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Theorem vdwlem2 15893
Description: Lemma for vdw 15905. (Contributed by Mario Carneiro, 12-Sep-2014.)
Hypotheses
Ref Expression
vdwlem2.r (𝜑𝑅 ∈ Fin)
vdwlem2.k (𝜑𝐾 ∈ ℕ0)
vdwlem2.w (𝜑𝑊 ∈ ℕ)
vdwlem2.n (𝜑𝑁 ∈ ℕ)
vdwlem2.f (𝜑𝐹:(1...𝑀)⟶𝑅)
vdwlem2.m (𝜑𝑀 ∈ (ℤ‘(𝑊 + 𝑁)))
vdwlem2.g 𝐺 = (𝑥 ∈ (1...𝑊) ↦ (𝐹‘(𝑥 + 𝑁)))
Assertion
Ref Expression
vdwlem2 (𝜑 → (𝐾 MonoAP 𝐺𝐾 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.
StepHypRef Expression
1 id 22 . . . . . 6 (𝑎 ∈ ℕ → 𝑎 ∈ ℕ)
2 vdwlem2.n . . . . . 6 (𝜑𝑁 ∈ ℕ)
3 nnaddcl 11244 . . . . . 6 ((𝑎 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑎 + 𝑁) ∈ ℕ)
41, 2, 3syl2anr 584 . . . . 5 ((𝜑𝑎 ∈ ℕ) → (𝑎 + 𝑁) ∈ ℕ)
5 simpllr 760 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑎 ∈ ℕ)
65nncnd 11238 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑎 ∈ ℂ)
72ad3antrrr 709 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑁 ∈ ℕ)
87nncnd 11238 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑁 ∈ ℂ)
9 elfznn0 12640 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ (0...(𝐾 − 1)) → 𝑚 ∈ ℕ0)
109adantl 467 . . . . . . . . . . . . . . . 16 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℕ0)
1110nn0cnd 11555 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℂ)
12 simplrl 762 . . . . . . . . . . . . . . . 16 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑑 ∈ ℕ)
1312nncnd 11238 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑑 ∈ ℂ)
1411, 13mulcld 10262 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · 𝑑) ∈ ℂ)
156, 8, 14add32d 10465 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + 𝑁) + (𝑚 · 𝑑)) = ((𝑎 + (𝑚 · 𝑑)) + 𝑁))
16 oveq1 6800 . . . . . . . . . . . . . . 15 (𝑥 = (𝑎 + (𝑚 · 𝑑)) → (𝑥 + 𝑁) = ((𝑎 + (𝑚 · 𝑑)) + 𝑁))
1716eleq1d 2835 . . . . . . . . . . . . . 14 (𝑥 = (𝑎 + (𝑚 · 𝑑)) → ((𝑥 + 𝑁) ∈ (1...𝑀) ↔ ((𝑎 + (𝑚 · 𝑑)) + 𝑁) ∈ (1...𝑀)))
18 elfznn 12577 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (1...𝑊) → 𝑥 ∈ ℕ)
19 nnaddcl 11244 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑥 + 𝑁) ∈ ℕ)
2018, 2, 19syl2anr 584 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (1...𝑊)) → (𝑥 + 𝑁) ∈ ℕ)
21 nnuz 11925 . . . . . . . . . . . . . . . . . 18 ℕ = (ℤ‘1)
2220, 21syl6eleq 2860 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (1...𝑊)) → (𝑥 + 𝑁) ∈ (ℤ‘1))
23 vdwlem2.m . . . . . . . . . . . . . . . . . . 19 (𝜑𝑀 ∈ (ℤ‘(𝑊 + 𝑁)))
2423adantr 466 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (1...𝑊)) → 𝑀 ∈ (ℤ‘(𝑊 + 𝑁)))
25 elfzuz3 12546 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (1...𝑊) → 𝑊 ∈ (ℤ𝑥))
262nnzd 11683 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ ℤ)
27 eluzadd 11917 . . . . . . . . . . . . . . . . . . 19 ((𝑊 ∈ (ℤ𝑥) ∧ 𝑁 ∈ ℤ) → (𝑊 + 𝑁) ∈ (ℤ‘(𝑥 + 𝑁)))
2825, 26, 27syl2anr 584 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (1...𝑊)) → (𝑊 + 𝑁) ∈ (ℤ‘(𝑥 + 𝑁)))
29 uztrn 11905 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ (ℤ‘(𝑊 + 𝑁)) ∧ (𝑊 + 𝑁) ∈ (ℤ‘(𝑥 + 𝑁))) → 𝑀 ∈ (ℤ‘(𝑥 + 𝑁)))
3024, 28, 29syl2anc 573 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (1...𝑊)) → 𝑀 ∈ (ℤ‘(𝑥 + 𝑁)))
31 elfzuzb 12543 . . . . . . . . . . . . . . . . 17 ((𝑥 + 𝑁) ∈ (1...𝑀) ↔ ((𝑥 + 𝑁) ∈ (ℤ‘1) ∧ 𝑀 ∈ (ℤ‘(𝑥 + 𝑁))))
3222, 30, 31sylanbrc 572 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (1...𝑊)) → (𝑥 + 𝑁) ∈ (1...𝑀))
3332ralrimiva 3115 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑥 ∈ (1...𝑊)(𝑥 + 𝑁) ∈ (1...𝑀))
3433ad3antrrr 709 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ∀𝑥 ∈ (1...𝑊)(𝑥 + 𝑁) ∈ (1...𝑀))
35 simplrr 763 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))
36 eqid 2771 . . . . . . . . . . . . . . . . . . . 20 (𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑚 · 𝑑))
37 oveq1 6800 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑚 → (𝑛 · 𝑑) = (𝑚 · 𝑑))
3837oveq2d 6809 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑚 → (𝑎 + (𝑛 · 𝑑)) = (𝑎 + (𝑚 · 𝑑)))
3938eqeq2d 2781 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → ((𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑)) ↔ (𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑚 · 𝑑))))
