aks6d1c6isolem2 - Mathbox for metakunt

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Theorem aks6d1c6isolem2 42176

Description: Lemma to construct the group homomorphism for the AKS Theorem. (Contributed by metakunt, 14-May-2025.)

**Hypotheses**
Ref	Expression
aks6d1c6isolem1.1	⊢ (𝜑 → 𝑅 ∈ CMnd)
aks6d1c6isolem1.2	⊢ (𝜑 → 𝐾 ∈ ℕ)
aks6d1c6isolem1.3	⊢ 𝑈 = {𝑎 ∈ (Base‘𝑅) ∣ ∃𝑖 ∈ (Base‘𝑅)(𝑖(+_g‘𝑅)𝑎) = (0_g‘𝑅)}
aks6d1c6isolem1.4	⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀))
aks6d1c6isolem1.5	⊢ (𝜑 → 𝑀 ∈ (𝑅 PrimRoots 𝐾))

**Assertion**
Ref	Expression
aks6d1c6isolem2	⊢ (𝜑 → 𝐹 ∈ (ℤ_ring GrpHom ((𝑅 ↾_s 𝑈) ↾_s ran 𝐹)))

Distinct variable groups: 𝑥,𝑀 𝑅,𝑎,𝑖 𝑥,𝑅 𝑥,𝑈 𝜑,𝑥 Allowed substitution hints: 𝜑(𝑖,𝑎) 𝑈(𝑖,𝑎) 𝐹(𝑥,𝑖,𝑎) 𝐾(𝑥,𝑖,𝑎) 𝑀(𝑖,𝑎)

