Theorem aks6d1c6isolem1 42450

Description: Lemma to construct the map out of the quotient for AKS. (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) = (0g‘𝑅)}
aks6d1c6isolem1.4	⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥(.g‘(𝑅 ↾s 𝑈))𝑀))
aks6d1c6isolem1.5	⊢ (𝜑 → 𝑀 ∈ (𝑅 PrimRoots 𝐾))

Assertion

Ref	Expression
aks6d1c6isolem1	⊢ (𝜑 → ((𝑅 ↾s 𝑈) ↾s ran 𝐹) ∈ Grp)

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 aks6d1c6isolem1

Dummy variables 𝑐 𝑑 𝑓 𝑔 𝑦 𝑒 𝑧 ℎ 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2737	. 2 ⊢ (𝜑 → ((𝑅 ↾s 𝑈) ↾s ran 𝐹) = ((𝑅 ↾s 𝑈) ↾s ran 𝐹))
2		eqidd 2737	. 2 ⊢ (𝜑 → (0g‘(𝑅 ↾s 𝑈)) = (0g‘(𝑅 ↾s 𝑈)))
3		eqidd 2737	. 2 ⊢ (𝜑 → (+g‘(𝑅 ↾s 𝑈)) = (+g‘(𝑅 ↾s 𝑈)))
4		eqid 2736	. . . . 5 ⊢ (Base‘(𝑅 ↾s 𝑈)) = (Base‘(𝑅 ↾s 𝑈))
5		eqid 2736	. . . . 5 ⊢ (.g‘(𝑅 ↾s 𝑈)) = (.g‘(𝑅 ↾s 𝑈))
6		aks6d1c6isolem1.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ CMnd)
7		aks6d1c6isolem1.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℕ)
8		aks6d1c6isolem1.3	. . . . . . . . 9 ⊢ 𝑈 = {𝑎 ∈ (Base‘𝑅) ∣ ∃𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑎) = (0g‘𝑅)}
9	6, 7, 8	primrootsunit 42374	. . . . . . . 8 ⊢ (𝜑 → ((𝑅 PrimRoots 𝐾) = ((𝑅 ↾s 𝑈) PrimRoots 𝐾) ∧ (𝑅 ↾s 𝑈) ∈ Abel))
10	9	simprd 495	. . . . . . 7 ⊢ (𝜑 → (𝑅 ↾s 𝑈) ∈ Abel)
11	10	ablgrpd 19717	. . . . . 6 ⊢ (𝜑 → (𝑅 ↾s 𝑈) ∈ Grp)
12	11	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → (𝑅 ↾s 𝑈) ∈ Grp)
13		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → 𝑥 ∈ ℤ)
14		aks6d1c6isolem1.5	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (𝑅 PrimRoots 𝐾))
15	9	simpld 494	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 PrimRoots 𝐾) = ((𝑅 ↾s 𝑈) PrimRoots 𝐾))
16	14, 15	eleqtrd 2838	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ((𝑅 ↾s 𝑈) PrimRoots 𝐾))
17	10	ablcmnd 19719	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ↾s 𝑈) ∈ CMnd)
18	7	nnnn0d 12464	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
19	17, 18, 5	isprimroot 42369	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 ∈ ((𝑅 ↾s 𝑈) PrimRoots 𝐾) ↔ (𝑀 ∈ (Base‘(𝑅 ↾s 𝑈)) ∧ (𝐾(.g‘(𝑅 ↾s 𝑈))𝑀) = (0g‘(𝑅 ↾s 𝑈)) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘(𝑅 ↾s 𝑈))𝑀) = (0g‘(𝑅 ↾s 𝑈)) → 𝐾 ∥ 𝑙))))
20	19	biimpd 229	. . . . . . . 8 ⊢ (𝜑 → (𝑀 ∈ ((𝑅 ↾s 𝑈) PrimRoots 𝐾) → (𝑀 ∈ (Base‘(𝑅 ↾s 𝑈)) ∧ (𝐾(.g‘(𝑅 ↾s 𝑈))𝑀) = (0g‘(𝑅 ↾s 𝑈)) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘(𝑅 ↾s 𝑈))𝑀) = (0g‘(𝑅 ↾s 𝑈)) → 𝐾 ∥ 𝑙))))
21	16, 20	mpd 15	. . . . . . 7 ⊢ (𝜑 → (𝑀 ∈ (Base‘(𝑅 ↾s 𝑈)) ∧ (𝐾(.g‘(𝑅 ↾s 𝑈))𝑀) = (0g‘(𝑅 ↾s 𝑈)) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘(𝑅 ↾s 𝑈))𝑀) = (0g‘(𝑅 ↾s 𝑈)) → 𝐾 ∥ 𝑙)))
