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Theorem vdwlem2 17020
Description: Lemma for vdw 17032. (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 12289 . . . . . 6 ((𝑎 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑎 + 𝑁) ∈ ℕ)
41, 2, 3syl2anr 597 . . . . 5 ((𝜑𝑎 ∈ ℕ) → (𝑎 + 𝑁) ∈ ℕ)
5 simpllr 776 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑎 ∈ ℕ)
65nncnd 12282 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑎 ∈ ℂ)
72ad3antrrr 730 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑁 ∈ ℕ)
87nncnd 12282 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑁 ∈ ℂ)
9 elfznn0 13660 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ (0...(𝐾 − 1)) → 𝑚 ∈ ℕ0)
109adantl 481 . . . . . . . . . . . . . . . 16 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℕ0)
1110nn0cnd 12589 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℂ)
12 simplrl 777 . . . . . . . . . . . . . . . 16 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑑 ∈ ℕ)
1312nncnd 12282 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑑 ∈ ℂ)
1411, 13mulcld 11281 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · 𝑑) ∈ ℂ)
156, 8, 14add32d 11489 . . . . . . . . . . . . 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 7438 . . . . . . . . . . . . . . 15 (𝑥 = (𝑎 + (𝑚 · 𝑑)) → (𝑥 + 𝑁) = ((𝑎 + (𝑚 · 𝑑)) + 𝑁))
1716eleq1d 2826 . . . . . . . . . . . . . 14 (𝑥 = (𝑎 + (𝑚 · 𝑑)) → ((𝑥 + 𝑁) ∈ (1...𝑀) ↔ ((𝑎 + (𝑚 · 𝑑)) + 𝑁) ∈ (1...𝑀)))
18 elfznn 13593 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (1...𝑊) → 𝑥 ∈ ℕ)
19 nnaddcl 12289 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑥 + 𝑁) ∈ ℕ)
2018, 2, 19syl2anr 597 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (1...𝑊)) → (𝑥 + 𝑁) ∈ ℕ)
21 nnuz 12921 . . . . . . . . . . . . . . . . . 18 ℕ = (ℤ‘1)
2220, 21eleqtrdi 2851 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (1...𝑊)) → (𝑥 + 𝑁) ∈ (ℤ‘1))
23 vdwlem2.m . . . . . . . . . . . . . . . . . 18 (𝜑𝑀 ∈ (ℤ‘(𝑊 + 𝑁)))
24 elfzuz3 13561 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (1...𝑊) → 𝑊 ∈ (ℤ𝑥))
252nnzd 12640 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ ℤ)
26 eluzadd 12907 . . . . . . . . . . . . . . . . . . 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 12896 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ (ℤ‘(𝑊 + 𝑁)) ∧ (𝑊 + 𝑁) ∈ (ℤ‘(𝑥 + 𝑁))) → 𝑀 ∈ (ℤ‘(𝑥 + 𝑁)))
2923, 27, 28syl2an2r 685 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (1...𝑊)) → 𝑀 ∈ (ℤ‘(𝑥 + 𝑁)))
30 elfzuzb 13558 . . . . . . . . . . . . . . . . 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 3146 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑥 ∈ (1...𝑊)(𝑥 + 𝑁) ∈ (1...𝑀))
3332ad3antrrr 730 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ∀𝑥 ∈ (1...𝑊)(𝑥 + 𝑁) ∈ (1...𝑀))
34 simplrr 778 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))
35 eqid 2737 . . . . . . . . . . . . . . . . . . . 20 (𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑚 · 𝑑))
36 oveq1 7438 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑚 → (𝑛 · 𝑑) = (𝑚 · 𝑑))
3736oveq2d 7447 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → (𝑎 + (𝑛 · 𝑑)) = (𝑎 + (𝑚 · 𝑑)))
3837rspceeqv 3645 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 ∈ (0...(𝐾 − 1)) ∧ (𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑚 · 𝑑))) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑)))
3935, 38mpan2 691 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ (0...(𝐾 − 1)) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑)))
4039adantl 481 . . . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → 𝐾 ∈ ℕ0)
4342adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐾 ∈ ℕ0)
44 vdwapval 17011 . . . . . . . . . . . . . . . . . . 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 1373 . . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎 + (𝑚 · 𝑑)) ∈ (𝑎(AP‘𝐾)𝑑))
4734, 46sseldd 3984 . . . . . . . . . . . . . . . 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 7104 . . . . . . . . . . . . . . . . . . . . 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 7134 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐺:(1...𝑊)⟶𝑅)
