Theorem rhmquskerlem 34042

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 2763	. 2 ⊢ (Base‘𝑄) = (Base‘𝑄)
2		eqid 2763	. 2 ⊢ (1r‘𝑄) = (1r‘𝑄)
3		eqid 2763	. 2 ⊢ (1r‘𝐻) = (1r‘𝐻)
4		eqid 2763	. 2 ⊢ (.r‘𝑄) = (.r‘𝑄)
5		eqid 2763	. 2 ⊢ (.r‘𝐻) = (.r‘𝐻)
6		rhmqusker.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐺 RingHom 𝐻))
7		rhmrcl1 20734	. . . . 5 ⊢ (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐺 ∈ Ring)
8	6, 7	syl 18	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Ring)
9		rhmqusker.k	. . . . . 6 ⊢ 𝐾 = (◡𝐹 “ { 0 })
10		eqid 2763	. . . . . . . 8 ⊢ (LIdeal‘𝐺) = (LIdeal‘𝐺)
11		rhmqusker.1	. . . . . . . 8 ⊢ 0 = (0g‘𝐻)
12	10, 11	kerlidl 21599	. . . . . . 7 ⊢ (𝐹 ∈ (𝐺 RingHom 𝐻) → (◡𝐹 “ { 0 }) ∈ (LIdeal‘𝐺))
13	6, 12	syl 18	. . . . . 6 ⊢ (𝜑 → (◡𝐹 “ { 0 }) ∈ (LIdeal‘𝐺))
14	9, 13	eqeltrid 2868	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (LIdeal‘𝐺))
15		rhmquskerlem.2	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ CRing)
16	10	crng2idl 21603	. . . . . 6 ⊢ (𝐺 ∈ CRing → (LIdeal‘𝐺) = (2Ideal‘𝐺))
17	15, 16	syl 18	. . . . 5 ⊢ (𝜑 → (LIdeal‘𝐺) = (2Ideal‘𝐺))
18	14, 17	eleqtrd 2866	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (2Ideal‘𝐺))
19		rhmqusker.q	. . . . 5 ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))
20		eqid 2763	. . . . 5 ⊢ (2Ideal‘𝐺) = (2Ideal‘𝐺)
21		eqid 2763	. . . . 5 ⊢ (1r‘𝐺) = (1r‘𝐺)
22	19, 20, 21	qus1 21595	. . . 4 ⊢ ((𝐺 ∈ Ring ∧ 𝐾 ∈ (2Ideal‘𝐺)) → (𝑄 ∈ Ring ∧ [(1r‘𝐺)](𝐺 ~QG 𝐾) = (1r‘𝑄)))
23	8, 18, 22	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑄 ∈ Ring ∧ [(1r‘𝐺)](𝐺 ~QG 𝐾) = (1r‘𝑄)))
24	23	simpld 500	. 2 ⊢ (𝜑 → 𝑄 ∈ Ring)
25		rhmrcl2 20735	. . 3 ⊢ (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐻 ∈ Ring)
26	6, 25	syl 18	. 2 ⊢ (𝜑 → 𝐻 ∈ Ring)
27		rhmghm 20741	. . . . 5 ⊢ (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
28	6, 27	syl 18	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))
29		rhmquskerlem.j	. . . 4 ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))
30		eqid 2763	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
31	30, 21	ringidcl 20521	. . . . 5 ⊢ (𝐺 ∈ Ring → (1r‘𝐺) ∈ (Base‘𝐺))
32	8, 31	syl 18	. . . 4 ⊢ (𝜑 → (1r‘𝐺) ∈ (Base‘𝐺))
33	11, 28, 9, 19, 29, 32	ghmquskerlem1 19524	. . 3 ⊢ (𝜑 → (𝐽‘[(1r‘𝐺)](𝐺 ~QG 𝐾)) = (𝐹‘(1r‘𝐺)))
34	23	simprd 501	. . . 4 ⊢ (𝜑 → [(1r‘𝐺)](𝐺 ~QG 𝐾) = (1r‘𝑄))
35	34	fveq2d 6909	. . 3 ⊢ (𝜑 → (𝐽‘[(1r‘𝐺)](𝐺 ~QG 𝐾)) = (𝐽‘(1r‘𝑄)))
36	21, 3	rhm1 20751	. . . 4 ⊢ (𝐹 ∈ (𝐺 RingHom 𝐻) → (𝐹‘(1r‘𝐺)) = (1r‘𝐻))
37	6, 36	syl 18	. . 3 ⊢ (𝜑 → (𝐹‘(1r‘𝐺)) = (1r‘𝐻))
38	33, 35, 37	3eqtr3d 2806	. 2 ⊢ (𝜑 → (𝐽‘(1r‘𝑄)) = (1r‘𝐻))
39	6	ad6antr 749	. . . . . . 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 2764	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
42		ovexd 7475	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 ~QG 𝐾) ∈ V)
43	40, 41, 42, 15	qusbas 17743	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) = (Base‘𝑄))
44	11	ghmker 19482	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
45	28, 44	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
46	9, 45	eqeltrid 2868	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ (NrmSGrp‘𝐺))
47		nsgsubg 19394	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (NrmSGrp‘𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
48		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ~QG 𝐾) = (𝐺 ~QG 𝐾)
49	30, 48	eqger 19416	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
50	46, 47, 49	3syl 19	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
51	50	qsss 8821	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) ⊆ 𝒫 (Base‘𝐺))
52	43, 51	eqsstrrd 3948	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝑄) ⊆ 𝒫 (Base‘𝐺))
53	52	sselda 3913	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ 𝒫 (Base‘𝐺))
54	53	elpwid 4569	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) → 𝑟 ⊆ (Base‘𝐺))
55	54	ad5antr 747	. . . . . . . 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 796	. . . . . . . 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 3914	. . . . . . 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 3913	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ∈ 𝒫 (Base‘𝐺))
59	58	elpwid 4569	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ⊆ (Base‘𝐺))
60	59	adantlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ⊆ (Base‘𝐺))
61	60	ad4antr 745	. . . . . . . 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 781	. . . . . . . 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 3914	. . . . . . 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 2763	. . . . . . . 8 ⊢ (.r‘𝐺) = (.r‘𝐺)
65	30, 64, 5	rhmmul 20747	. . . . . . 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 1399	. . . . . 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 749	. . . . . . . . . . 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 800	. . . . . . . . . . . 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 749	. . . . . . . . . . . 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 2867	. . . . . . . . . . 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 8842	. . . . . . . . . . 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 1399	. . . . . . . . . 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 798	. . . . . . . . . . . 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 2867	. . . . . . . . . . 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 8842	. . . . . . . . . . 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 1399	. . . . . . . . . 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 7458	. . . . . . . . 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 749	. . . . . . . . . 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 749	. . . . . . . . . 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 21607	. . . . . . . . 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 2799	. . . . . . . 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 6909	. . . . . . 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 18	. . . . . . . 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 18	. . . . . . . . 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 20511	. . . . . . . 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 19524	. . . . . . 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 2800	. . . . . 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 788	. . . . . . 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 490	. . . . . . 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 7458	. . . . . 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 2808	. . . . 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 745	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥 ∈ 𝑟) ∧ (𝐽‘𝑟) = (𝐹‘𝑥)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
93		simpllr 788	. . . . . 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 19526	. . . . 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 3174	. . . 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 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
97		simplr 781	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑟 ∈ (Base‘𝑄))
98	11, 96, 9, 19, 29, 97	ghmquskerlem2 19526	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → ∃𝑥 ∈ 𝑟 (𝐽‘𝑟) = (𝐹‘𝑥))
99	95, 98	r19.29a 3174	. . 3 ⊢ (((𝜑 ∧ 𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → (𝐽‘(𝑟(.r‘𝑄)𝑠)) = ((𝐽‘𝑟)(.r‘𝐻)(𝐽‘𝑠)))
100	99	anasss 472	. 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 19527	. 2 ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpHom 𝐻))
102	1, 2, 3, 4, 5, 24, 26, 38, 100, 101	isrhm2d 20748	1 ⊢ (𝜑 → 𝐽 ∈ (𝑄 RingHom 𝐻))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900  𝒫 cpw 4560  {csn 4588  ∪ cuni 4870   ↦ cmpt 5189  ^◡ccnv 5654   “ cima 5658  ‘cfv 6555  (class class class)co 7440   Er wer 8739  [cec 8740   / cqs 8741  Basecbs 17413  ._rcmulr 17455  0_gc0g 17636   /_s cqus 17703  SubGrpcsubg 19356  NrmSGrpcnsg 19357   ~_QG cqg 19358   GrpHom cghm 19453  1_rcur 20434  Ringcrg 20486  CRingccrg 20487   RingHom crh 20726  LIdealclidl 21511  2Idealc2idl 21569