22	21	simp1d 1142	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (Base‘(𝑅 ↾s 𝑈)))
23	22	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → 𝑀 ∈ (Base‘(𝑅 ↾s 𝑈)))
24	4, 5, 12, 13, 23	mulgcld 19028	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → (𝑥(.g‘(𝑅 ↾s 𝑈))𝑀) ∈ (Base‘(𝑅 ↾s 𝑈)))
25		aks6d1c6isolem1.4	. . . 4 ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥(.g‘(𝑅 ↾s 𝑈))𝑀))
26	24, 25	fmptd 7059	. . 3 ⊢ (𝜑 → 𝐹:ℤ⟶(Base‘(𝑅 ↾s 𝑈)))
27		frn 6669	. . 3 ⊢ (𝐹:ℤ⟶(Base‘(𝑅 ↾s 𝑈)) → ran 𝐹 ⊆ (Base‘(𝑅 ↾s 𝑈)))
28	26, 27	syl 17	. 2 ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘(𝑅 ↾s 𝑈)))
29		0zd 12502	. . . 4 ⊢ (𝜑 → 0 ∈ ℤ)
30		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 0) → 𝑐 = 0)
31	30	fveqeq2d 6842	. . . 4 ⊢ ((𝜑 ∧ 𝑐 = 0) → ((𝐹‘𝑐) = (0g‘(𝑅 ↾s 𝑈)) ↔ (𝐹‘0) = (0g‘(𝑅 ↾s 𝑈))))
32	25	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥(.g‘(𝑅 ↾s 𝑈))𝑀)))
33		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 0) → 𝑥 = 0)
34	33	oveq1d 7373	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 0) → (𝑥(.g‘(𝑅 ↾s 𝑈))𝑀) = (0(.g‘(𝑅 ↾s 𝑈))𝑀))
35		eqid 2736	. . . . . . . . 9 ⊢ (0g‘(𝑅 ↾s 𝑈)) = (0g‘(𝑅 ↾s 𝑈))
36	4, 35, 5	mulg0 19006	. . . . . . . 8 ⊢ (𝑀 ∈ (Base‘(𝑅 ↾s 𝑈)) → (0(.g‘(𝑅 ↾s 𝑈))𝑀) = (0g‘(𝑅 ↾s 𝑈)))
37	22, 36	syl 17	. . . . . . 7 ⊢ (𝜑 → (0(.g‘(𝑅 ↾s 𝑈))𝑀) = (0g‘(𝑅 ↾s 𝑈)))
38	37	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 0) → (0(.g‘(𝑅 ↾s 𝑈))𝑀) = (0g‘(𝑅 ↾s 𝑈)))
39	34, 38	eqtrd 2771	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 0) → (𝑥(.g‘(𝑅 ↾s 𝑈))𝑀) = (0g‘(𝑅 ↾s 𝑈)))
40		fvexd 6849	. . . . 5 ⊢ (𝜑 → (0g‘(𝑅 ↾s 𝑈)) ∈ V)
41	32, 39, 29, 40	fvmptd 6948	. . . 4 ⊢ (𝜑 → (𝐹‘0) = (0g‘(𝑅 ↾s 𝑈)))
42	29, 31, 41	rspcedvd 3578	. . 3 ⊢ (𝜑 → ∃𝑐 ∈ ℤ (𝐹‘𝑐) = (0g‘(𝑅 ↾s 𝑈)))
43	26	ffnd 6663	. . . 4 ⊢ (𝜑 → 𝐹 Fn ℤ)
44		fvelrnb 6894	. . . 4 ⊢ (𝐹 Fn ℤ → ((0g‘(𝑅 ↾s 𝑈)) ∈ ran 𝐹 ↔ ∃𝑐 ∈ ℤ (𝐹‘𝑐) = (0g‘(𝑅 ↾s 𝑈))))
45	43, 44	syl 17	. . 3 ⊢ (𝜑 → ((0g‘(𝑅 ↾s 𝑈)) ∈ ran 𝐹 ↔ ∃𝑐 ∈ ℤ (𝐹‘𝑐) = (0g‘(𝑅 ↾s 𝑈))))
46	42, 45	mpbird 257	. 2 ⊢ (𝜑 → (0g‘(𝑅 ↾s 𝑈)) ∈ ran 𝐹)
47		fvelrnb 6894	. . . . . . 7 ⊢ (𝐹 Fn ℤ → (𝑦 ∈ ran 𝐹 ↔ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦))
48	43, 47	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦))
49	48	biimpd 229	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ ran 𝐹 → ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦))
50	49	imp 406	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦)
51	50	3adant3 1132	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹 ∧ 𝑧 ∈ ran 𝐹) → ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦)
52		simpl1 1192	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹 ∧ 𝑧 ∈ ran 𝐹) ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) → 𝜑)