4039rspcev 3460 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 ∈ (0...(𝐾 − 1)) ∧ (𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑚 · 𝑑))) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑)))
4136, 40mpan2 671 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ (0...(𝐾 − 1)) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑)))
4241adantl 467 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑)))
43 vdwlem2.k . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐾 ∈ ℕ0)
4443ad2antrr 705 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → 𝐾 ∈ ℕ0)
4544adantr 466 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐾 ∈ ℕ0)
46 vdwapval 15884 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ ℕ0𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → ((𝑎 + (𝑚 · 𝑑)) ∈ (𝑎(AP‘𝐾)𝑑) ↔ ∃𝑛 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑))))
4745, 5, 12, 46syl3anc 1476 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑎 ∈ 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))))
4842, 47mpbird 247 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎 + (𝑚 · 𝑑)) ∈ (𝑎(AP‘𝐾)𝑑))
4935, 48sseldd 3753 . . . . . . . . . . . . . . . 16 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎 + (𝑚 · 𝑑)) ∈ (𝐺 “ {𝑐}))
50 vdwlem2.f . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐹:(1...𝑀)⟶𝑅)
5150ffvelrnda 6502 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑥 + 𝑁) ∈ (1...𝑀)) → (𝐹‘(𝑥 + 𝑁)) ∈ 𝑅)
5232, 51syldan 579 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ (1...𝑊)) → (𝐹‘(𝑥 + 𝑁)) ∈ 𝑅)
53 vdwlem2.g . . . . . . . . . . . . . . . . . . . 20 𝐺 = (𝑥 ∈ (1...𝑊) ↦ (𝐹‘(𝑥 + 𝑁)))
5452, 53fmptd 6527 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐺:(1...𝑊)⟶𝑅)
55 ffn 6185 . . . . . . . . . . . . . . . . . . 19 (𝐺:(1...𝑊)⟶𝑅𝐺 Fn (1...𝑊))
5654, 55syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑𝐺 Fn (1...𝑊))
5756ad3antrrr 709 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐺 Fn (1...𝑊))
58 fniniseg 6481 . . . . . . . . . . . . . . . . 17 (𝐺 Fn (1...𝑊) → ((𝑎 + (𝑚 · 𝑑)) ∈ (𝐺 “ {𝑐}) ↔ ((𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊) ∧ (𝐺‘(𝑎 + (𝑚 · 𝑑))) = 𝑐)))
5957, 58syl 17 . . . . . . . . . . . . . . . 16 ((((𝜑𝑎 ∈ 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)))
6049, 59mpbid 222 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊) ∧ (𝐺‘(𝑎 + (𝑚 · 𝑑))) = 𝑐))
6160simpld 482 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊))
6217, 34, 61rspcdva 3466 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + (𝑚 · 𝑑)) + 𝑁) ∈ (1...𝑀))
6315, 62eqeltrd 2850 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + 𝑁) + (𝑚 · 𝑑)) ∈ (1...𝑀))
6415fveq2d 6336 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)))
6516fveq2d 6336 . . . . . . . . . . . . . . 15 (𝑥 = (𝑎 + (𝑚 · 𝑑)) → (𝐹‘(𝑥 + 𝑁)) = (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)))
66 fvex 6342 . . . . . . . . . . . . . . 15 (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)) ∈ V
6765, 53, 66fvmpt 6424 . . . . . . . . . . . . . 14 ((𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊) → (𝐺‘(𝑎 + (𝑚 · 𝑑))) = (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)))
6861, 67syl 17 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐺‘(𝑎 + (𝑚 · 𝑑))) = (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)))
6960simprd 483 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐺‘(𝑎 + (𝑚 · 𝑑))) = 𝑐)
7064, 68, 693eqtr2d 2811 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = 𝑐)
7163, 70jca 501 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ 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))
72 eleq1 2838 . . . . . . . . . . . 12 (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → (𝑥 ∈ (1...𝑀) ↔ ((𝑎 + 𝑁) + (𝑚 · 𝑑)) ∈ (1...𝑀)))
73 fveq2 6332 . . . . . . . . . . . . 13 (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → (𝐹𝑥) = (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))))
