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Theorem vdwlem2 16321
Description: Lemma for vdw 16333. (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 11663 . . . . . 6 ((𝑎 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑎 + 𝑁) ∈ ℕ)
41, 2, 3syl2anr 598 . . . . 5 ((𝜑𝑎 ∈ ℕ) → (𝑎 + 𝑁) ∈ ℕ)
5 simpllr 774 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑎 ∈ ℕ)
65nncnd 11657 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑎 ∈ ℂ)
72ad3antrrr 728 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑁 ∈ ℕ)
87nncnd 11657 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑁 ∈ ℂ)
9 elfznn0 13003 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ (0...(𝐾 − 1)) → 𝑚 ∈ ℕ0)
109adantl 484 . . . . . . . . . . . . . . . 16 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℕ0)
1110nn0cnd 11960 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℂ)
12 simplrl 775 . . . . . . . . . . . . . . . 16 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑑 ∈ ℕ)
1312nncnd 11657 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑑 ∈ ℂ)
1411, 13mulcld 10664 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · 𝑑) ∈ ℂ)
156, 8, 14add32d 10870 . . . . . . . . . . . . 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 7166 . . . . . . . . . . . . . . 15 (𝑥 = (𝑎 + (𝑚 · 𝑑)) → (𝑥 + 𝑁) = ((𝑎 + (𝑚 · 𝑑)) + 𝑁))
1716eleq1d 2900 . . . . . . . . . . . . . 14 (𝑥 = (𝑎 + (𝑚 · 𝑑)) → ((𝑥 + 𝑁) ∈ (1...𝑀) ↔ ((𝑎 + (𝑚 · 𝑑)) + 𝑁) ∈ (1...𝑀)))
18 elfznn 12939 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (1...𝑊) → 𝑥 ∈ ℕ)
19 nnaddcl 11663 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑥 + 𝑁) ∈ ℕ)
2018, 2, 19syl2anr 598 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (1...𝑊)) → (𝑥 + 𝑁) ∈ ℕ)
21 nnuz 12284 . . . . . . . . . . . . . . . . . 18 ℕ = (ℤ‘1)
2220, 21eleqtrdi 2926 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (1...𝑊)) → (𝑥 + 𝑁) ∈ (ℤ‘1))
23 vdwlem2.m . . . . . . . . . . . . . . . . . 18 (𝜑𝑀 ∈ (ℤ‘(𝑊 + 𝑁)))
24 elfzuz3 12908 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (1...𝑊) → 𝑊 ∈ (ℤ𝑥))
252nnzd 12089 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ ℤ)
26 eluzadd 12276 . . . . . . . . . . . . . . . . . . 19 ((𝑊 ∈ (ℤ𝑥) ∧ 𝑁 ∈ ℤ) → (𝑊 + 𝑁) ∈ (ℤ‘(𝑥 + 𝑁)))
2724, 25, 26syl2anr 598 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (1...𝑊)) → (𝑊 + 𝑁) ∈ (ℤ‘(𝑥 + 𝑁)))
28 uztrn 12264 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ (ℤ‘(𝑊 + 𝑁)) ∧ (𝑊 + 𝑁) ∈ (ℤ‘(𝑥 + 𝑁))) → 𝑀 ∈ (ℤ‘(𝑥 + 𝑁)))
2923, 27, 28syl2an2r 683 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (1...𝑊)) → 𝑀 ∈ (ℤ‘(𝑥 + 𝑁)))
30 elfzuzb 12905 . . . . . . . . . . . . . . . . 17 ((𝑥 + 𝑁) ∈ (1...𝑀) ↔ ((𝑥 + 𝑁) ∈ (ℤ‘1) ∧ 𝑀 ∈ (ℤ‘(𝑥 + 𝑁))))
3122, 29, 30sylanbrc 585 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (1...𝑊)) → (𝑥 + 𝑁) ∈ (1...𝑀))
3231ralrimiva 3185 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑥 ∈ (1...𝑊)(𝑥 + 𝑁) ∈ (1...𝑀))
3332ad3antrrr 728 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ∀𝑥 ∈ (1...𝑊)(𝑥 + 𝑁) ∈ (1...𝑀))
34 simplrr 776 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))
35 eqid 2824 . . . . . . . . . . . . . . . . . . . 20 (𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑚 · 𝑑))
36 oveq1 7166 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑚 → (𝑛 · 𝑑) = (𝑚 · 𝑑))
3736oveq2d 7175 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → (𝑎 + (𝑛 · 𝑑)) = (𝑎 + (𝑚 · 𝑑)))
3837rspceeqv 3641 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 ∈ (0...(𝐾 − 1)) ∧ (𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑚 · 𝑑))) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑)))