Proof of Theorem aks6d1c6isolem2 Dummy variables 𝑣 𝑤 𝑧 𝑦 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zringbas 21464	. 2 ⊢ ℤ = (Base‘ℤ_ring)
2		eqid 2737	. 2 ⊢ (Base‘((𝑅 ↾_s 𝑈) ↾_s ran 𝐹)) = (Base‘((𝑅 ↾_s 𝑈) ↾_s ran 𝐹))
3		zringplusg 21465	. 2 ⊢ + = (+_g‘ℤ_ring)
4		aks6d1c6isolem1.4	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀))
5		zex 12622	. . . . . 6 ⊢ ℤ ∈ V
6	5	mptex 7243	. . . . 5 ⊢ (𝑥 ∈ ℤ ↦ (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀)) ∈ V
7	4, 6	eqeltri 2837	. . . 4 ⊢ 𝐹 ∈ V
8	7	rnex 7932	. . 3 ⊢ ran 𝐹 ∈ V
9		eqid 2737	. . . 4 ⊢ ((𝑅 ↾_s 𝑈) ↾_s ran 𝐹) = ((𝑅 ↾_s 𝑈) ↾_s ran 𝐹)
10		eqid 2737	. . . 4 ⊢ (+_g‘(𝑅 ↾_s 𝑈)) = (+_g‘(𝑅 ↾_s 𝑈))
11	9, 10	ressplusg 17334	. . 3 ⊢ (ran 𝐹 ∈ V → (+_g‘(𝑅 ↾_s 𝑈)) = (+_g‘((𝑅 ↾_s 𝑈) ↾_s ran 𝐹)))
12	8, 11	ax-mp 5	. 2 ⊢ (+_g‘(𝑅 ↾_s 𝑈)) = (+_g‘((𝑅 ↾_s 𝑈) ↾_s ran 𝐹))
13		zringring 21460	. . . 4 ⊢ ℤ_ring ∈ Ring
14	13	a1i 11	. . 3 ⊢ (𝜑 → ℤ_ring ∈ Ring)
15		ringgrp 20235	. . 3 ⊢ (ℤ_ring ∈ Ring → ℤ_ring ∈ Grp)
16	14, 15	syl 17	. 2 ⊢ (𝜑 → ℤ_ring ∈ Grp)
17		aks6d1c6isolem1.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ CMnd)
18		aks6d1c6isolem1.2	. . 3 ⊢ (𝜑 → 𝐾 ∈ ℕ)
19		aks6d1c6isolem1.3	. . 3 ⊢ 𝑈 = {𝑎 ∈ (Base‘𝑅) ∣ ∃𝑖 ∈ (Base‘𝑅)(𝑖(+_g‘𝑅)𝑎) = (0_g‘𝑅)}
20		aks6d1c6isolem1.5	. . 3 ⊢ (𝜑 → 𝑀 ∈ (𝑅 PrimRoots 𝐾))
21	17, 18, 19, 4, 20	aks6d1c6isolem1 42175	. 2 ⊢ (𝜑 → ((𝑅 ↾_s 𝑈) ↾_s ran 𝐹) ∈ Grp)
22		ovexd 7466	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀) ∈ V)
23	22, 4	fmptd 7134	. . . . 5 ⊢ (𝜑 → 𝐹:ℤ⟶V)
24		ffn 6736	. . . . 5 ⊢ (𝐹:ℤ⟶V → 𝐹 Fn ℤ)
25	23, 24	syl 17	. . . 4 ⊢ (𝜑 → 𝐹 Fn ℤ)
26		dffn3 6748	. . . 4 ⊢ (𝐹 Fn ℤ ↔ 𝐹:ℤ⟶ran 𝐹)
27	25, 26	sylib 218	. . 3 ⊢ (𝜑 → 𝐹:ℤ⟶ran 𝐹)
28		fvelrnb 6969	. . . . . . . . . . 11 ⊢ (𝐹 Fn ℤ → (𝑤 ∈ ran 𝐹 ↔ ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤))
29	25, 28	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤))
30	29	biimpd 229	. . . . . . . . 9 ⊢ (𝜑 → (𝑤 ∈ ran 𝐹 → ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤))
31	30	imp 406	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ran 𝐹) → ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤)
32		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤) ∧ 𝑧 ∈ ℤ) ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑧) = 𝑤)
33	32	eqcomd 2743	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤) ∧ 𝑧 ∈ ℤ) ∧ (𝐹‘𝑧) = 𝑤) → 𝑤 = (𝐹‘𝑧))
34		simplll 775	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤) ∧ 𝑧 ∈ ℤ) ∧ (𝐹‘𝑧) = 𝑤) → 𝜑)
35		simplr 769	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤) ∧ 𝑧 ∈ ℤ) ∧ (𝐹‘𝑧) = 𝑤) → 𝑧 ∈ ℤ)
36	34, 35	jca 511	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤) ∧ 𝑧 ∈ ℤ) ∧ (𝐹‘𝑧) = 𝑤) → (𝜑 ∧ 𝑧 ∈ ℤ))
37	4	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ ℤ) → 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀)))
38		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ ℤ) ∧ 𝑥 = 𝑧) → 𝑥 = 𝑧)
39	38	oveq1d 7446	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ ℤ) ∧ 𝑥 = 𝑧) → (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀) = (𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀))
40		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ ℤ) → 𝑧 ∈ ℤ)
41		ovexd 7466	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ ℤ) → (𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀) ∈ V)
42	37, 39, 40, 41	fvmptd 7023	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ ℤ) → (𝐹‘𝑧) = (𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀))
43		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (Base‘(𝑅 ↾_s 𝑈)) = (Base‘(𝑅 ↾_s 𝑈))
44		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (._g‘(𝑅 ↾_s 𝑈)) = (._g‘(𝑅 ↾_s 𝑈))
45	17, 18, 19	primrootsunit 42099	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑅 PrimRoots 𝐾) = ((𝑅 ↾_s 𝑈) PrimRoots 𝐾) ∧ (𝑅 ↾_s 𝑈) ∈ Abel))
46	45	simprd 495	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑅 ↾_s 𝑈) ∈ Abel)
47	46	ablgrpd 19804	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑅 ↾_s 𝑈) ∈ Grp)