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-tpos 8268  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-er 8742  df-ec 8744  df-qs 8748  df-map 8874  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-sup 9458  df-inf 9459  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-3 12430  df-4 12431  df-5 12432  df-6 12433  df-7 12434  df-8 12435  df-9 12436  df-n0 12631  df-z 12718  df-dec 12839  df-uz 12990  df-fz 13664  df-struct 17351  df-sets 17368  df-slot 17386  df-ndx 17398  df-base 17414  df-ress 17435  df-plusg 17467  df-mulr 17468  df-sca 17470  df-vsca 17471  df-ip 17472  df-tset 17473  df-ple 17474  df-ds 17476  df-0g 17638  df-imas 17706  df-qus 17707  df-mgm 18842  df-sgrp 18934  df-mnd 18950  df-mhm 19004  df-submnd 19005  df-grp 19173  df-minusg 19174  df-sbg 19175  df-subg 19359  df-nsg 19360  df-eqg 19361  df-ghm 19454  df-cmn 20023  df-abl 20024  df-mgp 20388  df-rng 20402  df-ur 20435  df-ring 20488  df-cring 20489  df-oppr 20594  df-rhm 20729  df-subrg 20849  df-lmod 21164  df-lss 21234  df-lsp 21274  df-sra 21475  df-rgmod 21476  df-lidl 21513  df-rsp 21514  df-2idl 21570

This theorem is used by:  rhmqusker  34043  algextdeglem4  34419