Theorem vdwmc2 16309

Description: Expand out the definition of an arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)

Hypotheses

Ref	Expression
vdwmc.1	⊢ 𝑋 ∈ V
vdwmc.2	⊢ (𝜑 → 𝐾 ∈ ℕ0)
vdwmc.3	⊢ (𝜑 → 𝐹:𝑋⟶𝑅)
vdwmc2.4	⊢ (𝜑 → 𝐴 ∈ 𝑋)

Assertion

Ref	Expression
vdwmc2	⊢ (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐})))

Distinct variable groups:   𝑎,𝑐,𝑑,𝑚,𝐹   𝐾,𝑎,𝑐,𝑑,𝑚   𝜑,𝑐   𝑅,𝑎,𝑐,𝑑   𝜑,𝑎,𝑑

Allowed substitution hints:   𝜑(𝑚)   𝐴(𝑚,𝑎,𝑐,𝑑)   𝑅(𝑚)   𝑋(𝑚,𝑎,𝑐,𝑑)

Proof of Theorem vdwmc2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vdwmc.1	. . 3 ⊢ 𝑋 ∈ V
2		vdwmc.2	. . 3 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
3		vdwmc.3	. . 3 ⊢ (𝜑 → 𝐹:𝑋⟶𝑅)
4	1, 2, 3	vdwmc 16308	. 2 ⊢ (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐})))
5		vdwapid1 16305	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℕ ∧ 𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → 𝑎 ∈ (𝑎(AP‘𝐾)𝑑))
6	5	ne0d 4300	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℕ ∧ 𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → (𝑎(AP‘𝐾)𝑑) ≠ ∅)
7	6	3expb 1116	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℕ ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → (𝑎(AP‘𝐾)𝑑) ≠ ∅)
8	7	adantll 712	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → (𝑎(AP‘𝐾)𝑑) ≠ ∅)
9		ssn0 4353	. . . . . . . . . 10 ⊢ (((𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}) ∧ (𝑎(AP‘𝐾)𝑑) ≠ ∅) → (◡𝐹 “ {𝑐}) ≠ ∅)
10	9	expcom 416	. . . . . . . . 9 ⊢ ((𝑎(AP‘𝐾)𝑑) ≠ ∅ → ((𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}) → (◡𝐹 “ {𝑐}) ≠ ∅))
11	8, 10	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → ((𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}) → (◡𝐹 “ {𝑐}) ≠ ∅))
12		disjsn 4640	. . . . . . . . . 10 ⊢ ((𝑅 ∩ {𝑐}) = ∅ ↔ ¬ 𝑐 ∈ 𝑅)
13	3	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ) → 𝐹:𝑋⟶𝑅)
14		fimacnvdisj 6551	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝑋⟶𝑅 ∧ (𝑅 ∩ {𝑐}) = ∅) → (◡𝐹 “ {𝑐}) = ∅)
15	14	ex 415	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋⟶𝑅 → ((𝑅 ∩ {𝑐}) = ∅ → (◡𝐹 “ {𝑐}) = ∅))
16	13, 15	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ) → ((𝑅 ∩ {𝑐}) = ∅ → (◡𝐹 “ {𝑐}) = ∅))
17	16	adantr 483	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → ((𝑅 ∩ {𝑐}) = ∅ → (◡𝐹 “ {𝑐}) = ∅))
18	12, 17	syl5bir 245	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → (¬ 𝑐 ∈ 𝑅 → (◡𝐹 “ {𝑐}) = ∅))
19	18	necon1ad 3033	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → ((◡𝐹 “ {𝑐}) ≠ ∅ → 𝑐 ∈ 𝑅))
20	11, 19	syld 47	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ ℕ) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → ((𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}) → 𝑐 ∈ 𝑅))
21	20	rexlimdvva 3294	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ) → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}) → 𝑐 ∈ 𝑅))
22	21	pm4.71rd 565	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ) → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}) ↔ (𝑐 ∈ 𝑅 ∧ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}))))
23	22	exbidv 1918	. . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ) → (∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}) ↔ ∃𝑐(𝑐 ∈ 𝑅 ∧ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}))))
24		df-rex 3144	. . . 4 ⊢ (∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}) ↔ ∃𝑐(𝑐 ∈ 𝑅 ∧ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐})))