7473eqeq1d 2773 . . . . . . . . . . . 12 (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → ((𝐹𝑥) = 𝑐 ↔ (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = 𝑐))
7572, 74anbi12d 616 . . . . . . . . . . 11 (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → ((𝑥 ∈ (1...𝑀) ∧ (𝐹𝑥) = 𝑐) ↔ (((𝑎 + 𝑁) + (𝑚 · 𝑑)) ∈ (1...𝑀) ∧ (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = 𝑐)))
7671, 75syl5ibrcom 237 . . . . . . . . . 10 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → (𝑥 ∈ (1...𝑀) ∧ (𝐹𝑥) = 𝑐)))
7776rexlimdva 3179 . . . . . . . . 9 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → (∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → (𝑥 ∈ (1...𝑀) ∧ (𝐹𝑥) = 𝑐)))
784adantr 466 . . . . . . . . . 10 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → (𝑎 + 𝑁) ∈ ℕ)
79 simprl 754 . . . . . . . . . 10 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → 𝑑 ∈ ℕ)
80 vdwapval 15884 . . . . . . . . . 10 ((𝐾 ∈ ℕ0 ∧ (𝑎 + 𝑁) ∈ ℕ ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑))))
8144, 78, 79, 80syl3anc 1476 . . . . . . . . 9 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → (𝑥 ∈ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑))))
82 ffn 6185 . . . . . . . . . . . 12 (𝐹:(1...𝑀)⟶𝑅𝐹 Fn (1...𝑀))
8350, 82syl 17 . . . . . . . . . . 11 (𝜑𝐹 Fn (1...𝑀))
8483ad2antrr 705 . . . . . . . . . 10 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → 𝐹 Fn (1...𝑀))
85 fniniseg 6481 . . . . . . . . . 10 (𝐹 Fn (1...𝑀) → (𝑥 ∈ (𝐹 “ {𝑐}) ↔ (𝑥 ∈ (1...𝑀) ∧ (𝐹𝑥) = 𝑐)))
8684, 85syl 17 . . . . . . . . 9 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → (𝑥 ∈ (𝐹 “ {𝑐}) ↔ (𝑥 ∈ (1...𝑀) ∧ (𝐹𝑥) = 𝑐)))
8777, 81, 863imtr4d 283 . . . . . . . 8 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → (𝑥 ∈ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) → 𝑥 ∈ (𝐹 “ {𝑐})))
8887ssrdv 3758 . . . . . . 7 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))
8988expr 444 . . . . . 6 (((𝜑𝑎 ∈ ℕ) ∧ 𝑑 ∈ ℕ) → ((𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}) → ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
9089reximdva 3165 . . . . 5 ((𝜑𝑎 ∈ ℕ) → (∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}) → ∃𝑑 ∈ ℕ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
91 oveq1 6800 . . . . . . . 8 (𝑏 = (𝑎 + 𝑁) → (𝑏(AP‘𝐾)𝑑) = ((𝑎 + 𝑁)(AP‘𝐾)𝑑))
9291sseq1d 3781 . . . . . . 7 (𝑏 = (𝑎 + 𝑁) → ((𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
9392rexbidv 3200 . . . . . 6 (𝑏 = (𝑎 + 𝑁) → (∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ ∃𝑑 ∈ ℕ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
9493rspcev 3460 . . . . 5 (((𝑎 + 𝑁) ∈ ℕ ∧ ∃𝑑 ∈ ℕ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})) → ∃𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))
954, 90, 94syl6an 663 . . . 4 ((𝜑𝑎 ∈ ℕ) → (∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}) → ∃𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
9695rexlimdva 3179 . . 3 (𝜑 → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}) → ∃𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
9796eximdv 1998 . 2 (𝜑 → (∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}) → ∃𝑐𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
98 ovex 6823 . . 3 (1...𝑊) ∈ V
9998, 43, 54vdwmc 15889 . 2 (𝜑 → (𝐾 MonoAP 𝐺 ↔ ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐})))
100 ovex 6823 . . 3 (1...𝑀) ∈ V
101100, 43, 50vdwmc 15889 . 2 (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
10297, 99, 1013imtr4d 283 1 (𝜑 → (𝐾 MonoAP 𝐺𝐾 MonoAP 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wex 1852  wcel 2145  wral 3061  wrex 3062  wss 3723  {csn 4316   class class class wbr 4786  cmpt 4863  ccnv 5248  cima 5252   Fn wfn 6026  wf 6027  cfv 6031  (class class class)co 6793  Fincfn 8109  0cc0 10138  1c1 10139   + caddc 10141   · cmul 10143  cmin 10468  cn 11222  0cn0 11494  cz 11579  cuz 11888  ...cfz 12533  APcvdwa 15876   MonoAP cvdwm 15877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-n0 11495  df-z 11580  df-uz 11889  df-fz 12534  df-vdwap 15879  df-vdwmc 15880
This theorem is referenced by:  vdwlem9  15900
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