3935, 38mpan2 689 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ (0...(𝐾 − 1)) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑)))
4039adantl 484 . . . . . . . . . . . . . . . . . 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)))
41 vdwlem2.k . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐾 ∈ ℕ0)
4241ad2antrr 724 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → 𝐾 ∈ ℕ0)
4342adantr 483 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐾 ∈ ℕ0)
44 vdwapval 16312 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ ℕ0𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → ((𝑎 + (𝑚 · 𝑑)) ∈ (𝑎(AP‘𝐾)𝑑) ↔ ∃𝑛 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑))))
4543, 5, 12, 44syl3anc 1367 . . . . . . . . . . . . . . . . . 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))))
4640, 45mpbird 259 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎 + (𝑚 · 𝑑)) ∈ (𝑎(AP‘𝐾)𝑑))
4734, 46sseldd 3971 . . . . . . . . . . . . . . . 16 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎 + (𝑚 · 𝑑)) ∈ (𝐺 “ {𝑐}))
48 vdwlem2.f . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐹:(1...𝑀)⟶𝑅)
4948ffvelrnda 6854 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑥 + 𝑁) ∈ (1...𝑀)) → (𝐹‘(𝑥 + 𝑁)) ∈ 𝑅)
5031, 49syldan 593 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ (1...𝑊)) → (𝐹‘(𝑥 + 𝑁)) ∈ 𝑅)
51 vdwlem2.g . . . . . . . . . . . . . . . . . . . 20 𝐺 = (𝑥 ∈ (1...𝑊) ↦ (𝐹‘(𝑥 + 𝑁)))
5250, 51fmptd 6881 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐺:(1...𝑊)⟶𝑅)
5352ffnd 6518 . . . . . . . . . . . . . . . . . 18 (𝜑𝐺 Fn (1...𝑊))
5453ad3antrrr 728 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐺 Fn (1...𝑊))
55 fniniseg 6833 . . . . . . . . . . . . . . . . 17 (𝐺 Fn (1...𝑊) → ((𝑎 + (𝑚 · 𝑑)) ∈ (𝐺 “ {𝑐}) ↔ ((𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊) ∧ (𝐺‘(𝑎 + (𝑚 · 𝑑))) = 𝑐)))
5654, 55syl 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)))
5747, 56mpbid 234 . . . . . . . . . . . . . . 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))
5857simpld 497 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊))
5917, 33, 58rspcdva 3628 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + (𝑚 · 𝑑)) + 𝑁) ∈ (1...𝑀))
6015, 59eqeltrd 2916 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + 𝑁) + (𝑚 · 𝑑)) ∈ (1...𝑀))
6115fveq2d 6677 . . . . . . . . . . . . 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)))
6216fveq2d 6677 . . . . . . . . . . . . . . 15 (𝑥 = (𝑎 + (𝑚 · 𝑑)) → (𝐹‘(𝑥 + 𝑁)) = (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)))
63 fvex 6686 . . . . . . . . . . . . . . 15 (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)) ∈ V
6462, 51, 63fvmpt 6771 . . . . . . . . . . . . . 14 ((𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊) → (𝐺‘(𝑎 + (𝑚 · 𝑑))) = (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)))
6558, 64syl 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)))
6657simprd 498 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐺‘(𝑎 + (𝑚 · 𝑑))) = 𝑐)
6761, 65, 663eqtr2d 2865 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = 𝑐)
6860, 67jca 514 . . . . . . . . . . 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))
69 eleq1 2903 . . . . . . . . . . . 12 (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → (𝑥 ∈ (1...𝑀) ↔ ((𝑎 + 𝑁) + (𝑚 · 𝑑)) ∈ (1...𝑀)))
70 fveqeq2 6682 . . . . . . . . . . . 12 (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → ((𝐹𝑥) = 𝑐 ↔ (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = 𝑐))
7169, 70anbi12d 632 . . . . . . . . . . 11 (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → ((𝑥 ∈ (1...𝑀) ∧ (𝐹𝑥) = 𝑐) ↔ (((𝑎 + 𝑁) + (𝑚 · 𝑑)) ∈ (1...𝑀) ∧ (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = 𝑐)))