25	23, 24	syl6bbr 291	. . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ) → (∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}) ↔ ∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐})))
26		vdwmc2.4	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
27	3, 26	ffvelrnd 6846	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝑅)
28	27	ne0d 4300	. . . . . 6 ⊢ (𝜑 → 𝑅 ≠ ∅)
29		1nn 11643	. . . . . . . . 9 ⊢ 1 ∈ ℕ
30	29	ne0ii 4302	. . . . . . . 8 ⊢ ℕ ≠ ∅
31		simpllr 774	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐾 = 0) ∧ 𝑎 ∈ ℕ) ∧ 𝑑 ∈ ℕ) → 𝐾 = 0)
32	31	fveq2d 6668	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐾 = 0) ∧ 𝑎 ∈ ℕ) ∧ 𝑑 ∈ ℕ) → (AP‘𝐾) = (AP‘0))
33	32	oveqd 7167	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐾 = 0) ∧ 𝑎 ∈ ℕ) ∧ 𝑑 ∈ ℕ) → (𝑎(AP‘𝐾)𝑑) = (𝑎(AP‘0)𝑑))
34		vdwap0 16306	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → (𝑎(AP‘0)𝑑) = ∅)
35	34	adantll 712	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐾 = 0) ∧ 𝑎 ∈ ℕ) ∧ 𝑑 ∈ ℕ) → (𝑎(AP‘0)𝑑) = ∅)
36	33, 35	eqtrd 2856	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐾 = 0) ∧ 𝑎 ∈ ℕ) ∧ 𝑑 ∈ ℕ) → (𝑎(AP‘𝐾)𝑑) = ∅)
37		0ss 4349	. . . . . . . . . . . 12 ⊢ ∅ ⊆ (◡𝐹 “ {𝑐})
38	36, 37	eqsstrdi 4020	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐾 = 0) ∧ 𝑎 ∈ ℕ) ∧ 𝑑 ∈ ℕ) → (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}))
39	38	ralrimiva 3182	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐾 = 0) ∧ 𝑎 ∈ ℕ) → ∀𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}))
40		r19.2z 4439	. . . . . . . . . 10 ⊢ ((ℕ ≠ ∅ ∧ ∀𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐})) → ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}))
41	30, 39, 40	sylancr 589	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐾 = 0) ∧ 𝑎 ∈ ℕ) → ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}))
42	41	ralrimiva 3182	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 = 0) → ∀𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}))
43		r19.2z 4439	. . . . . . . 8 ⊢ ((ℕ ≠ ∅ ∧ ∀𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐})) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}))
44	30, 42, 43	sylancr 589	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 = 0) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}))
45	44	ralrimivw 3183	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 = 0) → ∀𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}))
46		r19.2z 4439	. . . . . 6 ⊢ ((𝑅 ≠ ∅ ∧ ∀𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐})) → ∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}))
47	28, 45, 46	syl2an2r 683	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 = 0) → ∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}))
48		rexex 3240	. . . . 5 ⊢ (∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}) → ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}))
49	47, 48	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝐾 = 0) → ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}))
50	49, 47	2thd 267	. . 3 ⊢ ((𝜑 ∧ 𝐾 = 0) → (∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}) ↔ ∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐})))
51		elnn0 11893	. . . 4 ⊢ (𝐾 ∈ ℕ0 ↔ (𝐾 ∈ ℕ ∨ 𝐾 = 0))
52	2, 51	sylib 220	. . 3 ⊢ (𝜑 → (𝐾 ∈ ℕ ∨ 𝐾 = 0))
53	25, 50, 52	mpjaodan 955	. 2 ⊢ (𝜑 → (∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}) ↔ ∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐})))
