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Theorem vdwlem2 16683

Description: Lemma for vdw 16695. (Contributed by Mario Carneiro, 12-Sep-2014.)

**Hypotheses**
Ref	Expression
vdwlem2.r	⊢ (𝜑 → 𝑅 ∈ Fin)
vdwlem2.k	⊢ (𝜑 → 𝐾 ∈ ℕ₀)
vdwlem2.w	⊢ (𝜑 → 𝑊 ∈ ℕ)
vdwlem2.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
vdwlem2.f	⊢ (𝜑 → 𝐹:(1...𝑀)⟶𝑅)
vdwlem2.m	⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘(𝑊 + 𝑁)))
vdwlem2.g	⊢ 𝐺 = (𝑥 ∈ (1...𝑊) ↦ (𝐹‘(𝑥 + 𝑁)))

**Assertion**
Ref	Expression
vdwlem2	⊢ (𝜑 → (𝐾 MonoAP 𝐺 → 𝐾 MonoAP 𝐹))

Distinct variable groups: 𝑥,𝐹 𝑥,𝐾 𝑥,𝑀 𝜑,𝑥 𝑥,𝐺 𝑥,𝑁 𝑥,𝑅 𝑥,𝑊

Proof of Theorem vdwlem2 Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . . . 6 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℕ)
2		vdwlem2.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
3		nnaddcl 11996	. . . . . 6 ⊢ ((𝑎 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑎 + 𝑁) ∈ ℕ)
4	1, 2, 3	syl2anr 597	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → (𝑎 + 𝑁) ∈ ℕ)
5		simpllr 773	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑎 ∈ ℕ)
6	5	nncnd 11989	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑎 ∈ ℂ)
7	2	ad3antrrr 727	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑁 ∈ ℕ)
8	7	nncnd 11989	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑁 ∈ ℂ)
9		elfznn0 13349	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (0...(𝐾 − 1)) → 𝑚 ∈ ℕ₀)
10	9	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℕ₀)
11	10	nn0cnd 12295	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℂ)
12		simplrl 774	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑑 ∈ ℕ)
13	12	nncnd 11989	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑑 ∈ ℂ)
14	11, 13	mulcld 10995	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · 𝑑) ∈ ℂ)
15	6, 8, 14	add32d 11202	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + 𝑁) + (𝑚 · 𝑑)) = ((𝑎 + (𝑚 · 𝑑)) + 𝑁))
16		oveq1 7282	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑎 + (𝑚 · 𝑑)) → (𝑥 + 𝑁) = ((𝑎 + (𝑚 · 𝑑)) + 𝑁))
17	16	eleq1d 2823	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑎 + (𝑚 · 𝑑)) → ((𝑥 + 𝑁) ∈ (1...𝑀) ↔ ((𝑎 + (𝑚 · 𝑑)) + 𝑁) ∈ (1...𝑀)))
18		elfznn 13285	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (1...𝑊) → 𝑥 ∈ ℕ)
19		nnaddcl 11996	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑥 + 𝑁) ∈ ℕ)
20	18, 2, 19	syl2anr 597	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑊)) → (𝑥 + 𝑁) ∈ ℕ)
21		nnuz 12621	. . . . . . . . . . . . . . . . . 18 ⊢ ℕ = (ℤ_≥‘1)
22	20, 21	eleqtrdi 2849	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑊)) → (𝑥 + 𝑁) ∈ (ℤ_≥‘1))
23		vdwlem2.m	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘(𝑊 + 𝑁)))
24		elfzuz3 13253	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (1...𝑊) → 𝑊 ∈ (ℤ_≥‘𝑥))
25	2	nnzd 12425	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℤ)
26		eluzadd 12613	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑊 ∈ (ℤ_≥‘𝑥) ∧ 𝑁 ∈ ℤ) → (𝑊 + 𝑁) ∈ (ℤ_≥‘(𝑥 + 𝑁)))
27	24, 25, 26	syl2anr 597	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑊)) → (𝑊 + 𝑁) ∈ (ℤ_≥‘(𝑥 + 𝑁)))
28		uztrn 12600	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ (ℤ_≥‘(𝑊 + 𝑁)) ∧ (𝑊 + 𝑁) ∈ (ℤ_≥‘(𝑥 + 𝑁))) → 𝑀 ∈ (ℤ_≥‘(𝑥 + 𝑁)))
29	23, 27, 28	syl2an2r 682	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑊)) → 𝑀 ∈ (ℤ_≥‘(𝑥 + 𝑁)))
30		elfzuzb 13250	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 + 𝑁) ∈ (1...𝑀) ↔ ((𝑥 + 𝑁) ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ (ℤ_≥‘(𝑥 + 𝑁))))
31	22, 29, 30	sylanbrc 583	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑊)) → (𝑥 + 𝑁) ∈ (1...𝑀))
32	31	ralrimiva 3103	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ (1...𝑊)(𝑥 + 𝑁) ∈ (1...𝑀))
33	32	ad3antrrr 727	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ∀𝑥 ∈ (1...𝑊)(𝑥 + 𝑁) ∈ (1...𝑀))
34		simplrr 775	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))
35		eqid 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑚 · 𝑑))
36		oveq1 7282	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑚 → (𝑛 · 𝑑) = (𝑚 · 𝑑))
