aks6d1c6isolem2 - Mathbox for metakunt

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Theorem aks6d1c6isolem2 43002

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 21655	. 2 ⊢ ℤ = (Base‘ℤ_ring)
2		eqid 2765	. 2 ⊢ (Base‘((𝑅 ↾_s 𝑈) ↾_s ran 𝐹)) = (Base‘((𝑅 ↾_s 𝑈) ↾_s ran 𝐹))
3		zringplusg 21656	. 2 ⊢ + = (+_g‘ℤ_ring)
4		aks6d1c6isolem1.4	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀))
5		zex 12617	. . . . . 6 ⊢ ℤ ∈ V
6	5	mptex 7228	. . . . 5 ⊢ (𝑥 ∈ ℤ ↦ (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀)) ∈ V
7	4, 6	eqeltri 2861	. . . 4 ⊢ 𝐹 ∈ V
8	7	rnex 7913	. . 3 ⊢ ran 𝐹 ∈ V
9		eqid 2765	. . . 4 ⊢ ((𝑅 ↾_s 𝑈) ↾_s ran 𝐹) = ((𝑅 ↾_s 𝑈) ↾_s ran 𝐹)
10		eqid 2765	. . . 4 ⊢ (+_g‘(𝑅 ↾_s 𝑈)) = (+_g‘(𝑅 ↾_s 𝑈))
11	9, 10	ressplusg 17368	. . 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 21651	. . . 4 ⊢ ℤ_ring ∈ Ring
14	13	a1i 11	. . 3 ⊢ (𝜑 → ℤ_ring ∈ Ring)
15		ringgrp 20366	. . 3 ⊢ (ℤ_ring ∈ Ring → ℤ_ring ∈ Grp)
16	14, 15	syl 18	. 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 43001	. 2 ⊢ (𝜑 → ((𝑅 ↾_s 𝑈) ↾_s ran 𝐹) ∈ Grp)
22		ovexd 7454	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀) ∈ V)
23	22, 4	fmptd 7113	. . . . 5 ⊢ (𝜑 → 𝐹:ℤ⟶V)
24		ffn 6709	. . . . 5 ⊢ (𝐹:ℤ⟶V → 𝐹 Fn ℤ)
25	23, 24	syl 18	. . . 4 ⊢ (𝜑 → 𝐹 Fn ℤ)
26		dffn3 6722	. . . 4 ⊢ (𝐹 Fn ℤ ↔ 𝐹:ℤ⟶ran 𝐹)
27	25, 26	sylib 221	. . 3 ⊢ (𝜑 → 𝐹:ℤ⟶ran 𝐹)
28		fvelrnb 6945	. . . . . . . . . . 11 ⊢ (𝐹 Fn ℤ → (𝑤 ∈ ran 𝐹 ↔ ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤))
29	25, 28	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤))
30	29	biimpd 232	. . . . . . . . 9 ⊢ (𝜑 → (𝑤 ∈ ran 𝐹 → ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤))
31	30	imp 412	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ran 𝐹) → ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤)
32		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤) ∧ 𝑧 ∈ ℤ) ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑧) = 𝑤)
33	32	eqcomd 2771	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤) ∧ 𝑧 ∈ ℤ) ∧ (𝐹‘𝑧) = 𝑤) → 𝑤 = (𝐹‘𝑧))
34		simplll 787	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤) ∧ 𝑧 ∈ ℤ) ∧ (𝐹‘𝑧) = 𝑤) → 𝜑)
35		simplr 781	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤) ∧ 𝑧 ∈ ℤ) ∧ (𝐹‘𝑧) = 𝑤) → 𝑧 ∈ ℤ)
36	34, 35	jca 521	. . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ ℤ) ∧ 𝑥 = 𝑧) → 𝑥 = 𝑧)
39	38	oveq1d 7434	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ ℤ) ∧ 𝑥 = 𝑧) → (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀) = (𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀))
40		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ ℤ) → 𝑧 ∈ ℤ)
41		ovexd 7454	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ ℤ) → (𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀) ∈ V)
42	37, 39, 40, 41	fvmptd 7001	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ ℤ) → (𝐹‘𝑧) = (𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀))
43		eqid 2765	. . . . . . . . . . . . . . . 16 ⊢ (Base‘(𝑅 ↾_s 𝑈)) = (Base‘(𝑅 ↾_s 𝑈))
44		eqid 2765	. . . . . . . . . . . . . . . 16 ⊢ (._g‘(𝑅 ↾_s 𝑈)) = (._g‘(𝑅 ↾_s 𝑈))
45	17, 18, 19	primrootsunit 42925	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑅 PrimRoots 𝐾) = ((𝑅 ↾_s 𝑈) PrimRoots 𝐾) ∧ (𝑅 ↾_s 𝑈) ∈ Abel))
46	45	simprd 501	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑅 ↾_s 𝑈) ∈ Abel)
47	46	ablgrpd 19902	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑅 ↾_s 𝑈) ∈ Grp)
48	47	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ ℤ) → (𝑅 ↾_s 𝑈) ∈ Grp)
