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Theorem vdwlem8 16921

Description: Lemma for vdw 16927. (Contributed by Mario Carneiro, 18-Aug-2014.)

**Hypotheses**
Ref	Expression
vdwlem8.r	⊢ (𝜑 → 𝑅 ∈ Fin)
vdwlem8.k	⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘2))
vdwlem8.w	⊢ (𝜑 → 𝑊 ∈ ℕ)
vdwlem8.f	⊢ (𝜑 → 𝐹:(1...(2 · 𝑊))⟶𝑅)
vdwlem8.c	⊢ 𝐶 ∈ V
vdwlem8.a	⊢ (𝜑 → 𝐴 ∈ ℕ)
vdwlem8.d	⊢ (𝜑 → 𝐷 ∈ ℕ)
vdwlem8.s	⊢ (𝜑 → (𝐴(AP‘𝐾)𝐷) ⊆ (^◡𝐺 “ {𝐶}))
vdwlem8.g	⊢ 𝐺 = (𝑥 ∈ (1...𝑊) ↦ (𝐹‘(𝑥 + 𝑊)))

**Assertion**
Ref	Expression
vdwlem8	⊢ (𝜑 → ⟨1, 𝐾⟩ PolyAP 𝐹)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐷 𝑥,𝐹 𝜑,𝑥 𝑥,𝐶 𝑥,𝐾 𝑥,𝑊 Allowed substitution hints: 𝑅(𝑥) 𝐺(𝑥)

Proof of Theorem vdwlem8 Dummy variables 𝑎 𝑑 𝑖 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vdwlem8.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℕ)
2	1	nncnd 12228	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
3		vdwlem8.d	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ ℕ)
4	3	nncnd 12228	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ ℂ)
5	2, 4	addcomd 11416	. . . . . . . 8 ⊢ (𝜑 → (𝐴 + 𝐷) = (𝐷 + 𝐴))
6	5	oveq2d 7425	. . . . . . 7 ⊢ (𝜑 → (𝑊 − (𝐴 + 𝐷)) = (𝑊 − (𝐷 + 𝐴)))
7		vdwlem8.w	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ ℕ)
8	7	nncnd 12228	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ ℂ)
9	8, 4, 2	subsub4d 11602	. . . . . . 7 ⊢ (𝜑 → ((𝑊 − 𝐷) − 𝐴) = (𝑊 − (𝐷 + 𝐴)))
10	6, 9	eqtr4d 2776	. . . . . 6 ⊢ (𝜑 → (𝑊 − (𝐴 + 𝐷)) = ((𝑊 − 𝐷) − 𝐴))
11	10	oveq2d 7425	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝐴) + (𝑊 − (𝐴 + 𝐷))) = ((𝐴 + 𝐴) + ((𝑊 − 𝐷) − 𝐴)))
12	8, 4	subcld 11571	. . . . . 6 ⊢ (𝜑 → (𝑊 − 𝐷) ∈ ℂ)
13	2, 2, 12	ppncand 11611	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝐴) + ((𝑊 − 𝐷) − 𝐴)) = (𝐴 + (𝑊 − 𝐷)))
14	11, 13	eqtrd 2773	. . . 4 ⊢ (𝜑 → ((𝐴 + 𝐴) + (𝑊 − (𝐴 + 𝐷))) = (𝐴 + (𝑊 − 𝐷)))
15	1, 1	nnaddcld 12264	. . . . 5 ⊢ (𝜑 → (𝐴 + 𝐴) ∈ ℕ)
16		vdwlem8.s	. . . . . . . 8 ⊢ (𝜑 → (𝐴(AP‘𝐾)𝐷) ⊆ (^◡𝐺 “ {𝐶}))
17		cnvimass 6081	. . . . . . . . 9 ⊢ (^◡𝐺 “ {𝐶}) ⊆ dom 𝐺
18		fvex 6905	. . . . . . . . . 10 ⊢ (𝐹‘(𝑥 + 𝑊)) ∈ V
19		vdwlem8.g	. . . . . . . . . 10 ⊢ 𝐺 = (𝑥 ∈ (1...𝑊) ↦ (𝐹‘(𝑥 + 𝑊)))
20	18, 19	dmmpti 6695	. . . . . . . . 9 ⊢ dom 𝐺 = (1...𝑊)
21	17, 20	sseqtri 4019	. . . . . . . 8 ⊢ (^◡𝐺 “ {𝐶}) ⊆ (1...𝑊)
22	16, 21	sstrdi 3995	. . . . . . 7 ⊢ (𝜑 → (𝐴(AP‘𝐾)𝐷) ⊆ (1...𝑊))
23		ssun2 4174	. . . . . . . . 9 ⊢ ((𝐴 + 𝐷)(AP‘(𝐾 − 1))𝐷) ⊆ ({𝐴} ∪ ((𝐴 + 𝐷)(AP‘(𝐾 − 1))𝐷))
24		vdwlem8.k	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘2))
25		uz2m1nn 12907	. . . . . . . . . . 11 ⊢ (𝐾 ∈ (ℤ_≥‘2) → (𝐾 − 1) ∈ ℕ)
26	24, 25	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾 − 1) ∈ ℕ)
27	1, 3	nnaddcld 12264	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 + 𝐷) ∈ ℕ)
28		vdwapid1 16908	. . . . . . . . . 10 ⊢ (((𝐾 − 1) ∈ ℕ ∧ (𝐴 + 𝐷) ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴 + 𝐷) ∈ ((𝐴 + 𝐷)(AP‘(𝐾 − 1))𝐷))
29	26, 27, 3, 28	syl3anc 1372	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 + 𝐷) ∈ ((𝐴 + 𝐷)(AP‘(𝐾 − 1))𝐷))
30	23, 29	sselid 3981	. . . . . . . 8 ⊢ (𝜑 → (𝐴 + 𝐷) ∈ ({𝐴} ∪ ((𝐴 + 𝐷)(AP‘(𝐾 − 1))𝐷)))