37	36	oveq2d 7291	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑚 → (𝑎 + (𝑛 · 𝑑)) = (𝑎 + (𝑚 · 𝑑)))
38	37	rspceeqv 3575	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ (0...(𝐾 − 1)) ∧ (𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑚 · 𝑑))) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑)))
39	35, 38	mpan2 688	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ (0...(𝐾 − 1)) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑)))
40	39	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑)))
41		vdwlem2.k	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
42	41	ad2antrr 723	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) → 𝐾 ∈ ℕ₀)
43	42	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐾 ∈ ℕ₀)
44		vdwapval 16674	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → ((𝑎 + (𝑚 · 𝑑)) ∈ (𝑎(AP‘𝐾)𝑑) ↔ ∃𝑛 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑))))
45	43, 5, 12, 44	syl3anc 1370	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + (𝑚 · 𝑑)) ∈ (𝑎(AP‘𝐾)𝑑) ↔ ∃𝑛 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑))))
46	40, 45	mpbird 256	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎 + (𝑚 · 𝑑)) ∈ (𝑎(AP‘𝐾)𝑑))
47	34, 46	sseldd 3922	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐺 “ {𝑐}))
48		vdwlem2.f	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐹:(1...𝑀)⟶𝑅)
49	48	ffvelrnda 6961	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑥 + 𝑁) ∈ (1...𝑀)) → (𝐹‘(𝑥 + 𝑁)) ∈ 𝑅)
50	31, 49	syldan 591	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑊)) → (𝐹‘(𝑥 + 𝑁)) ∈ 𝑅)
51		vdwlem2.g	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐺 = (𝑥 ∈ (1...𝑊) ↦ (𝐹‘(𝑥 + 𝑁)))
52	50, 51	fmptd 6988	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐺:(1...𝑊)⟶𝑅)
53	52	ffnd 6601	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐺 Fn (1...𝑊))
54	53	ad3antrrr 727	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐺 Fn (1...𝑊))
55		fniniseg 6937	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 Fn (1...𝑊) → ((𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐺 “ {𝑐}) ↔ ((𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊) ∧ (𝐺‘(𝑎 + (𝑚 · 𝑑))) = 𝑐)))
56	54, 55	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐺 “ {𝑐}) ↔ ((𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊) ∧ (𝐺‘(𝑎 + (𝑚 · 𝑑))) = 𝑐)))
57	47, 56	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊) ∧ (𝐺‘(𝑎 + (𝑚 · 𝑑))) = 𝑐))
58	57	simpld 495	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊))
59	17, 33, 58	rspcdva 3562	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + (𝑚 · 𝑑)) + 𝑁) ∈ (1...𝑀))
60	15, 59	eqeltrd 2839	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + 𝑁) + (𝑚 · 𝑑)) ∈ (1...𝑀))
61	15	fveq2d 6778	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)))
62	16	fveq2d 6778	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑎 + (𝑚 · 𝑑)) → (𝐹‘(𝑥 + 𝑁)) = (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)))
63		fvex 6787	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)) ∈ V
64	62, 51, 63	fvmpt 6875	. . . . . . . . . . . . . 14 ⊢ ((𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊) → (𝐺‘(𝑎 + (𝑚 · 𝑑))) = (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)))
65	58, 64	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐺‘(𝑎 + (𝑚 · 𝑑))) = (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)))
66	57	simprd 496	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐺‘(𝑎 + (𝑚 · 𝑑))) = 𝑐)
67	61, 65, 66	3eqtr2d 2784	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = 𝑐)
68	60, 67	jca 512	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (((𝑎 + 𝑁) + (𝑚 · 𝑑)) ∈ (1...𝑀) ∧ (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = 𝑐))
69		eleq1 2826	. . . . . . . . . . . 12 ⊢ (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → (𝑥 ∈ (1...𝑀) ↔ ((𝑎 + 𝑁) + (𝑚 · 𝑑)) ∈ (1...𝑀)))
70		fveqeq2 6783	. . . . . . . . . . . 12 ⊢ (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → ((𝐹‘𝑥) = 𝑐 ↔ (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = 𝑐))