49	45	simpld 500	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑅 PrimRoots 𝐾) = ((𝑅 ↾_s 𝑈) PrimRoots 𝐾))
50	20, 49	eleqtrd 2867	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ ((𝑅 ↾_s 𝑈) PrimRoots 𝐾))
51	46	ablcmnd 19904	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑅 ↾_s 𝑈) ∈ CMnd)
52	18	nnnn0d 12582	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
53	51, 52, 44	isprimroot 42920	. . . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . . . . 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 1160	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈)))
57	56	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ ℤ) → 𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈)))
58	43, 44, 48, 40, 57	mulgcld 19208	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ ℤ) → (𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀) ∈ (Base‘(𝑅 ↾_s 𝑈)))
59	42, 58	eqeltrd 2865	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ ℤ) → (𝐹‘𝑧) ∈ (Base‘(𝑅 ↾_s 𝑈)))
60	36, 59	syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤) ∧ 𝑧 ∈ ℤ) ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑧) ∈ (Base‘(𝑅 ↾_s 𝑈)))
61	33, 60	eqeltrd 2865	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤) ∧ 𝑧 ∈ ℤ) ∧ (𝐹‘𝑧) = 𝑤) → 𝑤 ∈ (Base‘(𝑅 ↾_s 𝑈)))
62		nfv 1947	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧(𝐹‘𝑣) = 𝑤
63		nfv 1947	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑣(𝐹‘𝑧) = 𝑤
64		fveqeq2 6894	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑧 → ((𝐹‘𝑣) = 𝑤 ↔ (𝐹‘𝑧) = 𝑤))
65	62, 63, 64	cbvrexw 3310	. . . . . . . . . . . . 13 ⊢ (∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤 ↔ ∃𝑧 ∈ ℤ (𝐹‘𝑧) = 𝑤)
66	65	bilani 510	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤) → ∃𝑧 ∈ ℤ (𝐹‘𝑧) = 𝑤)
67	61, 66	r19.29a 3175	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤) → 𝑤 ∈ (Base‘(𝑅 ↾_s 𝑈)))
68	67	ex 418	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤 → 𝑤 ∈ (Base‘(𝑅 ↾_s 𝑈))))
69	68	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ ran 𝐹) → (∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤 → 𝑤 ∈ (Base‘(𝑅 ↾_s 𝑈))))
70	69	imp 412	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ ran 𝐹) ∧ ∃𝑣 ∈ ℤ (𝐹‘𝑣) = 𝑤) → 𝑤 ∈ (Base‘(𝑅 ↾_s 𝑈)))
71	31, 70	mpdan 700	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ran 𝐹) → 𝑤 ∈ (Base‘(𝑅 ↾_s 𝑈)))
72	71	ex 418	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ ran 𝐹 → 𝑤 ∈ (Base‘(𝑅 ↾_s 𝑈))))
73	72	ssrdv 3944	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘(𝑅 ↾_s 𝑈)))
74	9, 43	ressbas2 17322	. . . . 5 ⊢ (ran 𝐹 ⊆ (Base‘(𝑅 ↾_s 𝑈)) → ran 𝐹 = (Base‘((𝑅 ↾_s 𝑈) ↾_s ran 𝐹)))
75	73, 74	syl 18	. . . 4 ⊢ (𝜑 → ran 𝐹 = (Base‘((𝑅 ↾_s 𝑈) ↾_s ran 𝐹)))
76	75	feq3d 6694	. . 3 ⊢ (𝜑 → (𝐹:ℤ⟶ran 𝐹 ↔ 𝐹:ℤ⟶(Base‘((𝑅 ↾_s 𝑈) ↾_s ran 𝐹))))
77	27, 76	mpbid 235	. 2 ⊢ (𝜑 → 𝐹:ℤ⟶(Base‘((𝑅 ↾_s 𝑈) ↾_s ran 𝐹)))
78	4	a1i 11	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀)))
79		simpr 490	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ 𝑥 = (𝑦 + 𝑧)) → 𝑥 = (𝑦 + 𝑧))
80	79	oveq1d 7434	. . . 4 ⊢ (((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ 𝑥 = (𝑦 + 𝑧)) → (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀) = ((𝑦 + 𝑧)(._g‘(𝑅 ↾_s 𝑈))𝑀))
81		simprl 783	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → 𝑦 ∈ ℤ)
82		simprr 785	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → 𝑧 ∈ ℤ)
83	81, 82	zaddcld 12722	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝑦 + 𝑧) ∈ ℤ)
84		ovexd 7454	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → ((𝑦 + 𝑧)(._g‘(𝑅 ↾_s 𝑈))𝑀) ∈ V)
