rhmquskerlem - Mathbox for Thierry Arnoux

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Theorem rhmquskerlem 33185

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 = (0_g‘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 2725	. 2 ⊢ (Base‘𝑄) = (Base‘𝑄)
2		eqid 2725	. 2 ⊢ (1_r‘𝑄) = (1_r‘𝑄)
3		eqid 2725	. 2 ⊢ (1_r‘𝐻) = (1_r‘𝐻)
4		eqid 2725	. 2 ⊢ (._r‘𝑄) = (._r‘𝑄)
5		eqid 2725	. 2 ⊢ (._r‘𝐻) = (._r‘𝐻)
6		rhmqusker.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐺 RingHom 𝐻))
7		rhmrcl1 20414	. . . . 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 2725	. . . . . . . 8 ⊢ (LIdeal‘𝐺) = (LIdeal‘𝐺)
11		rhmqusker.1	. . . . . . . 8 ⊢ 0 = (0_g‘𝐻)
12	10, 11	kerlidl 33179	. . . . . . 7 ⊢ (𝐹 ∈ (𝐺 RingHom 𝐻) → (^◡𝐹 “ { 0 }) ∈ (LIdeal‘𝐺))
13	6, 12	syl 17	. . . . . 6 ⊢ (𝜑 → (^◡𝐹 “ { 0 }) ∈ (LIdeal‘𝐺))
14	9, 13	eqeltrid 2829	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (LIdeal‘𝐺))
15		rhmquskerlem.2	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ CRing)
16	10	crng2idl 21172	. . . . . 6 ⊢ (𝐺 ∈ CRing → (LIdeal‘𝐺) = (2Ideal‘𝐺))
17	15, 16	syl 17	. . . . 5 ⊢ (𝜑 → (LIdeal‘𝐺) = (2Ideal‘𝐺))
18	14, 17	eleqtrd 2827	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (2Ideal‘𝐺))
19		rhmqusker.q	. . . . 5 ⊢ 𝑄 = (𝐺 /_s (𝐺 ~_QG 𝐾))
20		eqid 2725	. . . . 5 ⊢ (2Ideal‘𝐺) = (2Ideal‘𝐺)
21		eqid 2725	. . . . 5 ⊢ (1_r‘𝐺) = (1_r‘𝐺)
22	19, 20, 21	qus1 21167	. . . 4 ⊢ ((𝐺 ∈ Ring ∧ 𝐾 ∈ (2Ideal‘𝐺)) → (𝑄 ∈ Ring ∧ [(1_r‘𝐺)](𝐺 ~_QG 𝐾) = (1_r‘𝑄)))
23	8, 18, 22	syl2anc 582	. . 3 ⊢ (𝜑 → (𝑄 ∈ Ring ∧ [(1_r‘𝐺)](𝐺 ~_QG 𝐾) = (1_r‘𝑄)))
24	23	simpld 493	. 2 ⊢ (𝜑 → 𝑄 ∈ Ring)
25		rhmrcl2 20415	. . 3 ⊢ (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐻 ∈ Ring)
26	6, 25	syl 17	. 2 ⊢ (𝜑 → 𝐻 ∈ Ring)
27		rhmghm 20422	. . . . 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 2725	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
31	30, 21	ringidcl 20201	. . . . 5 ⊢ (𝐺 ∈ Ring → (1_r‘𝐺) ∈ (Base‘𝐺))
32	8, 31	syl 17	. . . 4 ⊢ (𝜑 → (1_r‘𝐺) ∈ (Base‘𝐺))
33	11, 28, 9, 19, 29, 32	ghmquskerlem1 19233	. . 3 ⊢ (𝜑 → (𝐽‘[(1_r‘𝐺)](𝐺 ~_QG 𝐾)) = (𝐹‘(1_r‘𝐺)))
34	23	simprd 494	. . . 4 ⊢ (𝜑 → [(1_r‘𝐺)](𝐺 ~_QG 𝐾) = (1_r‘𝑄))
35	34	fveq2d 6894	. . 3 ⊢ (𝜑 → (𝐽‘[(1_r‘𝐺)](𝐺 ~_QG 𝐾)) = (𝐽‘(1_r‘𝑄)))
36	21, 3	rhm1 20427	. . . 4 ⊢ (𝐹 ∈ (𝐺 RingHom 𝐻) → (𝐹‘(1_r‘𝐺)) = (1_r‘𝐻))
37	6, 36	syl 17	. . 3 ⊢ (𝜑 → (𝐹‘(1_r‘𝐺)) = (1_r‘𝐻))
38	33, 35, 37	3eqtr3d 2773	. 2 ⊢ (𝜑 → (𝐽‘(1_r‘𝑄)) = (1_r‘𝐻))
39	6	ad6antr 734	. . . . . . 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 2726	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
42		ovexd 7448	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 ~_QG 𝐾) ∈ V)
43	40, 41, 42, 15	qusbas 17521	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~_QG 𝐾)) = (Base‘𝑄))
44	11	ghmker 19195	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (^◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
45	28, 44	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (^◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
46	9, 45	eqeltrid 2829	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ (NrmSGrp‘𝐺))
47		nsgsubg 19112	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (NrmSGrp‘𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
48		eqid 2725	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ~_QG 𝐾) = (𝐺 ~_QG 𝐾)
49	30, 48	eqger 19132	. . . . . . . . . . . . . 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 8790	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~_QG 𝐾)) ⊆ 𝒫 (Base‘𝐺))
52	43, 51	eqsstrrd 4013	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝑄) ⊆ 𝒫 (Base‘𝐺))
53	52	sselda 3973	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ 𝒫 (Base‘𝐺))
54	53	elpwid 4608	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ⊆ (Base‘𝐺))
55	54	ad5antr 732	. . . . . . . 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 782	. . . . . . . 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 3974	. . . . . . 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 3973	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ∈ 𝒫 (Base‘𝐺))
59	58	elpwid 4608	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ⊆ (Base‘𝐺))
60	59	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ⊆ (Base‘𝐺))