53		simpl3 1194	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹 ∧ 𝑧 ∈ ran 𝐹) ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) → 𝑧 ∈ ran 𝐹)
54	52, 53	jca 511	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹 ∧ 𝑧 ∈ ran 𝐹) ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) → (𝜑 ∧ 𝑧 ∈ ran 𝐹))
55		fvelrnb 6894	. . . . . . . 8 ⊢ (𝐹 Fn ℤ → (𝑧 ∈ ran 𝐹 ↔ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧))
56	43, 55	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ ran 𝐹 ↔ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧))
57	56	biimpd 229	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ ran 𝐹 → ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧))
58	57	imp 406	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐹) → ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧)
59	54, 58	syl 17	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹 ∧ 𝑧 ∈ ran 𝐹) ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) → ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧)
60		simpll1 1213	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝐹 ∧ 𝑧 ∈ ran 𝐹) ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) → 𝜑)
61		simplr 768	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝐹 ∧ 𝑧 ∈ ran 𝐹) ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) → ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦)
62		simpr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝐹 ∧ 𝑧 ∈ ran 𝐹) ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) → ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧)
63	60, 61, 62	3jca 1128	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝐹 ∧ 𝑧 ∈ ran 𝐹) ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) → (𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧))
64		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) ∧ 𝑔 ∈ ℤ) ∧ (𝐹‘𝑔) = 𝑧) → (𝐹‘𝑔) = 𝑧)
65	64	eqcomd 2742	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) ∧ 𝑔 ∈ ℤ) ∧ (𝐹‘𝑔) = 𝑧) → 𝑧 = (𝐹‘𝑔))
66	65	oveq2d 7374	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) ∧ 𝑔 ∈ ℤ) ∧ (𝐹‘𝑔) = 𝑧) → (𝑦(+g‘(𝑅 ↾s 𝑈))𝑧) = (𝑦(+g‘(𝑅 ↾s 𝑈))(𝐹‘𝑔)))
67		simpr 484	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) ∧ 𝑔 ∈ ℤ) ∧ 𝑓 ∈ ℤ) ∧ (𝐹‘𝑓) = 𝑦) → (𝐹‘𝑓) = 𝑦)
68	67	eqcomd 2742	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) ∧ 𝑔 ∈ ℤ) ∧ 𝑓 ∈ ℤ) ∧ (𝐹‘𝑓) = 𝑦) → 𝑦 = (𝐹‘𝑓))
69	68	oveq1d 7373	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) ∧ 𝑔 ∈ ℤ) ∧ 𝑓 ∈ ℤ) ∧ (𝐹‘𝑓) = 𝑦) → (𝑦(+g‘(𝑅 ↾s 𝑈))(𝐹‘𝑔)) = ((𝐹‘𝑓)(+g‘(𝑅 ↾s 𝑈))(𝐹‘𝑔)))
70		simpll1 1213	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) ∧ 𝑔 ∈ ℤ) ∧ 𝑓 ∈ ℤ) → 𝜑)
