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Theorem vdwlem2 15610
Description: Lemma for vdw 15622. (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 10986 . . . . . 6 ((𝑎 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑎 + 𝑁) ∈ ℕ)
41, 2, 3syl2anr 495 . . . . 5 ((𝜑𝑎 ∈ ℕ) → (𝑎 + 𝑁) ∈ ℕ)
5 simpllr 798 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑎 ∈ ℕ)
65nncnd 10980 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑎 ∈ ℂ)
72ad3antrrr 765 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑁 ∈ ℕ)
87nncnd 10980 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑁 ∈ ℂ)
9 elfznn0 12374 . . . . . . . . . . . . . . . . 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 11297 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℂ)
12 simplrl 799 . . . . . . . . . . . . . . . 16 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑑 ∈ ℕ)
1312nncnd 10980 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑑 ∈ ℂ)
1411, 13mulcld 10004 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · 𝑑) ∈ ℂ)
156, 8, 14add32d 10207 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + 𝑁) + (𝑚 · 𝑑)) = ((𝑎 + (𝑚 · 𝑑)) + 𝑁))
16 simplrr 800 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))
17 eqid 2621 . . . . . . . . . . . . . . . . . . . 20 (𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑚 · 𝑑))
18 oveq1 6611 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑚 → (𝑛 · 𝑑) = (𝑚 · 𝑑))
1918oveq2d 6620 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑚 → (𝑎 + (𝑛 · 𝑑)) = (𝑎 + (𝑚 · 𝑑)))
2019eqeq2d 2631 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚 → ((𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑)) ↔ (𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑚 · 𝑑))))
2120rspcev 3295 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 ∈ (0...(𝐾 − 1)) ∧ (𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑚 · 𝑑))) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑)))
2217, 21mpan2 706 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ (0...(𝐾 − 1)) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑)))
2322adantl 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)))
24 vdwlem2.k . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐾 ∈ ℕ0)
2524ad2antrr 761 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → 𝐾 ∈ ℕ0)
2625adantr 481 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐾 ∈ ℕ0)
27 vdwapval 15601 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ ℕ0𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → ((𝑎 + (𝑚 · 𝑑)) ∈ (𝑎(AP‘𝐾)𝑑) ↔ ∃𝑛 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑))))
2826, 5, 12, 27syl3anc 1323 . . . . . . . . . . . . . . . . . 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))))
2923, 28mpbird 247 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎 + (𝑚 · 𝑑)) ∈ (𝑎(AP‘𝐾)𝑑))
3016, 29sseldd 3584 . . . . . . . . . . . . . . . 16 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎 + (𝑚 · 𝑑)) ∈ (𝐺 “ {𝑐}))
31 elfznn 12312 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ (1...𝑊) → 𝑥 ∈ ℕ)
32 nnaddcl 10986 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑥 + 𝑁) ∈ ℕ)
3331, 2, 32syl2anr 495 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑥 ∈ (1...𝑊)) → (𝑥 + 𝑁) ∈ ℕ)
34 nnuz 11667 . . . . . . . . . . . . . . . . . . . . . . 23 ℕ = (ℤ‘1)