7268, 71syl5ibrcom 249 . . . . . . . . . 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)))
7372rexlimdva 3287 . . . . . . . . 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)))
744adantr 483 . . . . . . . . . 10 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → (𝑎 + 𝑁) ∈ ℕ)
75 simprl 769 . . . . . . . . . 10 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → 𝑑 ∈ ℕ)
76 vdwapval 16312 . . . . . . . . . 10 ((𝐾 ∈ ℕ0 ∧ (𝑎 + 𝑁) ∈ ℕ ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑))))
7742, 74, 75, 76syl3anc 1367 . . . . . . . . 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))))
7848ffnd 6518 . . . . . . . . . . 11 (𝜑𝐹 Fn (1...𝑀))
7978ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → 𝐹 Fn (1...𝑀))
80 fniniseg 6833 . . . . . . . . . 10 (𝐹 Fn (1...𝑀) → (𝑥 ∈ (𝐹 “ {𝑐}) ↔ (𝑥 ∈ (1...𝑀) ∧ (𝐹𝑥) = 𝑐)))
8179, 80syl 17 . . . . . . . . 9 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → (𝑥 ∈ (𝐹 “ {𝑐}) ↔ (𝑥 ∈ (1...𝑀) ∧ (𝐹𝑥) = 𝑐)))
8273, 77, 813imtr4d 296 . . . . . . . 8 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → (𝑥 ∈ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) → 𝑥 ∈ (𝐹 “ {𝑐})))
8382ssrdv 3976 . . . . . . 7 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))
8483expr 459 . . . . . 6 (((𝜑𝑎 ∈ ℕ) ∧ 𝑑 ∈ ℕ) → ((𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}) → ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
8584reximdva 3277 . . . . 5 ((𝜑𝑎 ∈ ℕ) → (∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}) → ∃𝑑 ∈ ℕ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
86 oveq1 7166 . . . . . . . 8 (𝑏 = (𝑎 + 𝑁) → (𝑏(AP‘𝐾)𝑑) = ((𝑎 + 𝑁)(AP‘𝐾)𝑑))
8786sseq1d 4001 . . . . . . 7 (𝑏 = (𝑎 + 𝑁) → ((𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
8887rexbidv 3300 . . . . . 6 (𝑏 = (𝑎 + 𝑁) → (∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ ∃𝑑 ∈ ℕ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
8988rspcev 3626 . . . . 5 (((𝑎 + 𝑁) ∈ ℕ ∧ ∃𝑑 ∈ ℕ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})) → ∃𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))
904, 85, 89syl6an 682 . . . 4 ((𝜑𝑎 ∈ ℕ) → (∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}) → ∃𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
9190rexlimdva 3287 . . 3 (𝜑 → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}) → ∃𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
9291eximdv 1917 . 2 (𝜑 → (∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}) → ∃𝑐𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
93 ovex 7192 . . 3 (1...𝑊) ∈ V
9493, 41, 52vdwmc 16317 . 2 (𝜑 → (𝐾 MonoAP 𝐺 ↔ ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐})))
95 ovex 7192 . . 3 (1...𝑀) ∈ V
9695, 41, 48vdwmc 16317 . 2 (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
9792, 94, 963imtr4d 296 1 (𝜑 → (𝐾 MonoAP 𝐺𝐾 MonoAP 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1536  wex 1779  wcel 2113  wral 3141  wrex 3142  wss 3939  {csn 4570   class class class wbr 5069  cmpt 5149  ccnv 5557  cima 5561   Fn wfn 6353  wf 6354  cfv 6358  (class class class)co 7159  Fincfn 8512  0cc0 10540  1c1 10541   + caddc 10543   · cmul 10545  cmin 10873  cn 11641  0cn0 11900  cz 11984  cuz 12246  ...cfz 12895  APcvdwa 16304   MonoAP cvdwm 16305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-1st 7692  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-er 8292  df-en 8513  df-dom 8514  df-sdom 8515  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-nn 11642  df-n0 11901  df-z 11985  df-uz 12247  df-fz 12896  df-vdwap 16307  df-vdwmc 16308
This theorem is referenced by:  vdwlem9  16328
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