	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → 𝐾 ∈ (LIdeal‘𝐺))
80	19, 30, 64, 4, 78, 79, 57, 63	qusmulcrng 21221	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) ∧ 𝑦 ∈ 𝑠) ∧ (𝐽‘𝑠) = (𝐹‘𝑦)) → ([𝑥](𝐺 ~QG 𝐾)(.r‘𝑄)[𝑦](𝐺 ~QG 𝐾)) = [(𝑥(.r‘𝐺)𝑦)](𝐺 ~QG 𝐾))
81	77, 80	eqtr2d 2767	. . . . . . . 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 𝐾) = (𝑟(.r‘𝑄)𝑠))
82	81	fveq2d 6826	. . . . . . 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 𝐾)) = (𝐽‘(𝑟(.r‘𝑄)𝑠)))
83	39, 27	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))
84	39, 7	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, 64, 84, 57, 63	ringcld 20178	. . . . . . . 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	11, 83, 9, 19, 29, 85	ghmquskerlem1 19195	. . . . . . 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 𝐾)) = (𝐹‘(𝑥(.r‘𝐺)𝑦)))
87	82, 86	eqtr3d 2768	. . . . . 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 7364	. . . . . 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	66, 87, 90	3eqtr4d 2776	. . . . 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	28	ad4antr 732	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
93		simpllr 775	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝑠 ∈ (Base‘𝑄))
94	11, 92, 9, 19, 29, 93	ghmquskerlem2 19197	. . . . 5 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → ∃𝑦 ∈ 𝑠 (𝐽‘𝑠) = (𝐹‘𝑦))
95	91, 94	r19.29a 3140	. . . 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)))
96	28	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
97		simplr 768	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑟 ∈ (Base‘𝑄))
98	11, 96, 9, 19, 29, 97	ghmquskerlem2 19197	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → ∃𝑥 ∈ 𝑟 (𝐽‘𝑟) = (𝐹‘𝑥))
99	95, 98	r19.29a 3140	. . 3 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → (𝐽‘(𝑟(.r‘𝑄)𝑠)) = ((𝐽‘𝑟)(.r‘𝐻)(𝐽‘𝑠)))
100	99	anasss 466	. 2 ⊢ ((𝜑 ∧ (𝑟 ∈ (Base‘𝑄) ∧ 𝑠 ∈ (Base‘𝑄))) → (𝐽‘(𝑟(.r‘𝑄)𝑠)) = ((𝐽‘𝑟)(.r‘𝐻)(𝐽‘𝑠)))
101	11, 28, 9, 19, 29	ghmquskerlem3 19198	. 2 ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpHom 𝐻))
102	1, 2, 3, 4, 5, 24, 26, 38, 100, 101	isrhm2d 20404	1 ⊢ (𝜑 → 𝐽 ∈ (𝑄 RingHom 𝐻))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ⊆ wss 3897  𝒫 cpw 4547  {csn 4573  ∪ cuni 4856   ↦ cmpt 5170  ^◡ccnv 5613   “ cima 5617  ‘cfv 6481  (class class class)co 7346   Er wer 8619  [cec 8620   / cqs 8621  Basecbs 17120  ._rcmulr 17162  0_gc0g 17343   /_s cqus 17409  SubGrpcsubg 19033  NrmSGrpcnsg 19034   ~_QG cqg 19035   GrpHom cghm 19124  1_rcur 20099  Ringcrg 20151  CRingccrg 20152   RingHom crh 20387  LIdealclidl 21143  2Idealc2idl 21186

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  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-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-tpos 8156  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-ec 8624  df-qs 8628  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-sup 9326  df-inf 9327  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-3 12189  df-4 12190  df-5 12191  df-6 12192  df-7 12193  df-8 12194  df-9 12195  df-n0 12382  df-z 12469  df-dec 12589  df-uz 12733  df-fz 13408  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-sca 17177  df-vsca 17178  df-ip 17179  df-tset 17180  df-ple 17181  df-ds 17183  df-0g 17345  df-imas 17412  df-qus 17413  df-mgm 18548  df-sgrp 18627  df-mnd 18643  df-mhm 18691  df-submnd 18692  df-grp 18849  df-minusg 18850  df-sbg 18851  df-subg 19036  df-nsg 19037  df-eqg 19038  df-ghm 19125  df-cmn 19694  df-abl 19695  df-mgp 20059  df-rng 20071  df-ur 20100  df-ring 20153  df-cring 20154  df-oppr 20255  df-rhm 20390  df-subrg 20485  df-lmod 20795  df-lss 20865  df-lsp 20905  df-sra 21107  df-rgmod 21108  df-lidl 21145  df-rsp 21146  df-2idl 21187

This theorem is referenced by:  rhmqusker  33391  algextdeglem4  33733