71	70	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) ∧ 𝑔 ∈ ℤ) ∧ 𝑓 ∈ ℤ) ∧ (𝐹‘𝑓) = 𝑦) → 𝜑)
72		simpllr 775	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) ∧ 𝑔 ∈ ℤ) ∧ 𝑓 ∈ ℤ) ∧ (𝐹‘𝑓) = 𝑦) → 𝑔 ∈ ℤ)
73		simplr 768	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) ∧ 𝑔 ∈ ℤ) ∧ 𝑓 ∈ ℤ) ∧ (𝐹‘𝑓) = 𝑦) → 𝑓 ∈ ℤ)
74	71, 72, 73	3jca 1128	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) ∧ 𝑔 ∈ ℤ) ∧ 𝑓 ∈ ℤ) ∧ (𝐹‘𝑓) = 𝑦) → (𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ))
75	25	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥(.g‘(𝑅 ↾s 𝑈))𝑀)))
76		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) ∧ 𝑥 = 𝑓) → 𝑥 = 𝑓)
77	76	oveq1d 7373	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) ∧ 𝑥 = 𝑓) → (𝑥(.g‘(𝑅 ↾s 𝑈))𝑀) = (𝑓(.g‘(𝑅 ↾s 𝑈))𝑀))
78		simp3 1138	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → 𝑓 ∈ ℤ)
79		ovexd 7393	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → (𝑓(.g‘(𝑅 ↾s 𝑈))𝑀) ∈ V)
80	75, 77, 78, 79	fvmptd 6948	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → (𝐹‘𝑓) = (𝑓(.g‘(𝑅 ↾s 𝑈))𝑀))
81		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) ∧ 𝑥 = 𝑔) → 𝑥 = 𝑔)
82	81	oveq1d 7373	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) ∧ 𝑥 = 𝑔) → (𝑥(.g‘(𝑅 ↾s 𝑈))𝑀) = (𝑔(.g‘(𝑅 ↾s 𝑈))𝑀))
83		simp2 1137	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → 𝑔 ∈ ℤ)
84		ovexd 7393	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → (𝑔(.g‘(𝑅 ↾s 𝑈))𝑀) ∈ V)
85	75, 82, 83, 84	fvmptd 6948	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → (𝐹‘𝑔) = (𝑔(.g‘(𝑅 ↾s 𝑈))𝑀))
86	80, 85	oveq12d 7376	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → ((𝐹‘𝑓)(+g‘(𝑅 ↾s 𝑈))(𝐹‘𝑔)) = ((𝑓(.g‘(𝑅 ↾s 𝑈))𝑀)(+g‘(𝑅 ↾s 𝑈))(𝑔(.g‘(𝑅 ↾s 𝑈))𝑀)))
87	11	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → (𝑅 ↾s 𝑈) ∈ Grp)
88	22	3ad2ant1 1133	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → 𝑀 ∈ (Base‘(𝑅 ↾s 𝑈)))
89	78, 83, 88	3jca 1128	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → (𝑓 ∈ ℤ ∧ 𝑔 ∈ ℤ ∧ 𝑀 ∈ (Base‘(𝑅 ↾s 𝑈))))
90		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (+g‘(𝑅 ↾s 𝑈)) = (+g‘(𝑅 ↾s 𝑈))
91	4, 5, 90	mulgdir 19038	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ↾s 𝑈) ∈ Grp ∧ (𝑓 ∈ ℤ ∧ 𝑔 ∈ ℤ ∧ 𝑀 ∈ (Base‘(𝑅 ↾s 𝑈)))) → ((𝑓 + 𝑔)(.g‘(𝑅 ↾s 𝑈))𝑀) = ((𝑓(.g‘(𝑅 ↾s 𝑈))𝑀)(+g‘(𝑅 ↾s 𝑈))(𝑔(.g‘(𝑅 ↾s 𝑈))𝑀)))
92	87, 89, 91	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → ((𝑓 + 𝑔)(.g‘(𝑅 ↾s 𝑈))𝑀) = ((𝑓(.g‘(𝑅 ↾s 𝑈))𝑀)(+g‘(𝑅 ↾s 𝑈))(𝑔(.g‘(𝑅 ↾s 𝑈))𝑀)))
93	78, 83	zaddcld 12602	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → (𝑓 + 𝑔) ∈ ℤ)
94		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) ∧ ℎ = (𝑓 + 𝑔)) → ℎ = (𝑓 + 𝑔))