31		eluz2nn 12868	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (ℤ_≥‘2) → 𝐾 ∈ ℕ)
32	24, 31	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ ℕ)
33	32	nncnd 12228	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ ℂ)
34		ax-1cn 11168	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
35		npcan 11469	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐾 − 1) + 1) = 𝐾)
36	33, 34, 35	sylancl 587	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐾 − 1) + 1) = 𝐾)
37	36	fveq2d 6896	. . . . . . . . . 10 ⊢ (𝜑 → (AP‘((𝐾 − 1) + 1)) = (AP‘𝐾))
38	37	oveqd 7426	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(AP‘((𝐾 − 1) + 1))𝐷) = (𝐴(AP‘𝐾)𝐷))
39	26	nnnn0d 12532	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾 − 1) ∈ ℕ₀)
40		vdwapun 16907	. . . . . . . . . 10 ⊢ (((𝐾 − 1) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(AP‘((𝐾 − 1) + 1))𝐷) = ({𝐴} ∪ ((𝐴 + 𝐷)(AP‘(𝐾 − 1))𝐷)))
41	39, 1, 3, 40	syl3anc 1372	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(AP‘((𝐾 − 1) + 1))𝐷) = ({𝐴} ∪ ((𝐴 + 𝐷)(AP‘(𝐾 − 1))𝐷)))
42	38, 41	eqtr3d 2775	. . . . . . . 8 ⊢ (𝜑 → (𝐴(AP‘𝐾)𝐷) = ({𝐴} ∪ ((𝐴 + 𝐷)(AP‘(𝐾 − 1))𝐷)))
43	30, 42	eleqtrrd 2837	. . . . . . 7 ⊢ (𝜑 → (𝐴 + 𝐷) ∈ (𝐴(AP‘𝐾)𝐷))
44	22, 43	sseldd 3984	. . . . . 6 ⊢ (𝜑 → (𝐴 + 𝐷) ∈ (1...𝑊))
45		elfzuz3 13498	. . . . . 6 ⊢ ((𝐴 + 𝐷) ∈ (1...𝑊) → 𝑊 ∈ (ℤ_≥‘(𝐴 + 𝐷)))
46		uznn0sub 12861	. . . . . 6 ⊢ (𝑊 ∈ (ℤ_≥‘(𝐴 + 𝐷)) → (𝑊 − (𝐴 + 𝐷)) ∈ ℕ₀)
47	44, 45, 46	3syl 18	. . . . 5 ⊢ (𝜑 → (𝑊 − (𝐴 + 𝐷)) ∈ ℕ₀)
48		nnnn0addcl 12502	. . . . 5 ⊢ (((𝐴 + 𝐴) ∈ ℕ ∧ (𝑊 − (𝐴 + 𝐷)) ∈ ℕ₀) → ((𝐴 + 𝐴) + (𝑊 − (𝐴 + 𝐷))) ∈ ℕ)
49	15, 47, 48	syl2anc 585	. . . 4 ⊢ (𝜑 → ((𝐴 + 𝐴) + (𝑊 − (𝐴 + 𝐷))) ∈ ℕ)
50	14, 49	eqeltrrd 2835	. . 3 ⊢ (𝜑 → (𝐴 + (𝑊 − 𝐷)) ∈ ℕ)
51		1nn 12223	. . . . . . . 8 ⊢ 1 ∈ ℕ
52		f1osng 6875	. . . . . . . 8 ⊢ ((1 ∈ ℕ ∧ 𝐷 ∈ ℕ) → {⟨1, 𝐷⟩}:{1}–1-1-onto→{𝐷})
53	51, 3, 52	sylancr 588	. . . . . . 7 ⊢ (𝜑 → {⟨1, 𝐷⟩}:{1}–1-1-onto→{𝐷})
54		f1of 6834	. . . . . . 7 ⊢ ({⟨1, 𝐷⟩}:{1}–1-1-onto→{𝐷} → {⟨1, 𝐷⟩}:{1}⟶{𝐷})
55	53, 54	syl 17	. . . . . 6 ⊢ (𝜑 → {⟨1, 𝐷⟩}:{1}⟶{𝐷})
56	3	snssd 4813	. . . . . 6 ⊢ (𝜑 → {𝐷} ⊆ ℕ)
57	55, 56	fssd 6736	. . . . 5 ⊢ (𝜑 → {⟨1, 𝐷⟩}:{1}⟶ℕ)
58		1z 12592	. . . . . . 7 ⊢ 1 ∈ ℤ
59		fzsn 13543	. . . . . . 7 ⊢ (1 ∈ ℤ → (1...1) = {1})
60	58, 59	ax-mp 5	. . . . . 6 ⊢ (1...1) = {1}
61	60	feq2i 6710	. . . . 5 ⊢ ({⟨1, 𝐷⟩}:(1...1)⟶ℕ ↔ {⟨1, 𝐷⟩}:{1}⟶ℕ)
62	57, 61	sylibr 233	. . . 4 ⊢ (𝜑 → {⟨1, 𝐷⟩}:(1...1)⟶ℕ)
63		nnex 12218	. . . . 5 ⊢ ℕ ∈ V
64		ovex 7442	. . . . 5 ⊢ (1...1) ∈ V
65	63, 64	elmap 8865	. . . 4 ⊢ ({⟨1, 𝐷⟩} ∈ (ℕ ↑_m (1...1)) ↔ {⟨1, 𝐷⟩}:(1...1)⟶ℕ)
66	62, 65	sylibr 233	. . 3 ⊢ (𝜑 → {⟨1, 𝐷⟩} ∈ (ℕ ↑_m (1...1)))
67	1, 7	nnaddcld 12264	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴 + 𝑊) ∈ ℕ)
68	67	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 + 𝑊) ∈ ℕ)
69		elfznn0 13594	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (0...(𝐾 − 1)) → 𝑚 ∈ ℕ₀)
70	3	nnnn0d 12532	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐷 ∈ ℕ₀)
71		nn0mulcl 12508	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝐷 ∈ ℕ₀) → (𝑚 · 𝐷) ∈ ℕ₀)