85	78, 80, 83, 84	fvmptd 7001	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝐹‘(𝑦 + 𝑧)) = ((𝑦 + 𝑧)(._g‘(𝑅 ↾_s 𝑈))𝑀))
86	47	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝑅 ↾_s 𝑈) ∈ Grp)
87	56	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → 𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈)))
88	81, 82, 87	3jca 1146	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ ∧ 𝑀 ∈ (Base‘(𝑅 ↾_s 𝑈))))
89	43, 44, 10	mulgdir 19218	. . . . 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)))
90	86, 88, 89	syl2anc 596	. . . 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)))
91		simpr 490	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
92	91	oveq1d 7434	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ 𝑥 = 𝑦) → (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀) = (𝑦(._g‘(𝑅 ↾_s 𝑈))𝑀))
93		ovexd 7454	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝑦(._g‘(𝑅 ↾_s 𝑈))𝑀) ∈ V)
94	78, 92, 81, 93	fvmptd 7001	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝐹‘𝑦) = (𝑦(._g‘(𝑅 ↾_s 𝑈))𝑀))
95		simpr 490	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ 𝑥 = 𝑧) → 𝑥 = 𝑧)
96	95	oveq1d 7434	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ 𝑥 = 𝑧) → (𝑥(._g‘(𝑅 ↾_s 𝑈))𝑀) = (𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀))
97		ovexd 7454	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀) ∈ V)
98	78, 96, 82, 97	fvmptd 7001	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝐹‘𝑧) = (𝑧(._g‘(𝑅 ↾_s 𝑈))𝑀))
99	94, 98	oveq12d 7437	. . . . 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)))
100	99	eqcomd 2771	. . . 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)))
101	90, 100	eqtrd 2800	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → ((𝑦 + 𝑧)(._g‘(𝑅 ↾_s 𝑈))𝑀) = ((𝐹‘𝑦)(+_g‘(𝑅 ↾_s 𝑈))(𝐹‘𝑧)))
102	85, 101	eqtrd 2800	. 2 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (𝐹‘(𝑦 + 𝑧)) = ((𝐹‘𝑦)(+_g‘(𝑅 ↾_s 𝑈))(𝐹‘𝑧)))
103	1, 2, 3, 12, 16, 21, 77, 102	isghmd 19341	1 ⊢ (𝜑 → 𝐹 ∈ (ℤ_ring GrpHom ((𝑅 ↾_s 𝑈) ↾_s ran 𝐹)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ∀wral 3081 ∃wrex 3091 {crab 3418 Vcvv 3457 ⊆ wss 3906 class class class wbr 5111 ↦ cmpt 5194 ran crn 5664 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 + caddc 11120 ℕcn 12250 ℕ₀cn0 12521 ℤcz 12608 ∥ cdvds 16334 Basecbs 17293 ↾_s cress 17314 +_gcplusg 17334 0_gc0g 17516 Grpcgrp 19046 ._gcmg 19179 GrpHom cghm 19329 CMndccmn 19896 Abelcabl 19897 Ringcrg 20361 ℤ_ringczring 21648 PrimRoots cprimroots 42918

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 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194 ax-addf 11196

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 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-nn 12251 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-7 12325 df-8 12326 df-9 12327 df-n0 12522 df-z 12609 df-dec 12730 df-uz 12881 df-fz 13554 df-seq 14058 df-struct 17231 df-sets 17248 df-slot 17266 df-ndx 17278 df-base 17294 df-ress 17315 df-plusg 17347 df-mulr 17348 df-starv 17349 df-tset 17353 df-ple 17354 df-ds 17356 df-unif 17357 df-0g 17518 df-mgm 18722 df-sgrp 18811 df-mnd 18827 df-submnd 18881 df-grp 19049 df-minusg 19050 df-mulg 19180 df-subg 19235 df-ghm 19330 df-cmn 19898 df-abl 19899 df-mgp 20263 df-rng 20277 df-ur 20310 df-ring 20363 df-cring 20364 df-subrng 20697 df-subrg 20721 df-cnfld 21575 df-zring 21649 df-primroots 42919

This theorem is used by: aks6d1c6lem5 43004

W3C validator