95	94	fveqeq2d 6842	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) ∧ ℎ = (𝑓 + 𝑔)) → ((𝐹‘ℎ) = ((𝑓 + 𝑔)(.g‘(𝑅 ↾s 𝑈))𝑀) ↔ (𝐹‘(𝑓 + 𝑔)) = ((𝑓 + 𝑔)(.g‘(𝑅 ↾s 𝑈))𝑀)))
96		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) ∧ 𝑥 = (𝑓 + 𝑔)) → 𝑥 = (𝑓 + 𝑔))
97	96	oveq1d 7373	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) ∧ 𝑥 = (𝑓 + 𝑔)) → (𝑥(.g‘(𝑅 ↾s 𝑈))𝑀) = ((𝑓 + 𝑔)(.g‘(𝑅 ↾s 𝑈))𝑀))
98		ovexd 7393	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → ((𝑓 + 𝑔)(.g‘(𝑅 ↾s 𝑈))𝑀) ∈ V)
99	75, 97, 93, 98	fvmptd 6948	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → (𝐹‘(𝑓 + 𝑔)) = ((𝑓 + 𝑔)(.g‘(𝑅 ↾s 𝑈))𝑀))
100	93, 95, 99	rspcedvd 3578	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → ∃ℎ ∈ ℤ (𝐹‘ℎ) = ((𝑓 + 𝑔)(.g‘(𝑅 ↾s 𝑈))𝑀))
101		fvelrnb 6894	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 Fn ℤ → (((𝑓 + 𝑔)(.g‘(𝑅 ↾s 𝑈))𝑀) ∈ ran 𝐹 ↔ ∃ℎ ∈ ℤ (𝐹‘ℎ) = ((𝑓 + 𝑔)(.g‘(𝑅 ↾s 𝑈))𝑀)))
102	43, 101	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((𝑓 + 𝑔)(.g‘(𝑅 ↾s 𝑈))𝑀) ∈ ran 𝐹 ↔ ∃ℎ ∈ ℤ (𝐹‘ℎ) = ((𝑓 + 𝑔)(.g‘(𝑅 ↾s 𝑈))𝑀)))
103	102	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → (((𝑓 + 𝑔)(.g‘(𝑅 ↾s 𝑈))𝑀) ∈ ran 𝐹 ↔ ∃ℎ ∈ ℤ (𝐹‘ℎ) = ((𝑓 + 𝑔)(.g‘(𝑅 ↾s 𝑈))𝑀)))
104	100, 103	mpbird 257	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → ((𝑓 + 𝑔)(.g‘(𝑅 ↾s 𝑈))𝑀) ∈ ran 𝐹)
105	92, 104	eqeltrrd 2837	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → ((𝑓(.g‘(𝑅 ↾s 𝑈))𝑀)(+g‘(𝑅 ↾s 𝑈))(𝑔(.g‘(𝑅 ↾s 𝑈))𝑀)) ∈ ran 𝐹)
106	86, 105	eqeltrd 2836	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → ((𝐹‘𝑓)(+g‘(𝑅 ↾s 𝑈))(𝐹‘𝑔)) ∈ ran 𝐹)
107	74, 106	syl 17	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) ∧ 𝑔 ∈ ℤ) ∧ 𝑓 ∈ ℤ) ∧ (𝐹‘𝑓) = 𝑦) → ((𝐹‘𝑓)(+g‘(𝑅 ↾s 𝑈))(𝐹‘𝑔)) ∈ ran 𝐹)
108	69, 107	eqeltrd 2836	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) ∧ 𝑔 ∈ ℤ) ∧ 𝑓 ∈ ℤ) ∧ (𝐹‘𝑓) = 𝑦) → (𝑦(+g‘(𝑅 ↾s 𝑈))(𝐹‘𝑔)) ∈ ran 𝐹)
109		simpl2 1193	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) ∧ 𝑔 ∈ ℤ) → ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦)
110		nfv 1915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑓(𝐹‘𝑑) = 𝑦
111		nfv 1915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑑(𝐹‘𝑓) = 𝑦
112		fveqeq2 6843	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑓 → ((𝐹‘𝑑) = 𝑦 ↔ (𝐹‘𝑓) = 𝑦))
113	110, 111, 112	cbvrexw 3279	. . . . . . . . . . . 12 ⊢ (∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ↔ ∃𝑓 ∈ ℤ (𝐹‘𝑓) = 𝑦)
114	113	biimpi 216	. . . . . . . . . . 11 ⊢ (∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 → ∃𝑓 ∈ ℤ (𝐹‘𝑓) = 𝑦)
115	109, 114	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) ∧ 𝑔 ∈ ℤ) → ∃𝑓 ∈ ℤ (𝐹‘𝑓) = 𝑦)