72	69, 70, 71	syl2anr 598	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · 𝐷) ∈ ℕ₀)
73		nnnn0addcl 12502	. . . . . . . . . . . . 13 ⊢ (((𝐴 + 𝑊) ∈ ℕ ∧ (𝑚 · 𝐷) ∈ ℕ₀) → ((𝐴 + 𝑊) + (𝑚 · 𝐷)) ∈ ℕ)
74	68, 72, 73	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐴 + 𝑊) + (𝑚 · 𝐷)) ∈ ℕ)
75		nnuz 12865	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
76	74, 75	eleqtrdi 2844	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐴 + 𝑊) + (𝑚 · 𝐷)) ∈ (ℤ_≥‘1))
77	16	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐴(AP‘𝐾)𝐷) ⊆ (^◡𝐺 “ {𝐶}))
78		eqid 2733	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑚 · 𝐷))
79		oveq1 7416	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑚 → (𝑛 · 𝐷) = (𝑚 · 𝐷))
80	79	oveq2d 7425	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑚 → (𝐴 + (𝑛 · 𝐷)) = (𝐴 + (𝑚 · 𝐷)))
81	80	rspceeqv 3634	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ (0...(𝐾 − 1)) ∧ (𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑚 · 𝐷))) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷)))
82	78, 81	mpan2 690	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (0...(𝐾 − 1)) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷)))
83	32	nnnn0d 12532	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
84		vdwapval 16906	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → ((𝐴 + (𝑚 · 𝐷)) ∈ (𝐴(AP‘𝐾)𝐷) ↔ ∃𝑛 ∈ (0...(𝐾 − 1))(𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷))))
85	83, 1, 3, 84	syl3anc 1372	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝐴 + (𝑚 · 𝐷)) ∈ (𝐴(AP‘𝐾)𝐷) ↔ ∃𝑛 ∈ (0...(𝐾 − 1))(𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷))))
86	85	biimpar 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ∃𝑛 ∈ (0...(𝐾 − 1))(𝐴 + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷))) → (𝐴 + (𝑚 · 𝐷)) ∈ (𝐴(AP‘𝐾)𝐷))
87	82, 86	sylan2 594	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 + (𝑚 · 𝐷)) ∈ (𝐴(AP‘𝐾)𝐷))
88	77, 87	sseldd 3984	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 + (𝑚 · 𝐷)) ∈ (^◡𝐺 “ {𝐶}))
89	18, 19	fnmpti 6694	. . . . . . . . . . . . . . . 16 ⊢ 𝐺 Fn (1...𝑊)
90		fniniseg 7062	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 Fn (1...𝑊) → ((𝐴 + (𝑚 · 𝐷)) ∈ (^◡𝐺 “ {𝐶}) ↔ ((𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑊) ∧ (𝐺‘(𝐴 + (𝑚 · 𝐷))) = 𝐶)))
91	89, 90	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 + (𝑚 · 𝐷)) ∈ (^◡𝐺 “ {𝐶}) ↔ ((𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑊) ∧ (𝐺‘(𝐴 + (𝑚 · 𝐷))) = 𝐶))
92	88, 91	sylib 217	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑊) ∧ (𝐺‘(𝐴 + (𝑚 · 𝐷))) = 𝐶))
93	92	simpld 496	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑊))
94		elfzuz3 13498	. . . . . . . . . . . . 13 ⊢ ((𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑊) → 𝑊 ∈ (ℤ_≥‘(𝐴 + (𝑚 · 𝐷))))
95		eluzelz 12832	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ (ℤ_≥‘(𝐴 + (𝑚 · 𝐷))) → 𝑊 ∈ ℤ)
96		eluzadd 12851	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ (ℤ_≥‘(𝐴 + (𝑚 · 𝐷))) ∧ 𝑊 ∈ ℤ) → (𝑊 + 𝑊) ∈ (ℤ_≥‘((𝐴 + (𝑚 · 𝐷)) + 𝑊)))
97	95, 96	mpdan 686	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ (ℤ_≥‘(𝐴 + (𝑚 · 𝐷))) → (𝑊 + 𝑊) ∈ (ℤ_≥‘((𝐴 + (𝑚 · 𝐷)) + 𝑊)))