48	47	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ ℤ) → (𝑅 ↾_s 𝑈) ∈ Grp)
49	45	simpld 494	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑅 PrimRoots 𝐾) = ((𝑅 ↾_s 𝑈) PrimRoots 𝐾))
50	20, 49	eleqtrd 2843	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ ((𝑅 ↾_s 𝑈) PrimRoots 𝐾))
51	46	ablcmnd 19806	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑅 ↾_s 𝑈) ∈ CMnd)
52	18	nnnn0d 12587	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
53	51, 52, 44	isprimroot 42094	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑀 ∈ ((𝑅 ↾_s 𝑈) PrimRoots 𝐾) ↔ (𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈)) ∧ (𝐾(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)) ∧ ∀𝑙 ∈ ℕ₀ ((𝑙(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)) → 𝐾 ∥ 𝑙))))
54	53	biimpd 229	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀 ∈ ((𝑅 ↾_s 𝑈) PrimRoots 𝐾) → (𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈)) ∧ (𝐾(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)) ∧ ∀𝑙 ∈ ℕ₀ ((𝑙(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)) → 𝐾 ∥ 𝑙))))
55	50, 54	mpd 15	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈)) ∧ (𝐾(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)) ∧ ∀𝑙 ∈ ℕ₀ ((𝑙(._g‘(𝑅 ↾_s 𝑈))𝑀) = (0_g‘(𝑅 ↾_s 𝑈)) → 𝐾 ∥ 𝑙)))
56	55	simp1d 1143	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈)))
57	56	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ ℤ) → 𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈)))
58	43, 44, 48, 40, 57	mulgcld 19114	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ ℤ) → (𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀) ∈ (Base‘(𝑅 ↾_s 𝑈)))
59	42, 58	eqeltrd 2841	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ ℤ) → (𝐹‘𝑧) ∈ (Base‘(𝑅 ↾_s 𝑈)))
60	36, 59	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤) ∧ 𝑧 ∈ ℤ) ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑧) ∈ (Base‘(𝑅 ↾_s 𝑈)))
61	33, 60	eqeltrd 2841	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤) ∧ 𝑧 ∈ ℤ) ∧ (𝐹‘𝑧) = 𝑤) → 𝑤 ∈ (Base‘(𝑅 ↾_s 𝑈)))
62		nfv 1914	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧(𝐹‘𝑣) = 𝑤
63		nfv 1914	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑣(𝐹‘𝑧) = 𝑤
64		fveqeq2 6915	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑧 → ((𝐹‘𝑣) = 𝑤 ↔ (𝐹‘𝑧) = 𝑤))
65	62, 63, 64	cbvrexw 3307	. . . . . . . . . . . . . 14 ⊢ (∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤 ↔ ∃𝑧 ∈ ℤ (𝐹‘𝑧) = 𝑤)
66	65	biimpi 216	. . . . . . . . . . . . 13 ⊢ (∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤 → ∃𝑧 ∈ ℤ (𝐹‘𝑧) = 𝑤)
67	66	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤) → ∃𝑧 ∈ ℤ (𝐹‘𝑧) = 𝑤)
68	61, 67	r19.29a 3162	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤) → 𝑤 ∈ (Base‘(𝑅 ↾_s 𝑈)))
69	68	ex 412	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤 → 𝑤 ∈ (Base‘(𝑅 ↾_s 𝑈))))
70	69	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ ran 𝐹) → (∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤 → 𝑤 ∈ (Base‘(𝑅 ↾_s 𝑈))))
71	70	imp 406	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐹) ∧ ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤) → 𝑤 ∈ (Base‘(𝑅 ↾_s 𝑈)))
72	31, 71	mpdan 687	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ran 𝐹) → 𝑤 ∈ (Base‘(𝑅 ↾_s 𝑈)))
73	72	ex 412	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ ran 𝐹 → 𝑤 ∈ (Base‘(𝑅 ↾_s 𝑈))))
74	73	ssrdv 3989	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘(𝑅 ↾_s 𝑈)))
75	9, 43	ressbas2 17283	. . . . 5 ⊢ (ran 𝐹 ⊆ (Base‘(𝑅 ↾_s 𝑈)) → ran 𝐹 = (Base‘((𝑅 ↾_s 𝑈) ↾_s ran 𝐹)))
76	74, 75	syl 17	. . . 4 ⊢ (𝜑 → ran 𝐹 = (Base‘((𝑅 ↾_s 𝑈) ↾_s ran 𝐹)))
77	76	feq3d 6723	. . 3 ⊢ (𝜑 → (𝐹:ℤ⟶ran 𝐹 ↔ 𝐹:ℤ⟶(Base‘((𝑅 ↾_s 𝑈) ↾_s ran 𝐹))))
78	27, 77	mpbid 232	. 2 ⊢ (𝜑 → 𝐹:ℤ⟶(Base‘((𝑅 ↾_s 𝑈) ↾_s ran 𝐹)))
79	4	a1i 11	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀)))
80		simpr 484	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ 𝑥 = (𝑦 + 𝑧)) → 𝑥 = (𝑦 + 𝑧))