116	108, 115	r19.29a 3144	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) ∧ 𝑔 ∈ ℤ) → (𝑦(+g‘(𝑅 ↾s 𝑈))(𝐹‘𝑔)) ∈ ran 𝐹)
117	116	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) ∧ 𝑔 ∈ ℤ) ∧ (𝐹‘𝑔) = 𝑧) → (𝑦(+g‘(𝑅 ↾s 𝑈))(𝐹‘𝑔)) ∈ ran 𝐹)
118	66, 117	eqeltrd 2836	. . . . . . 7 ⊢ ((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) ∧ 𝑔 ∈ ℤ) ∧ (𝐹‘𝑔) = 𝑧) → (𝑦(+g‘(𝑅 ↾s 𝑈))𝑧) ∈ ran 𝐹)
119		simp3 1138	. . . . . . . 8 ⊢ ((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) → ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧)
120		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑔(𝐹‘𝑒) = 𝑧
121		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑒(𝐹‘𝑔) = 𝑧
122		fveqeq2 6843	. . . . . . . . . 10 ⊢ (𝑒 = 𝑔 → ((𝐹‘𝑒) = 𝑧 ↔ (𝐹‘𝑔) = 𝑧))
123	120, 121, 122	cbvrexw 3279	. . . . . . . . 9 ⊢ (∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧 ↔ ∃𝑔 ∈ ℤ (𝐹‘𝑔) = 𝑧)
124	123	biimpi 216	. . . . . . . 8 ⊢ (∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧 → ∃𝑔 ∈ ℤ (𝐹‘𝑔) = 𝑧)
125	119, 124	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) → ∃𝑔 ∈ ℤ (𝐹‘𝑔) = 𝑧)
126	118, 125	r19.29a 3144	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) → (𝑦(+g‘(𝑅 ↾s 𝑈))𝑧) ∈ ran 𝐹)
127	63, 126	syl 17	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝐹 ∧ 𝑧 ∈ ran 𝐹) ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) → (𝑦(+g‘(𝑅 ↾s 𝑈))𝑧) ∈ ran 𝐹)
128	127	ex 412	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹 ∧ 𝑧 ∈ ran 𝐹) ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) → (∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧 → (𝑦(+g‘(𝑅 ↾s 𝑈))𝑧) ∈ ran 𝐹))
129	59, 128	mpd 15	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹 ∧ 𝑧 ∈ ran 𝐹) ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) → (𝑦(+g‘(𝑅 ↾s 𝑈))𝑧) ∈ ran 𝐹)
130	51, 129	mpdan 687	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹 ∧ 𝑧 ∈ ran 𝐹) → (𝑦(+g‘(𝑅 ↾s 𝑈))𝑧) ∈ ran 𝐹)
131		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) ∧ 𝑓 ∈ ℤ) ∧ (𝐹‘𝑓) = 𝑦) → (𝐹‘𝑓) = 𝑦)
132	131	eqcomd 2742	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) ∧ 𝑓 ∈ ℤ) ∧ (𝐹‘𝑓) = 𝑦) → 𝑦 = (𝐹‘𝑓))
133	132	fveq2d 6838	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) ∧ 𝑓 ∈ ℤ) ∧ (𝐹‘𝑓) = 𝑦) → ((invg‘(𝑅 ↾s 𝑈))‘𝑦) = ((invg‘(𝑅 ↾s 𝑈))‘(𝐹‘𝑓)))
134		simplll 774	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) ∧ 𝑓 ∈ ℤ) ∧ (𝐹‘𝑓) = 𝑦) → 𝜑)
135		simplr 768	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) ∧ 𝑓 ∈ ℤ) ∧ (𝐹‘𝑓) = 𝑦) → 𝑓 ∈ ℤ)
136	134, 135	jca 511	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) ∧ 𝑓 ∈ ℤ) ∧ (𝐹‘𝑓) = 𝑦) → (𝜑 ∧ 𝑓 ∈ ℤ))
137		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → 𝑓 ∈ ℤ)