98	93, 94, 97	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑊 + 𝑊) ∈ (ℤ_≥‘((𝐴 + (𝑚 · 𝐷)) + 𝑊)))
99	8	2timesd 12455	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 · 𝑊) = (𝑊 + 𝑊))
100	99	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (2 · 𝑊) = (𝑊 + 𝑊))
101	2	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐴 ∈ ℂ)
102	8	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑊 ∈ ℂ)
103	72	nn0cnd 12534	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · 𝐷) ∈ ℂ)
104	101, 102, 103	add32d 11441	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐴 + 𝑊) + (𝑚 · 𝐷)) = ((𝐴 + (𝑚 · 𝐷)) + 𝑊))
105	104	fveq2d 6896	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (ℤ_≥‘((𝐴 + 𝑊) + (𝑚 · 𝐷))) = (ℤ_≥‘((𝐴 + (𝑚 · 𝐷)) + 𝑊)))
106	98, 100, 105	3eltr4d 2849	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (2 · 𝑊) ∈ (ℤ_≥‘((𝐴 + 𝑊) + (𝑚 · 𝐷))))
107		elfzuzb 13495	. . . . . . . . . . 11 ⊢ (((𝐴 + 𝑊) + (𝑚 · 𝐷)) ∈ (1...(2 · 𝑊)) ↔ (((𝐴 + 𝑊) + (𝑚 · 𝐷)) ∈ (ℤ_≥‘1) ∧ (2 · 𝑊) ∈ (ℤ_≥‘((𝐴 + 𝑊) + (𝑚 · 𝐷)))))
108	76, 106, 107	sylanbrc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐴 + 𝑊) + (𝑚 · 𝐷)) ∈ (1...(2 · 𝑊)))
109	104	fveq2d 6896	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐹‘((𝐴 + 𝑊) + (𝑚 · 𝐷))) = (𝐹‘((𝐴 + (𝑚 · 𝐷)) + 𝑊)))
110		fvoveq1 7432	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝐴 + (𝑚 · 𝐷)) → (𝐹‘(𝑥 + 𝑊)) = (𝐹‘((𝐴 + (𝑚 · 𝐷)) + 𝑊)))
111		fvex 6905	. . . . . . . . . . . . 13 ⊢ (𝐹‘((𝐴 + (𝑚 · 𝐷)) + 𝑊)) ∈ V
112	110, 19, 111	fvmpt 6999	. . . . . . . . . . . 12 ⊢ ((𝐴 + (𝑚 · 𝐷)) ∈ (1...𝑊) → (𝐺‘(𝐴 + (𝑚 · 𝐷))) = (𝐹‘((𝐴 + (𝑚 · 𝐷)) + 𝑊)))
113	93, 112	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐺‘(𝐴 + (𝑚 · 𝐷))) = (𝐹‘((𝐴 + (𝑚 · 𝐷)) + 𝑊)))
114	92	simprd 497	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐺‘(𝐴 + (𝑚 · 𝐷))) = 𝐶)
115	109, 113, 114	3eqtr2d 2779	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐹‘((𝐴 + 𝑊) + (𝑚 · 𝐷))) = 𝐶)
116	108, 115	jca 513	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝐴 + 𝑊) + (𝑚 · 𝐷)) ∈ (1...(2 · 𝑊)) ∧ (𝐹‘((𝐴 + 𝑊) + (𝑚 · 𝐷))) = 𝐶))
117		eleq1 2822	. . . . . . . . . 10 ⊢ (𝑥 = ((𝐴 + 𝑊) + (𝑚 · 𝐷)) → (𝑥 ∈ (1...(2 · 𝑊)) ↔ ((𝐴 + 𝑊) + (𝑚 · 𝐷)) ∈ (1...(2 · 𝑊))))
118		fveqeq2 6901	. . . . . . . . . 10 ⊢ (𝑥 = ((𝐴 + 𝑊) + (𝑚 · 𝐷)) → ((𝐹‘𝑥) = 𝐶 ↔ (𝐹‘((𝐴 + 𝑊) + (𝑚 · 𝐷))) = 𝐶))
119	117, 118	anbi12d 632	. . . . . . . . 9 ⊢ (𝑥 = ((𝐴 + 𝑊) + (𝑚 · 𝐷)) → ((𝑥 ∈ (1...(2 · 𝑊)) ∧ (𝐹‘𝑥) = 𝐶) ↔ (((𝐴 + 𝑊) + (𝑚 · 𝐷)) ∈ (1...(2 · 𝑊)) ∧ (𝐹‘((𝐴 + 𝑊) + (𝑚 · 𝐷))) = 𝐶)))
120	116, 119	syl5ibrcom 246	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑥 = ((𝐴 + 𝑊) + (𝑚 · 𝐷)) → (𝑥 ∈ (1...(2 · 𝑊)) ∧ (𝐹‘𝑥) = 𝐶)))
121	120	rexlimdva 3156	. . . . . . 7 ⊢ (𝜑 → (∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝐴 + 𝑊) + (𝑚 · 𝐷)) → (𝑥 ∈ (1...(2 · 𝑊)) ∧ (𝐹‘𝑥) = 𝐶)))
122		vdwapval 16906	. . . . . . . 8 ⊢ ((𝐾 ∈ ℕ₀ ∧ (𝐴 + 𝑊) ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑥 ∈ ((𝐴 + 𝑊)(AP‘𝐾)𝐷) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝐴 + 𝑊) + (𝑚 · 𝐷))))
