Theorem aks6d1c6isolem1 42156

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 2736	. 2 ⊢ (𝜑 → ((𝑅 ↾s 𝑈) ↾s ran 𝐹) = ((𝑅 ↾s 𝑈) ↾s ran 𝐹))
2		eqidd 2736	. 2 ⊢ (𝜑 → (0g‘(𝑅 ↾s 𝑈)) = (0g‘(𝑅 ↾s 𝑈)))
3		eqidd 2736	. 2 ⊢ (𝜑 → (+g‘(𝑅 ↾s 𝑈)) = (+g‘(𝑅 ↾s 𝑈)))
4		eqid 2735	. . . . 5 ⊢ (Base‘(𝑅 ↾s 𝑈)) = (Base‘(𝑅 ↾s 𝑈))
5		eqid 2735	. . . . 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 42080	. . . . . . . 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 19819	. . . . . 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 2841	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ((𝑅 ↾s 𝑈) PrimRoots 𝐾))
17	10	ablcmnd 19821	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ↾s 𝑈) ∈ CMnd)
18	7	nnnn0d 12585	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
19	17, 18, 5	isprimroot 42075	. . . . . . . . 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 1141	. . . . . 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 19127	. . . 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 7134	. . 3 ⊢ (𝜑 → 𝐹:ℤ⟶(Base‘(𝑅 ↾s 𝑈)))
27		frn 6744	. . 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 12623	. . . 4 ⊢ (𝜑 → 0 ∈ ℤ)
30		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 0) → 𝑐 = 0)
31	30	fveqeq2d 6915	. . . 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 7446	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 0) → (𝑥(.g‘(𝑅 ↾s 𝑈))𝑀) = (0(.g‘(𝑅 ↾s 𝑈))𝑀))
35		eqid 2735	. . . . . . . . 9 ⊢ (0g‘(𝑅 ↾s 𝑈)) = (0g‘(𝑅 ↾s 𝑈))
36	4, 35, 5	mulg0 19105	. . . . . . . 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 2775	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 0) → (𝑥(.g‘(𝑅 ↾s 𝑈))𝑀) = (0g‘(𝑅 ↾s 𝑈)))
40		fvexd 6922	. . . . 5 ⊢ (𝜑 → (0g‘(𝑅 ↾s 𝑈)) ∈ V)
41	32, 39, 29, 40	fvmptd 7023	. . . 4 ⊢ (𝜑 → (𝐹‘0) = (0g‘(𝑅 ↾s 𝑈)))
42	29, 31, 41	rspcedvd 3624	. . 3 ⊢ (𝜑 → ∃𝑐 ∈ ℤ (𝐹‘𝑐) = (0g‘(𝑅 ↾s 𝑈)))
43	26	ffnd 6738	. . . 4 ⊢ (𝜑 → 𝐹 Fn ℤ)
44		fvelrnb 6969	. . . 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 6969	. . . . . . 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 1131	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹 ∧ 𝑧 ∈ ran 𝐹) → ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦)
52		simpl1 1190	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹 ∧ 𝑧 ∈ ran 𝐹) ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) → 𝜑)
53		simpl3 1192	. . . . . 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 6969	. . . . . . . 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 1211	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ ran 𝐹 ∧ 𝑧 ∈ ran 𝐹) ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) → 𝜑)
61		simplr 769	. . . . . . 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 1127	. . . . . 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 2741	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) ∧ 𝑔 ∈ ℤ) ∧ (𝐹‘𝑔) = 𝑧) → 𝑧 = (𝐹‘𝑔))
66	65	oveq2d 7447	. . . . . . . 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 2741	. . . . . . . . . . . 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 7446	. . . . . . . . . . 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 1211	. . . . . . . . . . . . . 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 776	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) ∧ 𝑔 ∈ ℤ) ∧ 𝑓 ∈ ℤ) ∧ (𝐹‘𝑓) = 𝑦) → 𝑔 ∈ ℤ)