138	137	znegcld 12600	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → -𝑓 ∈ ℤ)
139		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ ℤ) ∧ ℎ = -𝑓) → ℎ = -𝑓)
140	139	fveqeq2d 6842	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ ℤ) ∧ ℎ = -𝑓) → ((𝐹‘ℎ) = ((invg‘(𝑅 ↾s 𝑈))‘(𝐹‘𝑓)) ↔ (𝐹‘-𝑓) = ((invg‘(𝑅 ↾s 𝑈))‘(𝐹‘𝑓))))
141	25	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥(.g‘(𝑅 ↾s 𝑈))𝑀)))
142		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑓 ∈ ℤ) ∧ 𝑥 = -𝑓) → 𝑥 = -𝑓)
143	142	oveq1d 7373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ℤ) ∧ 𝑥 = -𝑓) → (𝑥(.g‘(𝑅 ↾s 𝑈))𝑀) = (-𝑓(.g‘(𝑅 ↾s 𝑈))𝑀))
144		ovexd 7393	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → (-𝑓(.g‘(𝑅 ↾s 𝑈))𝑀) ∈ V)
145	141, 143, 138, 144	fvmptd 6948	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → (𝐹‘-𝑓) = (-𝑓(.g‘(𝑅 ↾s 𝑈))𝑀))
146	11	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → (𝑅 ↾s 𝑈) ∈ Grp)
147	22	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → 𝑀 ∈ (Base‘(𝑅 ↾s 𝑈)))
148		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (invg‘(𝑅 ↾s 𝑈)) = (invg‘(𝑅 ↾s 𝑈))
149	4, 5, 148	mulgneg 19024	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ↾s 𝑈) ∈ Grp ∧ 𝑓 ∈ ℤ ∧ 𝑀 ∈ (Base‘(𝑅 ↾s 𝑈))) → (-𝑓(.g‘(𝑅 ↾s 𝑈))𝑀) = ((invg‘(𝑅 ↾s 𝑈))‘(𝑓(.g‘(𝑅 ↾s 𝑈))𝑀)))
150	146, 137, 147, 149	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → (-𝑓(.g‘(𝑅 ↾s 𝑈))𝑀) = ((invg‘(𝑅 ↾s 𝑈))‘(𝑓(.g‘(𝑅 ↾s 𝑈))𝑀)))
151		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑓 ∈ ℤ) ∧ 𝑥 = 𝑓) → 𝑥 = 𝑓)
152	151	oveq1d 7373	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑓 ∈ ℤ) ∧ 𝑥 = 𝑓) → (𝑥(.g‘(𝑅 ↾s 𝑈))𝑀) = (𝑓(.g‘(𝑅 ↾s 𝑈))𝑀))
153		ovexd 7393	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → (𝑓(.g‘(𝑅 ↾s 𝑈))𝑀) ∈ V)
154	141, 152, 137, 153	fvmptd 6948	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → (𝐹‘𝑓) = (𝑓(.g‘(𝑅 ↾s 𝑈))𝑀))
155	154	eqcomd 2742	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → (𝑓(.g‘(𝑅 ↾s 𝑈))𝑀) = (𝐹‘𝑓))
156	155	fveq2d 6838	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → ((invg‘(𝑅 ↾s 𝑈))‘(𝑓(.g‘(𝑅 ↾s 𝑈))𝑀)) = ((invg‘(𝑅 ↾s 𝑈))‘(𝐹‘𝑓)))
157	150, 156	eqtrd 2771	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → (-𝑓(.g‘(𝑅 ↾s 𝑈))𝑀) = ((invg‘(𝑅 ↾s 𝑈))‘(𝐹‘𝑓)))
158	145, 157	eqtrd 2771	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → (𝐹‘-𝑓) = ((invg‘(𝑅 ↾s 𝑈))‘(𝐹‘𝑓)))
159	138, 140, 158	rspcedvd 3578	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → ∃ℎ ∈ ℤ (𝐹‘ℎ) = ((invg‘(𝑅 ↾s 𝑈))‘(𝐹‘𝑓)))
160		fvelrnb 6894	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn ℤ → (((invg‘(𝑅 ↾s 𝑈))‘(𝐹‘𝑓)) ∈ ran 𝐹 ↔ ∃ℎ ∈ ℤ (𝐹‘ℎ) = ((invg‘(𝑅 ↾s 𝑈))‘(𝐹‘𝑓))))