123	83, 67, 3, 122	syl3anc 1372	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ((𝐴 + 𝑊)(AP‘𝐾)𝐷) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝐴 + 𝑊) + (𝑚 · 𝐷))))
124		vdwlem8.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(1...(2 · 𝑊))⟶𝑅)
125		ffn 6718	. . . . . . . 8 ⊢ (𝐹:(1...(2 · 𝑊))⟶𝑅 → 𝐹 Fn (1...(2 · 𝑊)))
126		fniniseg 7062	. . . . . . . 8 ⊢ (𝐹 Fn (1...(2 · 𝑊)) → (𝑥 ∈ (^◡𝐹 “ {𝐶}) ↔ (𝑥 ∈ (1...(2 · 𝑊)) ∧ (𝐹‘𝑥) = 𝐶)))
127	124, 125, 126	3syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (^◡𝐹 “ {𝐶}) ↔ (𝑥 ∈ (1...(2 · 𝑊)) ∧ (𝐹‘𝑥) = 𝐶)))
128	121, 123, 127	3imtr4d 294	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ((𝐴 + 𝑊)(AP‘𝐾)𝐷) → 𝑥 ∈ (^◡𝐹 “ {𝐶})))
129	128	ssrdv 3989	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝑊)(AP‘𝐾)𝐷) ⊆ (^◡𝐹 “ {𝐶}))
130		fvsng 7178	. . . . . . . . 9 ⊢ ((1 ∈ ℕ ∧ 𝐷 ∈ ℕ) → ({⟨1, 𝐷⟩}‘1) = 𝐷)
131	51, 3, 130	sylancr 588	. . . . . . . 8 ⊢ (𝜑 → ({⟨1, 𝐷⟩}‘1) = 𝐷)
132	131	oveq2d 7425	. . . . . . 7 ⊢ (𝜑 → ((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1)) = ((𝐴 + (𝑊 − 𝐷)) + 𝐷))
133	2, 12, 4	addassd 11236	. . . . . . 7 ⊢ (𝜑 → ((𝐴 + (𝑊 − 𝐷)) + 𝐷) = (𝐴 + ((𝑊 − 𝐷) + 𝐷)))
134	8, 4	npcand 11575	. . . . . . . 8 ⊢ (𝜑 → ((𝑊 − 𝐷) + 𝐷) = 𝑊)
135	134	oveq2d 7425	. . . . . . 7 ⊢ (𝜑 → (𝐴 + ((𝑊 − 𝐷) + 𝐷)) = (𝐴 + 𝑊))
136	132, 133, 135	3eqtrd 2777	. . . . . 6 ⊢ (𝜑 → ((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1)) = (𝐴 + 𝑊))
137	136, 131	oveq12d 7427	. . . . 5 ⊢ (𝜑 → (((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1))(AP‘𝐾)({⟨1, 𝐷⟩}‘1)) = ((𝐴 + 𝑊)(AP‘𝐾)𝐷))
138	136	fveq2d 6896	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1))) = (𝐹‘(𝐴 + 𝑊)))
139		vdwapid1 16908	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → 𝐴 ∈ (𝐴(AP‘𝐾)𝐷))
140	32, 1, 3, 139	syl3anc 1372	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ (𝐴(AP‘𝐾)𝐷))
141	16, 140	sseldd 3984	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ (^◡𝐺 “ {𝐶}))
142		fniniseg 7062	. . . . . . . . . . . 12 ⊢ (𝐺 Fn (1...𝑊) → (𝐴 ∈ (^◡𝐺 “ {𝐶}) ↔ (𝐴 ∈ (1...𝑊) ∧ (𝐺‘𝐴) = 𝐶)))
143	89, 142	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (^◡𝐺 “ {𝐶}) ↔ (𝐴 ∈ (1...𝑊) ∧ (𝐺‘𝐴) = 𝐶))
144	141, 143	sylib 217	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ (1...𝑊) ∧ (𝐺‘𝐴) = 𝐶))
145	144	simpld 496	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (1...𝑊))
146		fvoveq1 7432	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝐹‘(𝑥 + 𝑊)) = (𝐹‘(𝐴 + 𝑊)))
147		fvex 6905	. . . . . . . . . 10 ⊢ (𝐹‘(𝐴 + 𝑊)) ∈ V
148	146, 19, 147	fvmpt 6999	. . . . . . . . 9 ⊢ (𝐴 ∈ (1...𝑊) → (𝐺‘𝐴) = (𝐹‘(𝐴 + 𝑊)))
149	145, 148	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝐴) = (𝐹‘(𝐴 + 𝑊)))
150	144	simprd 497	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝐴) = 𝐶)
151	138, 149, 150	3eqtr2d 2779	. . . . . . 7 ⊢ (𝜑 → (𝐹‘((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1))) = 𝐶)
152	151	sneqd 4641	. . . . . 6 ⊢ (𝜑 → {(𝐹‘((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1)))} = {𝐶})
153	152	imaeq2d 6060	. . . . 5 ⊢ (𝜑 → (^◡𝐹 “ {(𝐹‘((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1)))}) = (^◡𝐹 “ {𝐶}))