61	60	ad4antr 730	. . . . . . . 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 767	. . . . . . . 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 3974	. . . . . . 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 2725	. . . . . . . 8 ⊢ (._r‘𝐺) = (._r‘𝐺)
65	30, 64, 5	rhmmul 20424	. . . . . . 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 1368	. . . . . 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 734	. . . . . . . . . . 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 786	. . . . . . . . . . . 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 734	. . . . . . . . . . . 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 2828	. . . . . . . . . . 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 8808	. . . . . . . . . . 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 1368	. . . . . . . . . 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 784	. . . . . . . . . . . 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 2828	. . . . . . . . . . 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 8808	. . . . . . . . . . 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 1368	. . . . . . . . . 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 7431	. . . . . . . . 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 734	. . . . . . . . . 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 734	. . . . . . . . . 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	qusmul 33159	. . . . . . . . 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 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 𝐾) = (𝑟(._r‘𝑄)𝑠))
82	81	fveq2d 6894	. . . . . . 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 20198	. . . . . . . 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 19233	. . . . . . 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 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 774	. . . . . . 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 483	. . . . . . 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 7431	. . . . . 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 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	28	ad4antr 730	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
93		simpllr 774	. . . . . 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 19235	. . . . 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 3152	. . . 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 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
97		simplr 767	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑟 ∈ (Base‘𝑄))
98	11, 96, 9, 19, 29, 97	ghmquskerlem2 19235	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → ∃𝑥 ∈ 𝑟 (𝐽‘𝑟) = (𝐹‘𝑥))
99	95, 98	r19.29a 3152	. . 3 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → (𝐽‘(𝑟(._r‘𝑄)𝑠)) = ((𝐽‘𝑟)(._r‘𝐻)(𝐽‘𝑠)))
100	99	anasss 465	. 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 19236	. 2 ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpHom 𝐻))
102	1, 2, 3, 4, 5, 24, 26, 38, 100, 101	isrhm2d 20425	1 ⊢ (𝜑 → 𝐽 ∈ (𝑄 RingHom 𝐻))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ⊆ wss 3941 𝒫 cpw 4599 {csn 4625 ∪ cuni 4904 ↦ cmpt 5227 ^◡ccnv 5672 “ cima 5676 ‘cfv 6543 (class class class)co 7413 Er wer 8715 [cec 8716 / cqs 8717 Basecbs 17174 ._rcmulr 17228 0_gc0g 17415 /_s cqus 17481 SubGrpcsubg 19074 NrmSGrpcnsg 19075 ~_QG cqg 19076 GrpHom cghm 19166 1_rcur 20120 Ringcrg 20172 CRingccrg 20173 RingHom crh 20407 LIdealclidl 21101 2Idealc2idl 21142

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-ec 8720 df-qs 8724 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9460 df-inf 9461 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-fz 13512 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 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 17417 df-imas 17484 df-qus 17485 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-mhm 18734 df-submnd 18735 df-grp 18892 df-minusg 18893 df-sbg 18894 df-subg 19077 df-nsg 19078 df-eqg 19079 df-ghm 19167 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-cring 20175 df-oppr 20272 df-rhm 20410 df-subrg 20507 df-lmod 20744 df-lss 20815 df-lsp 20855 df-sra 21057 df-rgmod 21058 df-lidl 21103 df-rsp 21104 df-2idl 21143

This theorem is referenced by: rhmqusker 33186 algextdeglem4 33441

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