73		simplr 769	. . . . . . . . . . . . 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 1127	. . . . . . . . . . . 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 7446	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) ∧ 𝑥 = 𝑓) → (𝑥(.g‘(𝑅 ↾s 𝑈))𝑀) = (𝑓(.g‘(𝑅 ↾s 𝑈))𝑀))
78		simp3 1137	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → 𝑓 ∈ ℤ)
79		ovexd 7466	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → (𝑓(.g‘(𝑅 ↾s 𝑈))𝑀) ∈ V)
80	75, 77, 78, 79	fvmptd 7023	. . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) ∧ 𝑥 = 𝑔) → (𝑥(.g‘(𝑅 ↾s 𝑈))𝑀) = (𝑔(.g‘(𝑅 ↾s 𝑈))𝑀))
83		simp2 1136	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → 𝑔 ∈ ℤ)
84		ovexd 7466	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → (𝑔(.g‘(𝑅 ↾s 𝑈))𝑀) ∈ V)
85	75, 82, 83, 84	fvmptd 7023	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → (𝐹‘𝑔) = (𝑔(.g‘(𝑅 ↾s 𝑈))𝑀))
86	80, 85	oveq12d 7449	. . . . . . . . . . . . 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 1132	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → (𝑅 ↾s 𝑈) ∈ Grp)
88	22	3ad2ant1 1132	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → 𝑀 ∈ (Base‘(𝑅 ↾s 𝑈)))
89	78, 83, 88	3jca 1127	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → (𝑓 ∈ ℤ ∧ 𝑔 ∈ ℤ ∧ 𝑀 ∈ (Base‘(𝑅 ↾s 𝑈))))
90		eqid 2735	. . . . . . . . . . . . . . . 16 ⊢ (+g‘(𝑅 ↾s 𝑈)) = (+g‘(𝑅 ↾s 𝑈))
91	4, 5, 90	mulgdir 19137	. . . . . . . . . . . . . . 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 12724	. . . . . . . . . . . . . . . 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 6915	. . . . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) ∧ 𝑥 = (𝑓 + 𝑔)) → (𝑥(.g‘(𝑅 ↾s 𝑈))𝑀) = ((𝑓 + 𝑔)(.g‘(𝑅 ↾s 𝑈))𝑀))
98		ovexd 7466	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → ((𝑓 + 𝑔)(.g‘(𝑅 ↾s 𝑈))𝑀) ∈ V)
99	75, 97, 93, 98	fvmptd 7023	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → (𝐹‘(𝑓 + 𝑔)) = ((𝑓 + 𝑔)(.g‘(𝑅 ↾s 𝑈))𝑀))
100	93, 95, 99	rspcedvd 3624	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ℤ ∧ 𝑓 ∈ ℤ) → ∃ℎ ∈ ℤ (𝐹‘ℎ) = ((𝑓 + 𝑔)(.g‘(𝑅 ↾s 𝑈))𝑀))
101		fvelrnb 6969	. . . . . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . 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 2839	. . . . . . . . . . . 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 2839	. . . . . . . . . 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 1191	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) ∧ 𝑔 ∈ ℤ) → ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦)
110		nfv 1912	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑓(𝐹‘𝑑) = 𝑦
111		nfv 1912	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑑(𝐹‘𝑓) = 𝑦
112		fveqeq2 6916	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑓 → ((𝐹‘𝑑) = 𝑦 ↔ (𝐹‘𝑓) = 𝑦))
113	110, 111, 112	cbvrexw 3305	. . . . . . . . . . . 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 3160	. . . . . . . . 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 2839	. . . . . . 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 1137	. . . . . . . 8 ⊢ ((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦 ∧ ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧) → ∃𝑒 ∈ ℤ (𝐹‘𝑒) = 𝑧)
120		nfv 1912	. . . . . . . . . 10 ⊢ Ⅎ𝑔(𝐹‘𝑒) = 𝑧