154	129, 137, 153	3sstr4d 4030	. . . 4 ⊢ (𝜑 → (((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1))(AP‘𝐾)({⟨1, 𝐷⟩}‘1)) ⊆ (^◡𝐹 “ {(𝐹‘((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1)))}))
155	154	ralrimivw 3151	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ (1...1)(((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1))(AP‘𝐾)({⟨1, 𝐷⟩}‘1)) ⊆ (^◡𝐹 “ {(𝐹‘((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1)))}))
156	151	mpteq2dv 5251	. . . . . . . 8 ⊢ (𝜑 → (𝑖 ∈ (1...1) ↦ (𝐹‘((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1)))) = (𝑖 ∈ (1...1) ↦ 𝐶))
157		fconstmpt 5739	. . . . . . . 8 ⊢ ((1...1) × {𝐶}) = (𝑖 ∈ (1...1) ↦ 𝐶)
158	156, 157	eqtr4di 2791	. . . . . . 7 ⊢ (𝜑 → (𝑖 ∈ (1...1) ↦ (𝐹‘((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1)))) = ((1...1) × {𝐶}))
159	158	rneqd 5938	. . . . . 6 ⊢ (𝜑 → ran (𝑖 ∈ (1...1) ↦ (𝐹‘((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1)))) = ran ((1...1) × {𝐶}))
160		elfz3 13511	. . . . . . . 8 ⊢ (1 ∈ ℤ → 1 ∈ (1...1))
161		ne0i 4335	. . . . . . . 8 ⊢ (1 ∈ (1...1) → (1...1) ≠ ∅)
162	58, 160, 161	mp2b 10	. . . . . . 7 ⊢ (1...1) ≠ ∅
163		rnxp 6170	. . . . . . 7 ⊢ ((1...1) ≠ ∅ → ran ((1...1) × {𝐶}) = {𝐶})
164	162, 163	ax-mp 5	. . . . . 6 ⊢ ran ((1...1) × {𝐶}) = {𝐶}
165	159, 164	eqtrdi 2789	. . . . 5 ⊢ (𝜑 → ran (𝑖 ∈ (1...1) ↦ (𝐹‘((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1)))) = {𝐶})
166	165	fveq2d 6896	. . . 4 ⊢ (𝜑 → (♯‘ran (𝑖 ∈ (1...1) ↦ (𝐹‘((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1))))) = (♯‘{𝐶}))
167		vdwlem8.c	. . . . 5 ⊢ 𝐶 ∈ V
168		hashsng 14329	. . . . 5 ⊢ (𝐶 ∈ V → (♯‘{𝐶}) = 1)
169	167, 168	ax-mp 5	. . . 4 ⊢ (♯‘{𝐶}) = 1
170	166, 169	eqtrdi 2789	. . 3 ⊢ (𝜑 → (♯‘ran (𝑖 ∈ (1...1) ↦ (𝐹‘((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1))))) = 1)
171		oveq1 7416	. . . . . . . 8 ⊢ (𝑎 = (𝐴 + (𝑊 − 𝐷)) → (𝑎 + (𝑑‘𝑖)) = ((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖)))
172	171	oveq1d 7424	. . . . . . 7 ⊢ (𝑎 = (𝐴 + (𝑊 − 𝐷)) → ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) = (((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)))
173		fvoveq1 7432	. . . . . . . . 9 ⊢ (𝑎 = (𝐴 + (𝑊 − 𝐷)) → (𝐹‘(𝑎 + (𝑑‘𝑖))) = (𝐹‘((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖))))
174	173	sneqd 4641	. . . . . . . 8 ⊢ (𝑎 = (𝐴 + (𝑊 − 𝐷)) → {(𝐹‘(𝑎 + (𝑑‘𝑖)))} = {(𝐹‘((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖)))})
175	174	imaeq2d 6060	. . . . . . 7 ⊢ (𝑎 = (𝐴 + (𝑊 − 𝐷)) → (^◡𝐹 “ {(𝐹‘(𝑎 + (𝑑‘𝑖)))}) = (^◡𝐹 “ {(𝐹‘((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖)))}))
176	172, 175	sseq12d 4016	. . . . . 6 ⊢ (𝑎 = (𝐴 + (𝑊 − 𝐷)) → (((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐹 “ {(𝐹‘(𝑎 + (𝑑‘𝑖)))}) ↔ (((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐹 “ {(𝐹‘((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖)))})))
177	176	ralbidv 3178	. . . . 5 ⊢ (𝑎 = (𝐴 + (𝑊 − 𝐷)) → (∀𝑖 ∈ (1...1)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐹 “ {(𝐹‘(𝑎 + (𝑑‘𝑖)))}) ↔ ∀𝑖 ∈ (1...1)(((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐹 “ {(𝐹‘((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖)))})))