54		vdwapval 16303	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ (𝑎(AP‘𝐾)𝑑) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (𝑎 + (𝑚 · 𝑑))))
55	54	3expb 1116	. . . . . . . 8 ⊢ ((𝐾 ∈ ℕ0 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → (𝑥 ∈ (𝑎(AP‘𝐾)𝑑) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (𝑎 + (𝑚 · 𝑑))))
56	2, 55	sylan 582	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → (𝑥 ∈ (𝑎(AP‘𝐾)𝑑) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (𝑎 + (𝑚 · 𝑑))))
57	56	imbi1d 344	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → ((𝑥 ∈ (𝑎(AP‘𝐾)𝑑) → 𝑥 ∈ (◡𝐹 “ {𝑐})) ↔ (∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (𝑎 + (𝑚 · 𝑑)) → 𝑥 ∈ (◡𝐹 “ {𝑐}))))
58	57	albidv 1917	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → (∀𝑥(𝑥 ∈ (𝑎(AP‘𝐾)𝑑) → 𝑥 ∈ (◡𝐹 “ {𝑐})) ↔ ∀𝑥(∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (𝑎 + (𝑚 · 𝑑)) → 𝑥 ∈ (◡𝐹 “ {𝑐}))))
59		dfss2 3954	. . . . 5 ⊢ ((𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}) ↔ ∀𝑥(𝑥 ∈ (𝑎(AP‘𝐾)𝑑) → 𝑥 ∈ (◡𝐹 “ {𝑐})))
60		ralcom4 3235	. . . . . 6 ⊢ (∀𝑚 ∈ (0...(𝐾 − 1))∀𝑥(𝑥 = (𝑎 + (𝑚 · 𝑑)) → 𝑥 ∈ (◡𝐹 “ {𝑐})) ↔ ∀𝑥∀𝑚 ∈ (0...(𝐾 − 1))(𝑥 = (𝑎 + (𝑚 · 𝑑)) → 𝑥 ∈ (◡𝐹 “ {𝑐})))
61		ovex 7183	. . . . . . . 8 ⊢ (𝑎 + (𝑚 · 𝑑)) ∈ V
62		eleq1 2900	. . . . . . . 8 ⊢ (𝑥 = (𝑎 + (𝑚 · 𝑑)) → (𝑥 ∈ (◡𝐹 “ {𝑐}) ↔ (𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐})))
63	61, 62	ceqsalv 3532	. . . . . . 7 ⊢ (∀𝑥(𝑥 = (𝑎 + (𝑚 · 𝑑)) → 𝑥 ∈ (◡𝐹 “ {𝑐})) ↔ (𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐}))
64	63	ralbii 3165	. . . . . 6 ⊢ (∀𝑚 ∈ (0...(𝐾 − 1))∀𝑥(𝑥 = (𝑎 + (𝑚 · 𝑑)) → 𝑥 ∈ (◡𝐹 “ {𝑐})) ↔ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐}))
65		r19.23v 3279	. . . . . . 7 ⊢ (∀𝑚 ∈ (0...(𝐾 − 1))(𝑥 = (𝑎 + (𝑚 · 𝑑)) → 𝑥 ∈ (◡𝐹 “ {𝑐})) ↔ (∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (𝑎 + (𝑚 · 𝑑)) → 𝑥 ∈ (◡𝐹 “ {𝑐})))
66	65	albii 1816	. . . . . 6 ⊢ (∀𝑥∀𝑚 ∈ (0...(𝐾 − 1))(𝑥 = (𝑎 + (𝑚 · 𝑑)) → 𝑥 ∈ (◡𝐹 “ {𝑐})) ↔ ∀𝑥(∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (𝑎 + (𝑚 · 𝑑)) → 𝑥 ∈ (◡𝐹 “ {𝑐})))
67	60, 64, 66	3bitr3i 303	. . . . 5 ⊢ (∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐}) ↔ ∀𝑥(∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = (𝑎 + (𝑚 · 𝑑)) → 𝑥 ∈ (◡𝐹 “ {𝑐})))
68	58, 59, 67	3bitr4g 316	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → ((𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}) ↔ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐})))
69	68	2rexbidva 3299	. . 3 ⊢ (𝜑 → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}) ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐})))
70	69	rexbidv 3297	. 2 ⊢ (𝜑 → (∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝐹 “ {𝑐}) ↔ ∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐})))
71	4, 53, 70	3bitrd 307	1 ⊢ (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083  ∀wal 1531   = wceq 1533  ∃wex 1776   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  Vcvv 3494   ∩ cin 3934   ⊆ wss 3935  ∅c0 4290  {csn 4560   class class class wbr 5058  ^◡ccnv 5548   “ cima 5552  ⟶wf 6345  ‘cfv 6349  (class class class)co 7150  0cc0 10531  1c1 10532   + caddc 10534   · cmul 10536   − cmin 10864  ℕcn 11632  ℕ₀cn0 11891  ...cfz 12886  APcvdwa 16295   MonoAP cvdwm 16296

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-n0 11892  df-z 11976  df-uz 12238  df-fz 12887  df-vdwap 16298  df-vdwmc 16299

This theorem is referenced by:  vdw  16324