aks6d1c6isolem2 - Mathbox for metakunt

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Theorem aks6d1c6isolem2 42163

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 21363	. 2 ⊢ ℤ = (Base‘ℤ_ring)
2		eqid 2729	. 2 ⊢ (Base‘((𝑅 ↾_s 𝑈) ↾_s ran 𝐹)) = (Base‘((𝑅 ↾_s 𝑈) ↾_s ran 𝐹))
3		zringplusg 21364	. 2 ⊢ + = (+_g‘ℤ_ring)
4		aks6d1c6isolem1.4	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀))
5		zex 12538	. . . . . 6 ⊢ ℤ ∈ V
6	5	mptex 7197	. . . . 5 ⊢ (𝑥 ∈ ℤ ↦ (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀)) ∈ V
7	4, 6	eqeltri 2824	. . . 4 ⊢ 𝐹 ∈ V
8	7	rnex 7886	. . 3 ⊢ ran 𝐹 ∈ V
9		eqid 2729	. . . 4 ⊢ ((𝑅 ↾_s 𝑈) ↾_s ran 𝐹) = ((𝑅 ↾_s 𝑈) ↾_s ran 𝐹)
10		eqid 2729	. . . 4 ⊢ (+_g‘(𝑅 ↾_s 𝑈)) = (+_g‘(𝑅 ↾_s 𝑈))
11	9, 10	ressplusg 17254	. . 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 21359	. . . 4 ⊢ ℤ_ring ∈ Ring
14	13	a1i 11	. . 3 ⊢ (𝜑 → ℤ_ring ∈ Ring)
15		ringgrp 20147	. . 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 42162	. 2 ⊢ (𝜑 → ((𝑅 ↾_s 𝑈) ↾_s ran 𝐹) ∈ Grp)
22		ovexd 7422	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀) ∈ V)
23	22, 4	fmptd 7086	. . . . 5 ⊢ (𝜑 → 𝐹:ℤ⟶V)
24		ffn 6688	. . . . 5 ⊢ (𝐹:ℤ⟶V → 𝐹 Fn ℤ)
25	23, 24	syl 17	. . . 4 ⊢ (𝜑 → 𝐹 Fn ℤ)
26		dffn3 6700	. . . 4 ⊢ (𝐹 Fn ℤ ↔ 𝐹:ℤ⟶ran 𝐹)
27	25, 26	sylib 218	. . 3 ⊢ (𝜑 → 𝐹:ℤ⟶ran 𝐹)
28		fvelrnb 6921	. . . . . . . . . . 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 2735	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤) ∧ 𝑧 ∈ ℤ) ∧ (𝐹‘𝑧) = 𝑤) → 𝑤 = (𝐹‘𝑧))
34		simplll 774	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤) ∧ 𝑧 ∈ ℤ) ∧ (𝐹‘𝑧) = 𝑤) → 𝜑)
35		simplr 768	. . . . . . . . . . . . . . 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 7402	. . . . . . . . . . . . . . . 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 7422	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ ℤ) → (𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀) ∈ V)
42	37, 39, 40, 41	fvmptd 6975	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ ℤ) → (𝐹‘𝑧) = (𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀))
43		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (Base‘(𝑅 ↾_s 𝑈)) = (Base‘(𝑅 ↾_s 𝑈))
44		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (._g‘(𝑅 ↾_s 𝑈)) = (._g‘(𝑅 ↾_s 𝑈))
45	17, 18, 19	primrootsunit 42086	. . . . . . . . . . . . . . . . . . 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 19716	. . . . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ ((𝑅 ↾_s 𝑈) PrimRoots 𝐾))
51	46	ablcmnd 19718	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑅 ↾_s 𝑈) ∈ CMnd)
52	18	nnnn0d 12503	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
53	51, 52, 44	isprimroot 42081	. . . . . . . . . . . . . . . . . . . 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 1142	. . . . . . . . . . . . . . . . 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 19028	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ ℤ) → (𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀) ∈ (Base‘(𝑅 ↾_s 𝑈)))
59	42, 58	eqeltrd 2828	. . . . . . . . . . . . . 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 2828	. . . . . . . . . . . 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 6867	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑧 → ((𝐹‘𝑣) = 𝑤 ↔ (𝐹‘𝑧) = 𝑤))
65	62, 63, 64	cbvrexw 3281	. . . . . . . . . . . . . 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 3141	. . . . . . . . . . 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 3952	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘(𝑅 ↾_s 𝑈)))
75	9, 43	ressbas2 17208	. . . . 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 6673	. . 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 7402	. . . 4 ⊢ (((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ 𝑥 = (𝑦 + 𝑧)) → (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀) = ((𝑦 + 𝑧)(._g‘(𝑅 ↾_s 𝑈))𝑀))
82		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → 𝑦 ∈ ℤ)
83		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → 𝑧 ∈ ℤ)
84	82, 83	zaddcld 12642	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝑦 + 𝑧) ∈ ℤ)
85		ovexd 7422	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → ((𝑦 + 𝑧)(._g‘(𝑅 ↾_s 𝑈))𝑀) ∈ V)
86	79, 81, 84, 85	fvmptd 6975	. . 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 1128	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ ∧ 𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈))))
90	43, 44, 10	mulgdir 19038	. . . . 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 7402	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ 𝑥 = 𝑦) → (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀) = (𝑦(._g‘(𝑅 ↾_s 𝑈))𝑀))
94		ovexd 7422	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝑦(._g‘(𝑅 ↾_s 𝑈))𝑀) ∈ V)
95	79, 93, 82, 94	fvmptd 6975	. . . . . 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 7402	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ 𝑥 = 𝑧) → (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀) = (𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀))
98		ovexd 7422	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀) ∈ V)
99	79, 97, 83, 98	fvmptd 6975	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝐹‘𝑧) = (𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀))
100	95, 99	oveq12d 7405	. . . . 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 2735	. . . 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 2764	. . 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 2764	. 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 19157	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 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 {crab 3405 Vcvv 3447 ⊆ wss 3914 class class class wbr 5107 ↦ cmpt 5188 ran crn 5639 Fn wfn 6506 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 + caddc 11071 ℕcn 12186 ℕ₀cn0 12442 ℤcz 12529 ∥ cdvds 16222 Basecbs 17179 ↾_s cress 17200 +_gcplusg 17220 0_gc0g 17402 Grpcgrp 18865 ._gcmg 18999 GrpHom cghm 19144 CMndccmn 19710 Abelcabl 19711 Ringcrg 20142 ℤ_ringczring 21356 PrimRoots cprimroots 42079

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 ax-addf 11147

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-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-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 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-seq 13967 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-starv 17235 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-0g 17404 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-grp 18868 df-minusg 18869 df-mulg 19000 df-subg 19055 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-subrng 20455 df-subrg 20479 df-cnfld 21265 df-zring 21357 df-primroots 42080

This theorem is referenced by: aks6d1c6lem5 42165

W3C validator