178	173	mpteq2dv 5251	. . . . . . 7 ⊢ (𝑎 = (𝐴 + (𝑊 − 𝐷)) → (𝑖 ∈ (1...1) ↦ (𝐹‘(𝑎 + (𝑑‘𝑖)))) = (𝑖 ∈ (1...1) ↦ (𝐹‘((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖)))))
179	178	rneqd 5938	. . . . . 6 ⊢ (𝑎 = (𝐴 + (𝑊 − 𝐷)) → ran (𝑖 ∈ (1...1) ↦ (𝐹‘(𝑎 + (𝑑‘𝑖)))) = ran (𝑖 ∈ (1...1) ↦ (𝐹‘((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖)))))
180	179	fveqeq2d 6900	. . . . 5 ⊢ (𝑎 = (𝐴 + (𝑊 − 𝐷)) → ((♯‘ran (𝑖 ∈ (1...1) ↦ (𝐹‘(𝑎 + (𝑑‘𝑖))))) = 1 ↔ (♯‘ran (𝑖 ∈ (1...1) ↦ (𝐹‘((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖))))) = 1))
181	177, 180	anbi12d 632	. . . 4 ⊢ (𝑎 = (𝐴 + (𝑊 − 𝐷)) → ((∀𝑖 ∈ (1...1)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐹 “ {(𝐹‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...1) ↦ (𝐹‘(𝑎 + (𝑑‘𝑖))))) = 1) ↔ (∀𝑖 ∈ (1...1)(((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐹 “ {(𝐹‘((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...1) ↦ (𝐹‘((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖))))) = 1)))
182		fveq1 6891	. . . . . . . . . 10 ⊢ (𝑑 = {⟨1, 𝐷⟩} → (𝑑‘𝑖) = ({⟨1, 𝐷⟩}‘𝑖))
183		elfz1eq 13512	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...1) → 𝑖 = 1)
184	183	fveq2d 6896	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...1) → ({⟨1, 𝐷⟩}‘𝑖) = ({⟨1, 𝐷⟩}‘1))
185	182, 184	sylan9eq 2793	. . . . . . . . 9 ⊢ ((𝑑 = {⟨1, 𝐷⟩} ∧ 𝑖 ∈ (1...1)) → (𝑑‘𝑖) = ({⟨1, 𝐷⟩}‘1))
186	185	oveq2d 7425	. . . . . . . 8 ⊢ ((𝑑 = {⟨1, 𝐷⟩} ∧ 𝑖 ∈ (1...1)) → ((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖)) = ((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1)))
187	186, 185	oveq12d 7427	. . . . . . 7 ⊢ ((𝑑 = {⟨1, 𝐷⟩} ∧ 𝑖 ∈ (1...1)) → (((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) = (((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1))(AP‘𝐾)({⟨1, 𝐷⟩}‘1)))
188	186	fveq2d 6896	. . . . . . . . 9 ⊢ ((𝑑 = {⟨1, 𝐷⟩} ∧ 𝑖 ∈ (1...1)) → (𝐹‘((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖))) = (𝐹‘((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1))))
189	188	sneqd 4641	. . . . . . . 8 ⊢ ((𝑑 = {⟨1, 𝐷⟩} ∧ 𝑖 ∈ (1...1)) → {(𝐹‘((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖)))} = {(𝐹‘((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1)))})
190	189	imaeq2d 6060	. . . . . . 7 ⊢ ((𝑑 = {⟨1, 𝐷⟩} ∧ 𝑖 ∈ (1...1)) → (^◡𝐹 “ {(𝐹‘((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖)))}) = (^◡𝐹 “ {(𝐹‘((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1)))}))
191	187, 190	sseq12d 4016	. . . . . 6 ⊢ ((𝑑 = {⟨1, 𝐷⟩} ∧ 𝑖 ∈ (1...1)) → ((((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐹 “ {(𝐹‘((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖)))}) ↔ (((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1))(AP‘𝐾)({⟨1, 𝐷⟩}‘1)) ⊆ (^◡𝐹 “ {(𝐹‘((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1)))})))
192	191	ralbidva 3176	. . . . 5 ⊢ (𝑑 = {⟨1, 𝐷⟩} → (∀𝑖 ∈ (1...1)(((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐹 “ {(𝐹‘((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖)))}) ↔ ∀𝑖 ∈ (1...1)(((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1))(AP‘𝐾)({⟨1, 𝐷⟩}‘1)) ⊆ (^◡𝐹 “ {(𝐹‘((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1)))})))
