aks6d1c6isolem2 - Mathbox for metakunt

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Theorem aks6d1c6isolem2 42368

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 21406	. 2 ⊢ ℤ = (Base‘ℤ_ring)
2		eqid 2734	. 2 ⊢ (Base‘((𝑅 ↾_s 𝑈) ↾_s ran 𝐹)) = (Base‘((𝑅 ↾_s 𝑈) ↾_s ran 𝐹))
3		zringplusg 21407	. 2 ⊢ + = (+_g‘ℤ_ring)
4		aks6d1c6isolem1.4	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀))
5		zex 12495	. . . . . 6 ⊢ ℤ ∈ V
6	5	mptex 7167	. . . . 5 ⊢ (𝑥 ∈ ℤ ↦ (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀)) ∈ V
7	4, 6	eqeltri 2830	. . . 4 ⊢ 𝐹 ∈ V
8	7	rnex 7850	. . 3 ⊢ ran 𝐹 ∈ V
9		eqid 2734	. . . 4 ⊢ ((𝑅 ↾_s 𝑈) ↾_s ran 𝐹) = ((𝑅 ↾_s 𝑈) ↾_s ran 𝐹)
10		eqid 2734	. . . 4 ⊢ (+_g‘(𝑅 ↾_s 𝑈)) = (+_g‘(𝑅 ↾_s 𝑈))
11	9, 10	ressplusg 17209	. . 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 21402	. . . 4 ⊢ ℤ_ring ∈ Ring
14	13	a1i 11	. . 3 ⊢ (𝜑 → ℤ_ring ∈ Ring)
15		ringgrp 20171	. . 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 42367	. 2 ⊢ (𝜑 → ((𝑅 ↾_s 𝑈) ↾_s ran 𝐹) ∈ Grp)
22		ovexd 7391	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀) ∈ V)
23	22, 4	fmptd 7057	. . . . 5 ⊢ (𝜑 → 𝐹:ℤ⟶V)
24		ffn 6660	. . . . 5 ⊢ (𝐹:ℤ⟶V → 𝐹 Fn ℤ)
25	23, 24	syl 17	. . . 4 ⊢ (𝜑 → 𝐹 Fn ℤ)
26		dffn3 6672	. . . 4 ⊢ (𝐹 Fn ℤ ↔ 𝐹:ℤ⟶ran 𝐹)
27	25, 26	sylib 218	. . 3 ⊢ (𝜑 → 𝐹:ℤ⟶ran 𝐹)
28		fvelrnb 6892	. . . . . . . . . . 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 2740	. . . . . . . . . . . . 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 7371	. . . . . . . . . . . . . . . 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 7391	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ ℤ) → (𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀) ∈ V)
42	37, 39, 40, 41	fvmptd 6946	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ ℤ) → (𝐹‘𝑧) = (𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀))
43		eqid 2734	. . . . . . . . . . . . . . . 16 ⊢ (Base‘(𝑅 ↾_s 𝑈)) = (Base‘(𝑅 ↾_s 𝑈))
44		eqid 2734	. . . . . . . . . . . . . . . 16 ⊢ (._g‘(𝑅 ↾_s 𝑈)) = (._g‘(𝑅 ↾_s 𝑈))
45	17, 18, 19	primrootsunit 42291	. . . . . . . . . . . . . . . . . . 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 19713	. . . . . . . . . . . . . . . . 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 2836	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ ((𝑅 ↾_s 𝑈) PrimRoots 𝐾))
51	46	ablcmnd 19715	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑅 ↾_s 𝑈) ∈ CMnd)
52	18	nnnn0d 12460	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
53	51, 52, 44	isprimroot 42286	. . . . . . . . . . . . . . . . . . . 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 19024	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ ℤ) → (𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀) ∈ (Base‘(𝑅 ↾_s 𝑈)))
59	42, 58	eqeltrd 2834	. . . . . . . . . . . . . 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 2834	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤) ∧ 𝑧 ∈ ℤ) ∧ (𝐹‘𝑧) = 𝑤) → 𝑤 ∈ (Base‘(𝑅 ↾_s 𝑈)))
62		nfv 1915	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧(𝐹‘𝑣) = 𝑤
63		nfv 1915	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑣(𝐹‘𝑧) = 𝑤
64		fveqeq2 6841	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑧 → ((𝐹‘𝑣) = 𝑤 ↔ (𝐹‘𝑧) = 𝑤))
65	62, 63, 64	cbvrexw 3277	. . . . . . . . . . . . . 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 3142	. . . . . . . . . . 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 3937	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘(𝑅 ↾_s 𝑈)))
75	9, 43	ressbas2 17163	. . . . 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 6645	. . 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 7371	. . . 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 12598	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝑦 + 𝑧) ∈ ℤ)
85		ovexd 7391	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → ((𝑦 + 𝑧)(._g‘(𝑅 ↾_s 𝑈))𝑀) ∈ V)
86	79, 81, 84, 85	fvmptd 6946	. . 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 19034	. . . . 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 7371	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ 𝑥 = 𝑦) → (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀) = (𝑦(._g‘(𝑅 ↾_s 𝑈))𝑀))
94		ovexd 7391	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝑦(._g‘(𝑅 ↾_s 𝑈))𝑀) ∈ V)
95	79, 93, 82, 94	fvmptd 6946	. . . . . 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 7371	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ 𝑥 = 𝑧) → (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀) = (𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀))
98		ovexd 7391	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀) ∈ V)
99	79, 97, 83, 98	fvmptd 6946	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝐹‘𝑧) = (𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀))
100	95, 99	oveq12d 7374	. . . . 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 2740	. . . 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 2769	. . 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 2769	. 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 19152	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 1541 ∈ wcel 2113 ∀wral 3049 ∃wrex 3058 {crab 3397 Vcvv 3438 ⊆ wss 3899 class class class wbr 5096 ↦ cmpt 5177 ran crn 5623 Fn wfn 6485 ⟶wf 6486 ‘cfv 6490 (class class class)co 7356 + caddc 11027 ℕcn 12143 ℕ₀cn0 12399 ℤcz 12486 ∥ cdvds 16177 Basecbs 17134 ↾_s cress 17155 +_gcplusg 17175 0_gc0g 17357 Grpcgrp 18861 ._gcmg 18995 GrpHom cghm 19139 CMndccmn 19707 Abelcabl 19708 Ringcrg 20166 ℤ_ringczring 21399 PrimRoots cprimroots 42284

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-addf 11103

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 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8763 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-fz 13422 df-seq 13923 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-starv 17190 df-tset 17194 df-ple 17195 df-ds 17197 df-unif 17198 df-0g 17359 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18707 df-grp 18864 df-minusg 18865 df-mulg 18996 df-subg 19051 df-ghm 19140 df-cmn 19709 df-abl 19710 df-mgp 20074 df-rng 20086 df-ur 20115 df-ring 20168 df-cring 20169 df-subrng 20477 df-subrg 20501 df-cnfld 21308 df-zring 21400 df-primroots 42285

This theorem is referenced by: aks6d1c6lem5 42370

W3C validator