5352ffnd 6737 . . . . . . . . . . . . . . . . . 18 (𝜑𝐺 Fn (1...𝑊))
5453ad3antrrr 730 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐺 Fn (1...𝑊))
55 fniniseg 7080 . . . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . . 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 494 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊))
5917, 33, 58rspcdva 3623 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + (𝑚 · 𝑑)) + 𝑁) ∈ (1...𝑀))
6015, 59eqeltrd 2841 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + 𝑁) + (𝑚 · 𝑑)) ∈ (1...𝑀))
6115fveq2d 6910 . . . . . . . . . . . . 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 6910 . . . . . . . . . . . . . . 15 (𝑥 = (𝑎 + (𝑚 · 𝑑)) → (𝐹‘(𝑥 + 𝑁)) = (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)))
63 fvex 6919 . . . . . . . . . . . . . . 15 (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)) ∈ V
6462, 51, 63fvmpt 7016 . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐺‘(𝑎 + (𝑚 · 𝑑))) = 𝑐)
6761, 65, 663eqtr2d 2783 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = 𝑐)
6860, 67jca 511 . . . . . . . . . . 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 2829 . . . . . . . . . . . 12 (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → (𝑥 ∈ (1...𝑀) ↔ ((𝑎 + 𝑁) + (𝑚 · 𝑑)) ∈ (1...𝑀)))
70 fveqeq2 6915 . . . . . . . . . . . 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 247 . . . . . . . . . 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 3155 . . . . . . . . 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 480 . . . . . . . . . 10 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → (𝑎 + 𝑁) ∈ ℕ)
75 simprl 771 . . . . . . . . . 10 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → 𝑑 ∈ ℕ)
76 vdwapval 17011 . . . . . . . . . 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 1373 . . . . . . . . 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 6737 . . . . . . . . . . 11 (𝜑𝐹 Fn (1...𝑀))
7978ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → 𝐹 Fn (1...𝑀))
80 fniniseg 7080 . . . . . . . . . 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 294 . . . . . . . 8 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → (𝑥 ∈ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) → 𝑥 ∈ (𝐹 “ {𝑐})))
8382ssrdv 3989 . . . . . . 7 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))
8483expr 456 . . . . . 6 (((𝜑𝑎 ∈ ℕ) ∧ 𝑑 ∈ ℕ) → ((𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}) → ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
8584reximdva 3168 . . . . 5 ((𝜑𝑎 ∈ ℕ) → (∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}) → ∃𝑑 ∈ ℕ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
86 oveq1 7438 . . . . . . . 8 (𝑏 = (𝑎 + 𝑁) → (𝑏(AP‘𝐾)𝑑) = ((𝑎 + 𝑁)(AP‘𝐾)𝑑))
8786sseq1d 4015 . . . . . . 7 (𝑏 = (𝑎 + 𝑁) → ((𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
8887rexbidv 3179 . . . . . 6 (𝑏 = (𝑎 + 𝑁) → (∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ ∃𝑑 ∈ ℕ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
8988rspcev 3622 . . . . 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 684 . . . 4 ((𝜑𝑎 ∈ ℕ) → (∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}) → ∃𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
9190rexlimdva 3155 . . 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 7464 . . 3 (1...𝑊) ∈ V
9493, 41, 52vdwmc 17016 . 2 (𝜑 → (𝐾 MonoAP 𝐺 ↔ ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐})))
95 ovex 7464 . . 3 (1...𝑀) ∈ V
9695, 41, 48vdwmc 17016 . 2 (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
9792, 94, 963imtr4d 294 1 (𝜑 → (𝐾 MonoAP 𝐺𝐾 MonoAP 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wral 3061  wrex 3070  wss 3951  {csn 4626   class class class wbr 5143  cmpt 5225  ccnv 5684  cima 5688   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  Fincfn 8985  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160  cmin 11492  cn 12266  0cn0 12526  cz 12613  cuz 12878  ...cfz 13547  APcvdwa 17003   MonoAP cvdwm 17004
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-vdwap 17006  df-vdwmc 17007
This theorem is referenced by:  vdwlem9  17027
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