193	188	mpteq2dva 5249	. . . . . . 7 ⊢ (𝑑 = {⟨1, 𝐷⟩} → (𝑖 ∈ (1...1) ↦ (𝐹‘((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖)))) = (𝑖 ∈ (1...1) ↦ (𝐹‘((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1)))))
194	193	rneqd 5938	. . . . . 6 ⊢ (𝑑 = {⟨1, 𝐷⟩} → ran (𝑖 ∈ (1...1) ↦ (𝐹‘((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖)))) = ran (𝑖 ∈ (1...1) ↦ (𝐹‘((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1)))))
195	194	fveqeq2d 6900	. . . . 5 ⊢ (𝑑 = {⟨1, 𝐷⟩} → ((♯‘ran (𝑖 ∈ (1...1) ↦ (𝐹‘((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖))))) = 1 ↔ (♯‘ran (𝑖 ∈ (1...1) ↦ (𝐹‘((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1))))) = 1))
196	192, 195	anbi12d 632	. . . 4 ⊢ (𝑑 = {⟨1, 𝐷⟩} → ((∀𝑖 ∈ (1...1)(((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐹 “ {(𝐹‘((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...1) ↦ (𝐹‘((𝐴 + (𝑊 − 𝐷)) + (𝑑‘𝑖))))) = 1) ↔ (∀𝑖 ∈ (1...1)(((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1))(AP‘𝐾)({⟨1, 𝐷⟩}‘1)) ⊆ (^◡𝐹 “ {(𝐹‘((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1)))}) ∧ (♯‘ran (𝑖 ∈ (1...1) ↦ (𝐹‘((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1))))) = 1)))
197	181, 196	rspc2ev 3625	. . 3 ⊢ (((𝐴 + (𝑊 − 𝐷)) ∈ ℕ ∧ {⟨1, 𝐷⟩} ∈ (ℕ ↑_m (1...1)) ∧ (∀𝑖 ∈ (1...1)(((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1))(AP‘𝐾)({⟨1, 𝐷⟩}‘1)) ⊆ (^◡𝐹 “ {(𝐹‘((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1)))}) ∧ (♯‘ran (𝑖 ∈ (1...1) ↦ (𝐹‘((𝐴 + (𝑊 − 𝐷)) + ({⟨1, 𝐷⟩}‘1))))) = 1)) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑_m (1...1))(∀𝑖 ∈ (1...1)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐹 “ {(𝐹‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...1) ↦ (𝐹‘(𝑎 + (𝑑‘𝑖))))) = 1))
198	50, 66, 155, 170, 197	syl112anc 1375	. 2 ⊢ (𝜑 → ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑_m (1...1))(∀𝑖 ∈ (1...1)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐹 “ {(𝐹‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...1) ↦ (𝐹‘(𝑎 + (𝑑‘𝑖))))) = 1))
199		ovex 7442	. . 3 ⊢ (1...(2 · 𝑊)) ∈ V
200	51	a1i 11	. . 3 ⊢ (𝜑 → 1 ∈ ℕ)
201		eqid 2733	. . 3 ⊢ (1...1) = (1...1)
202	199, 83, 124, 200, 201	vdwpc 16913	. 2 ⊢ (𝜑 → (⟨1, 𝐾⟩ PolyAP 𝐹 ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑_m (1...1))(∀𝑖 ∈ (1...1)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐹 “ {(𝐹‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...1) ↦ (𝐹‘(𝑎 + (𝑑‘𝑖))))) = 1)))
203	198, 202	mpbird 257	1 ⊢ (𝜑 → ⟨1, 𝐾⟩ PolyAP 𝐹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ∪ cun 3947 ⊆ wss 3949 ∅c0 4323 {csn 4629 ⟨cop 4635 class class class wbr 5149 ↦ cmpt 5232 × cxp 5675 ^◡ccnv 5676 dom cdm 5677 ran crn 5678 “ cima 5680 Fn wfn 6539 ⟶wf 6540 –1-1-onto→wf1o 6543 ‘cfv 6544 (class class class)co 7409 ↑_m cmap 8820 Fincfn 8939 ℂcc 11108 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 − cmin 11444 ℕcn 12212 2c2 12267 ℕ₀cn0 12472 ℤcz 12558 ℤ_≥cuz 12822 ...cfz 13484 ♯chash 14290 APcvdwa 16898 PolyAP cvdwp 16900

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-hash 14291 df-vdwap 16901 df-vdwpc 16903

This theorem is referenced by: vdwlem10 16923

W3C validator