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Theorem vdwlem2 16854
Description: Lemma for vdw 16866. (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 12176 . . . . . 6 ((𝑎 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑎 + 𝑁) ∈ ℕ)
41, 2, 3syl2anr 597 . . . . 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 12169 . . . . . . . . . . . . . 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 12169 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑁 ∈ ℂ)
9 elfznn0 13534 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ (0...(𝐾 − 1)) → 𝑚 ∈ ℕ0)
109adantl 482 . . . . . . . . . . . . . . . 16 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℕ0)
1110nn0cnd 12475 . . . . . . . . . . . . . . 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 12169 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑑 ∈ ℂ)
1411, 13mulcld 11175 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · 𝑑) ∈ ℂ)
156, 8, 14add32d 11382 . . . . . . . . . . . . 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 7364 . . . . . . . . . . . . . . 15 (𝑥 = (𝑎 + (𝑚 · 𝑑)) → (𝑥 + 𝑁) = ((𝑎 + (𝑚 · 𝑑)) + 𝑁))
1716eleq1d 2822 . . . . . . . . . . . . . 14 (𝑥 = (𝑎 + (𝑚 · 𝑑)) → ((𝑥 + 𝑁) ∈ (1...𝑀) ↔ ((𝑎 + (𝑚 · 𝑑)) + 𝑁) ∈ (1...𝑀)))
18 elfznn 13470 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (1...𝑊) → 𝑥 ∈ ℕ)
19 nnaddcl 12176 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑥 + 𝑁) ∈ ℕ)
2018, 2, 19syl2anr 597 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (1...𝑊)) → (𝑥 + 𝑁) ∈ ℕ)
21 nnuz 12806 . . . . . . . . . . . . . . . . . 18 ℕ = (ℤ‘1)
2220, 21eleqtrdi 2848 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (1...𝑊)) → (𝑥 + 𝑁) ∈ (ℤ‘1))
23 vdwlem2.m . . . . . . . . . . . . . . . . . 18 (𝜑𝑀 ∈ (ℤ‘(𝑊 + 𝑁)))
24 elfzuz3 13438 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (1...𝑊) → 𝑊 ∈ (ℤ𝑥))
252nnzd 12526 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ ℤ)
26 eluzadd 12792 . . . . . . . . . . . . . . . . . . 19 ((𝑊 ∈ (ℤ𝑥) ∧ 𝑁 ∈ ℤ) → (𝑊 + 𝑁) ∈ (ℤ‘(𝑥 + 𝑁)))
2724, 25, 26syl2anr 597 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (1...𝑊)) → (𝑊 + 𝑁) ∈ (ℤ‘(𝑥 + 𝑁)))
28 uztrn 12781 . . . . . . . . . . . . . . . . . 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 13435 . . . . . . . . . . . . . . . . 17 ((𝑥 + 𝑁) ∈ (1...𝑀) ↔ ((𝑥 + 𝑁) ∈ (ℤ‘1) ∧ 𝑀 ∈ (ℤ‘(𝑥 + 𝑁))))
3122, 29, 30sylanbrc 583 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (1...𝑊)) → (𝑥 + 𝑁) ∈ (1...𝑀))
3231ralrimiva 3143 . . . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . . . . . . . . 20 (𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑚 · 𝑑))
36 oveq1 7364 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑚 → (𝑛 · 𝑑) = (𝑚 · 𝑑))
3736oveq2d 7373 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → (𝑎 + (𝑛 · 𝑑)) = (𝑎 + (𝑚 · 𝑑)))
3837rspceeqv 3595 . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐾 ∈ ℕ0)
44 vdwapval 16845 . . . . . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎 + (𝑚 · 𝑑)) ∈ (𝑎(AP‘𝐾)𝑑))
4734, 46sseldd 3945 . . . . . . . . . . . . . . . 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)
4948ffvelcdmda 7035 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑥 + 𝑁) ∈ (1...𝑀)) → (𝐹‘(𝑥 + 𝑁)) ∈ 𝑅)
5031, 49syldan 591 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ (1...𝑊)) → (𝐹‘(𝑥 + 𝑁)) ∈ 𝑅)
51 vdwlem2.g . . . . . . . . . . . . . . . . . . . 20 𝐺 = (𝑥 ∈ (1...𝑊) ↦ (𝐹‘(𝑥 + 𝑁)))
5250, 51fmptd 7062 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐺:(1...𝑊)⟶𝑅)
5352ffnd 6669 . . . . . . . . . . . . . . . . . 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 7010 . . . . . . . . . . . . . . . . 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 231 . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊))
5917, 33, 58rspcdva 3582 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + (𝑚 · 𝑑)) + 𝑁) ∈ (1...𝑀))