71	69, 70	anbi12d 631	. . . . . . . . . . 11 ⊢ (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → ((𝑥 ∈ (1...𝑀) ∧ (𝐹‘𝑥) = 𝑐) ↔ (((𝑎 + 𝑁) + (𝑚 · 𝑑)) ∈ (1...𝑀) ∧ (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = 𝑐)))
72	68, 71	syl5ibrcom 246	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → (𝑥 ∈ (1...𝑀) ∧ (𝐹‘𝑥) = 𝑐)))
73	72	rexlimdva 3213	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) → (∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → (𝑥 ∈ (1...𝑀) ∧ (𝐹‘𝑥) = 𝑐)))
74	4	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) → (𝑎 + 𝑁) ∈ ℕ)
75		simprl 768	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) → 𝑑 ∈ ℕ)
76		vdwapval 16674	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℕ₀ ∧ (𝑎 + 𝑁) ∈ ℕ ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑))))
77	42, 74, 75, 76	syl3anc 1370	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) → (𝑥 ∈ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑))))
78	48	ffnd 6601	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn (1...𝑀))
79	78	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) → 𝐹 Fn (1...𝑀))
80		fniniseg 6937	. . . . . . . . . 10 ⊢ (𝐹 Fn (1...𝑀) → (𝑥 ∈ (^◡𝐹 “ {𝑐}) ↔ (𝑥 ∈ (1...𝑀) ∧ (𝐹‘𝑥) = 𝑐)))
81	79, 80	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) → (𝑥 ∈ (^◡𝐹 “ {𝑐}) ↔ (𝑥 ∈ (1...𝑀) ∧ (𝐹‘𝑥) = 𝑐)))
82	73, 77, 81	3imtr4d 294	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) → (𝑥 ∈ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) → 𝑥 ∈ (^◡𝐹 “ {𝑐})))
83	82	ssrdv 3927	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) → ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐}))
84	83	expr 457	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ 𝑑 ∈ ℕ) → ((𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}) → ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐})))
85	84	reximdva 3203	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → (∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}) → ∃𝑑 ∈ ℕ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐})))
86		oveq1 7282	. . . . . . . 8 ⊢ (𝑏 = (𝑎 + 𝑁) → (𝑏(AP‘𝐾)𝑑) = ((𝑎 + 𝑁)(AP‘𝐾)𝑑))
87	86	sseq1d 3952	. . . . . . 7 ⊢ (𝑏 = (𝑎 + 𝑁) → ((𝑏(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐}) ↔ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐})))
88	87	rexbidv 3226	. . . . . 6 ⊢ (𝑏 = (𝑎 + 𝑁) → (∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐}) ↔ ∃𝑑 ∈ ℕ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐})))
89	88	rspcev 3561	. . . . 5 ⊢ (((𝑎 + 𝑁) ∈ ℕ ∧ ∃𝑑 ∈ ℕ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐})) → ∃𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐}))
90	4, 85, 89	syl6an 681	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → (∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}) → ∃𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐})))
91	90	rexlimdva 3213	. . 3 ⊢ (𝜑 → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}) → ∃𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐})))
92	91	eximdv 1920	. 2 ⊢ (𝜑 → (∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}) → ∃𝑐∃𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐})))
93		ovex 7308	. . 3 ⊢ (1...𝑊) ∈ V
94	93, 41, 52	vdwmc 16679	. 2 ⊢ (𝜑 → (𝐾 MonoAP 𝐺 ↔ ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐})))
95		ovex 7308	. . 3 ⊢ (1...𝑀) ∈ V
96	95, 41, 48	vdwmc 16679	. 2 ⊢ (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐∃𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐})))
97	92, 94, 96	3imtr4d 294	1 ⊢ (𝜑 → (𝐾 MonoAP 𝐺 → 𝐾 MonoAP 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∃wex 1782 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 ⊆ wss 3887 {csn 4561 class class class wbr 5074 ↦ cmpt 5157 ^◡ccnv 5588 “ cima 5592 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 Fincfn 8733 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 − cmin 11205 ℕcn 11973 ℕ₀cn0 12233 ℤcz 12319 ℤ_≥cuz 12582 ...cfz 13239 APcvdwa 16666 MonoAP cvdwm 16667

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-vdwap 16669 df-vdwmc 16670

This theorem is referenced by: vdwlem9 16690

W3C validator