81	80	oveq1d 7446	. . . 4 ⊢ (((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ 𝑥 = (𝑦 + 𝑧)) → (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀) = ((𝑦 + 𝑧)(._g‘(𝑅 ↾_s 𝑈))𝑀))
82		simprl 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → 𝑦 ∈ ℤ)
83		simprr 773	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → 𝑧 ∈ ℤ)
84	82, 83	zaddcld 12726	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝑦 + 𝑧) ∈ ℤ)
85		ovexd 7466	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → ((𝑦 + 𝑧)(._g‘(𝑅 ↾_s 𝑈))𝑀) ∈ V)
86	79, 81, 84, 85	fvmptd 7023	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝐹‘(𝑦 + 𝑧)) = ((𝑦 + 𝑧)(._g‘(𝑅 ↾_s 𝑈))𝑀))
87	47	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝑅 ↾_s 𝑈) ∈ Grp)
88	56	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → 𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈)))
89	82, 83, 88	3jca 1129	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ ∧ 𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈))))
90	43, 44, 10	mulgdir 19124	. . . . 5 ⊢ (((𝑅 ↾_s 𝑈) ∈ Grp ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ ∧ 𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈)))) → ((𝑦 + 𝑧)(._g‘(𝑅 ↾_s 𝑈))𝑀) = ((𝑦(._g‘(𝑅 ↾_s 𝑈))𝑀)(+_g‘(𝑅 ↾_s 𝑈))(𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀)))
91	87, 89, 90	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → ((𝑦 + 𝑧)(._g‘(𝑅 ↾_s 𝑈))𝑀) = ((𝑦(._g‘(𝑅 ↾_s 𝑈))𝑀)(+_g‘(𝑅 ↾_s 𝑈))(𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀)))
92		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
93	92	oveq1d 7446	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ 𝑥 = 𝑦) → (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀) = (𝑦(._g‘(𝑅 ↾_s 𝑈))𝑀))
94		ovexd 7466	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝑦(._g‘(𝑅 ↾_s 𝑈))𝑀) ∈ V)
95	79, 93, 82, 94	fvmptd 7023	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝐹‘𝑦) = (𝑦(._g‘(𝑅 ↾_s 𝑈))𝑀))
96		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ 𝑥 = 𝑧) → 𝑥 = 𝑧)
97	96	oveq1d 7446	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ 𝑥 = 𝑧) → (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀) = (𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀))
98		ovexd 7466	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀) ∈ V)
99	79, 97, 83, 98	fvmptd 7023	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝐹‘𝑧) = (𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀))
100	95, 99	oveq12d 7449	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → ((𝐹‘𝑦)(+_g‘(𝑅 ↾_s 𝑈))(𝐹‘𝑧)) = ((𝑦(._g‘(𝑅 ↾_s 𝑈))𝑀)(+_g‘(𝑅 ↾_s 𝑈))(𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀)))
101	100	eqcomd 2743	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → ((𝑦(._g‘(𝑅 ↾_s 𝑈))𝑀)(+_g‘(𝑅 ↾_s 𝑈))(𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀)) = ((𝐹‘𝑦)(+_g‘(𝑅 ↾_s 𝑈))(𝐹‘𝑧)))
102	91, 101	eqtrd 2777	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → ((𝑦 + 𝑧)(._g‘(𝑅 ↾_s 𝑈))𝑀) = ((𝐹‘𝑦)(+_g‘(𝑅 ↾_s 𝑈))(𝐹‘𝑧)))
103	86, 102	eqtrd 2777	. 2 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝐹‘(𝑦 + 𝑧)) = ((𝐹‘𝑦)(+_g‘(𝑅 ↾_s 𝑈))(𝐹‘𝑧)))
104	1, 2, 3, 12, 16, 21, 78, 103	isghmd 19243	1 ⊢ (𝜑 → 𝐹 ∈ (ℤ_ring GrpHom ((𝑅 ↾_s 𝑈) ↾_s ran 𝐹)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 {crab 3436 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 ran crn 5686 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 + caddc 11158 ℕcn 12266 ℕ₀cn0 12526 ℤcz 12613 ∥ cdvds 16290 Basecbs 17247 ↾_s cress 17274 +_gcplusg 17297 0_gc0g 17484 Grpcgrp 18951 ._gcmg 19085 GrpHom cghm 19230 CMndccmn 19798 Abelcabl 19799 Ringcrg 20230 ℤ_ringczring 21457 PrimRoots cprimroots 42092

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-seq 14043 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-subrng 20546 df-subrg 20570 df-cnfld 21365 df-zring 21458 df-primroots 42093

This theorem is referenced by: aks6d1c6lem5 42178

W3C validator