6015, 59eqeltrd 2838 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + 𝑁) + (𝑚 · 𝑑)) ∈ (1...𝑀))
6115fveq2d 6846 . . . . . . . . . . . . 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 6846 . . . . . . . . . . . . . . 15 (𝑥 = (𝑎 + (𝑚 · 𝑑)) → (𝐹‘(𝑥 + 𝑁)) = (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)))
63 fvex 6855 . . . . . . . . . . . . . . 15 (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)) ∈ V
6462, 51, 63fvmpt 6948 . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐺‘(𝑎 + (𝑚 · 𝑑))) = 𝑐)
6761, 65, 663eqtr2d 2782 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = 𝑐)
6860, 67jca 512 . . . . . . . . . . 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 2825 . . . . . . . . . . . 12 (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → (𝑥 ∈ (1...𝑀) ↔ ((𝑎 + 𝑁) + (𝑚 · 𝑑)) ∈ (1...𝑀)))
70 fveqeq2 6851 . . . . . . . . . . . 12 (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → ((𝐹𝑥) = 𝑐 ↔ (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = 𝑐))
7169, 70anbi12d 631 . . . . . . . . . . 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 246 . . . . . . . . . 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 3152 . . . . . . . . 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 481 . . . . . . . . . 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 16845 . . . . . . . . . 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 1371 . . . . . . . . 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 6669 . . . . . . . . . . 11 (𝜑𝐹 Fn (1...𝑀))
7978ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → 𝐹 Fn (1...𝑀))
80 fniniseg 7010 . . . . . . . . . 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 293 . . . . . . . 8 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → (𝑥 ∈ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) → 𝑥 ∈ (𝐹 “ {𝑐})))
8382ssrdv 3950 . . . . . . 7 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))
8483expr 457 . . . . . 6 (((𝜑𝑎 ∈ ℕ) ∧ 𝑑 ∈ ℕ) → ((𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}) → ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
8584reximdva 3165 . . . . 5 ((𝜑𝑎 ∈ ℕ) → (∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}) → ∃𝑑 ∈ ℕ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
86 oveq1 7364 . . . . . . . 8 (𝑏 = (𝑎 + 𝑁) → (𝑏(AP‘𝐾)𝑑) = ((𝑎 + 𝑁)(AP‘𝐾)𝑑))
8786sseq1d 3975 . . . . . . 7 (𝑏 = (𝑎 + 𝑁) → ((𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
8887rexbidv 3175 . . . . . 6 (𝑏 = (𝑎 + 𝑁) → (∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ ∃𝑑 ∈ ℕ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
8988rspcev 3581 . . . . 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 3152 . . 3 (𝜑 → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}) → ∃𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
9291eximdv 1920 . 2 (𝜑 → (∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}) → ∃𝑐𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
93 ovex 7390 . . 3 (1...𝑊) ∈ V
9493, 41, 52vdwmc 16850 . 2 (𝜑 → (𝐾 MonoAP 𝐺 ↔ ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐})))
95 ovex 7390 . . 3 (1...𝑀) ∈ V
9695, 41, 48vdwmc 16850 . 2 (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
9792, 94, 963imtr4d 293 1 (𝜑 → (𝐾 MonoAP 𝐺𝐾 MonoAP 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  wral 3064  wrex 3073  wss 3910  {csn 4586   class class class wbr 5105  cmpt 5188  ccnv 5632  cima 5636   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  Fincfn 8883  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  cmin 11385  cn 12153  0cn0 12413  cz 12499  cuz 12763  ...cfz 13424  APcvdwa 16837   MonoAP cvdwm 16838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-vdwap 16840  df-vdwmc 16841
This theorem is referenced by:  vdwlem9  16861
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