121		nfv 1912	. . . . . . . . . 10 ⊢ Ⅎ𝑒(𝐹‘𝑔) = 𝑧
122		fveqeq2 6916	. . . . . . . . . 10 ⊢ (𝑒 = 𝑔 → ((𝐹‘𝑒) = 𝑧 ↔ (𝐹‘𝑔) = 𝑧))
123	120, 121, 122	cbvrexw 3305	. . . . . . . . 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 3160	. . . . . 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 2741	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) ∧ 𝑓 ∈ ℤ) ∧ (𝐹‘𝑓) = 𝑦) → 𝑦 = (𝐹‘𝑓))
133	132	fveq2d 6911	. . . . . . . 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 775	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∃𝑑 ∈ ℤ (𝐹‘𝑑) = 𝑦) ∧ 𝑓 ∈ ℤ) ∧ (𝐹‘𝑓) = 𝑦) → 𝜑)
135		simplr 769	. . . . . . . . . 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 12722	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → -𝑓 ∈ ℤ)
139		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ ℤ) ∧ ℎ = -𝑓) → ℎ = -𝑓)
140	139	fveqeq2d 6915	. . . . . . . . . . . 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 7446	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ℤ) ∧ 𝑥 = -𝑓) → (𝑥(.g‘(𝑅 ↾s 𝑈))𝑀) = (-𝑓(.g‘(𝑅 ↾s 𝑈))𝑀))
144		ovexd 7466	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → (-𝑓(.g‘(𝑅 ↾s 𝑈))𝑀) ∈ V)
145	141, 143, 138, 144	fvmptd 7023	. . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . . 16 ⊢ (invg‘(𝑅 ↾s 𝑈)) = (invg‘(𝑅 ↾s 𝑈))
149	4, 5, 148	mulgneg 19123	. . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑓 ∈ ℤ) ∧ 𝑥 = 𝑓) → (𝑥(.g‘(𝑅 ↾s 𝑈))𝑀) = (𝑓(.g‘(𝑅 ↾s 𝑈))𝑀))
153		ovexd 7466	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → (𝑓(.g‘(𝑅 ↾s 𝑈))𝑀) ∈ V)
154	141, 152, 137, 153	fvmptd 7023	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → (𝐹‘𝑓) = (𝑓(.g‘(𝑅 ↾s 𝑈))𝑀))
155	154	eqcomd 2741	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → (𝑓(.g‘(𝑅 ↾s 𝑈))𝑀) = (𝐹‘𝑓))
156	155	fveq2d 6911	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → ((invg‘(𝑅 ↾s 𝑈))‘(𝑓(.g‘(𝑅 ↾s 𝑈))𝑀)) = ((invg‘(𝑅 ↾s 𝑈))‘(𝐹‘𝑓)))
157	150, 156	eqtrd 2775	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → (-𝑓(.g‘(𝑅 ↾s 𝑈))𝑀) = ((invg‘(𝑅 ↾s 𝑈))‘(𝐹‘𝑓)))
158	145, 157	eqtrd 2775	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → (𝐹‘-𝑓) = ((invg‘(𝑅 ↾s 𝑈))‘(𝐹‘𝑓)))
159	138, 140, 158	rspcedvd 3624	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ ℤ) → ∃ℎ ∈ ℤ (𝐹‘ℎ) = ((invg‘(𝑅 ↾s 𝑈))‘(𝐹‘𝑓)))
160		fvelrnb 6969	. . . . . . . . . . . . 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 2839	. . . . . . 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 3160	. . . . . 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 19174	1 ⊢ (𝜑 → ((𝑅 ↾s 𝑈) ↾s ran 𝐹) ∈ Grp)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  {crab 3433  Vcvv 3478   ⊆ wss 3963   class class class wbr 5148   ↦ cmpt 5231  ran crn 5690   Fn wfn 6558  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  0cc0 11153   + caddc 11156  -cneg 11491  ℕcn 12264  ℕ₀cn0 12524  ℤcz 12611   ∥ cdvds 16287  Basecbs 17245   ↾_s cress 17274  +_gcplusg 17298  0_gc0g 17486  Grpcgrp 18964  inv_gcminusg 18965  ._gcmg 19098  CMndccmn 19813  Abelcabl 19814   PrimRoots cprimroots 42073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-seq 14040  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-grp 18967  df-minusg 18968  df-mulg 19099  df-subg 19154  df-cmn 19815  df-abl 19816  df-primroots 42074

This theorem is referenced by:  aks6d1c6isolem2  42157