161	43, 160	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (((invg‘(𝑅 ↾s 𝑈))‘(𝐹‘𝑓)) ∈ ran 𝐹 ↔ ∃ℎ ∈ ℤ (𝐹‘ℎ) = ((invg‘(𝑅 ↾s 𝑈))‘(𝐹‘𝑓))))
162	161	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → (((invg‘(𝑅 ↾s 𝑈))‘(𝐹‘𝑓)) ∈ ran 𝐹 ↔ ∃ℎ ∈ ℤ (𝐹‘ℎ) = ((invg‘(𝑅 ↾s 𝑈))‘(𝐹‘𝑓))))
163	159, 162	mpbird 257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → ((invg‘(𝑅 ↾s 𝑈))‘(𝐹‘𝑓)) ∈ ran 𝐹)
164	163	a1i 11	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) ∧ 𝑓 ∈ ℤ) ∧ (𝐹‘𝑓) = 𝑦) → ((𝜑 ∧ 𝑓 ∈ ℤ) → ((invg‘(𝑅 ↾s 𝑈))‘(𝐹‘𝑓)) ∈ ran 𝐹))
165	136, 164	mpd 15	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) ∧ 𝑓 ∈ ℤ) ∧ (𝐹‘𝑓) = 𝑦) → ((invg‘(𝑅 ↾s 𝑈))‘(𝐹‘𝑓)) ∈ ran 𝐹)
166	133, 165	eqeltrd 2836	. . . . . . 7 ⊢ ((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) ∧ 𝑓 ∈ ℤ) ∧ (𝐹‘𝑓) = 𝑦) → ((invg‘(𝑅 ↾s 𝑈))‘𝑦) ∈ ran 𝐹)
167	114	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) → ∃𝑓 ∈ ℤ (𝐹‘𝑓) = 𝑦)
168	166, 167	r19.29a 3144	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) → ((invg‘(𝑅 ↾s 𝑈))‘𝑦) ∈ ran 𝐹)
169	168	ex 412	. . . . 5 ⊢ (𝜑 → (∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 → ((invg‘(𝑅 ↾s 𝑈))‘𝑦) ∈ ran 𝐹))
170	169	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → (∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 → ((invg‘(𝑅 ↾s 𝑈))‘𝑦) ∈ ran 𝐹))
171	170	imp 406	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹) ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) → ((invg‘(𝑅 ↾s 𝑈))‘𝑦) ∈ ran 𝐹)
172	50, 171	mpdan 687	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → ((invg‘(𝑅 ↾s 𝑈))‘𝑦) ∈ ran 𝐹)
173	1, 2, 3, 28, 46, 130, 172, 11	issubgrpd 19075	1 ⊢ (𝜑 → ((𝑅 ↾s 𝑈) ↾s ran 𝐹) ∈ Grp)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  {crab 3399  Vcvv 3440   ⊆ wss 3901   class class class wbr 5098   ↦ cmpt 5179  ran crn 5625   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  0cc0 11028   + caddc 11031  -cneg 11367  ℕcn 12147  ℕ₀cn0 12403  ℤcz 12490   ∥ cdvds 16181  Basecbs 17138   ↾_s cress 17159  +_gcplusg 17179  0_gc0g 17361  Grpcgrp 18865  inv_gcminusg 18866  ._gcmg 18999  CMndccmn 19711  Abelcabl 19712   PrimRoots cprimroots 42367

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 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-er 8635  df-en 8886  df-dom 8887  df-sdom 8888  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-n0 12404  df-z 12491  df-uz 12754  df-fz 13426  df-seq 13927  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-ress 17160  df-plusg 17192  df-0g 17363  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-cmn 19713  df-abl 19714  df-primroots 42368

This theorem is referenced by:  aks6d1c6isolem2  42451