3533, 34syl6eleq 2708 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑥 ∈ (1...𝑊)) → (𝑥 + 𝑁) ∈ (ℤ‘1))
36 vdwlem2.m . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑀 ∈ (ℤ‘(𝑊 + 𝑁)))
3736adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑥 ∈ (1...𝑊)) → 𝑀 ∈ (ℤ‘(𝑊 + 𝑁)))
38 elfzuz3 12281 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ (1...𝑊) → 𝑊 ∈ (ℤ𝑥))
392nnzd 11425 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑁 ∈ ℤ)
40 eluzadd 11660 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑊 ∈ (ℤ𝑥) ∧ 𝑁 ∈ ℤ) → (𝑊 + 𝑁) ∈ (ℤ‘(𝑥 + 𝑁)))
4138, 39, 40syl2anr 495 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑥 ∈ (1...𝑊)) → (𝑊 + 𝑁) ∈ (ℤ‘(𝑥 + 𝑁)))
42 uztrn 11648 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑀 ∈ (ℤ‘(𝑊 + 𝑁)) ∧ (𝑊 + 𝑁) ∈ (ℤ‘(𝑥 + 𝑁))) → 𝑀 ∈ (ℤ‘(𝑥 + 𝑁)))
4337, 41, 42syl2anc 692 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑥 ∈ (1...𝑊)) → 𝑀 ∈ (ℤ‘(𝑥 + 𝑁)))
44 elfzuzb 12278 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 + 𝑁) ∈ (1...𝑀) ↔ ((𝑥 + 𝑁) ∈ (ℤ‘1) ∧ 𝑀 ∈ (ℤ‘(𝑥 + 𝑁))))
4535, 43, 44sylanbrc 697 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑥 ∈ (1...𝑊)) → (𝑥 + 𝑁) ∈ (1...𝑀))
46 vdwlem2.f . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐹:(1...𝑀)⟶𝑅)
4746ffvelrnda 6315 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑥 + 𝑁) ∈ (1...𝑀)) → (𝐹‘(𝑥 + 𝑁)) ∈ 𝑅)
4845, 47syldan 487 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ (1...𝑊)) → (𝐹‘(𝑥 + 𝑁)) ∈ 𝑅)
49 vdwlem2.g . . . . . . . . . . . . . . . . . . . 20 𝐺 = (𝑥 ∈ (1...𝑊) ↦ (𝐹‘(𝑥 + 𝑁)))
5048, 49fmptd 6340 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐺:(1...𝑊)⟶𝑅)
51 ffn 6002 . . . . . . . . . . . . . . . . . . 19 (𝐺:(1...𝑊)⟶𝑅𝐺 Fn (1...𝑊))
5250, 51syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑𝐺 Fn (1...𝑊))
5352ad3antrrr 765 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐺 Fn (1...𝑊))
54 fniniseg 6294 . . . . . . . . . . . . . . . . 17 (𝐺 Fn (1...𝑊) → ((𝑎 + (𝑚 · 𝑑)) ∈ (𝐺 “ {𝑐}) ↔ ((𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊) ∧ (𝐺‘(𝑎 + (𝑚 · 𝑑))) = 𝑐)))
5553, 54syl 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)))
5630, 55mpbid 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))
5756simpld 475 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊))
5845ralrimiva 2960 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑥 ∈ (1...𝑊)(𝑥 + 𝑁) ∈ (1...𝑀))
5958ad3antrrr 765 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ∀𝑥 ∈ (1...𝑊)(𝑥 + 𝑁) ∈ (1...𝑀))
60 oveq1 6611 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑎 + (𝑚 · 𝑑)) → (𝑥 + 𝑁) = ((𝑎 + (𝑚 · 𝑑)) + 𝑁))
6160eleq1d 2683 . . . . . . . . . . . . . . 15 (𝑥 = (𝑎 + (𝑚 · 𝑑)) → ((𝑥 + 𝑁) ∈ (1...𝑀) ↔ ((𝑎 + (𝑚 · 𝑑)) + 𝑁) ∈ (1...𝑀)))
6261rspcv 3291 . . . . . . . . . . . . . 14 ((𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊) → (∀𝑥 ∈ (1...𝑊)(𝑥 + 𝑁) ∈ (1...𝑀) → ((𝑎 + (𝑚 · 𝑑)) + 𝑁) ∈ (1...𝑀)))
6357, 59, 62sylc 65 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + (𝑚 · 𝑑)) + 𝑁) ∈ (1...𝑀))
6415, 63eqeltrd 2698 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + 𝑁) + (𝑚 · 𝑑)) ∈ (1...𝑀))
6515fveq2d 6152 . . . . . . . . . . . . 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)))
6660fveq2d 6152 . . . . . . . . . . . . . . 15 (𝑥 = (𝑎 + (𝑚 · 𝑑)) → (𝐹‘(𝑥 + 𝑁)) = (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)))
67 fvex 6158 . . . . . . . . . . . . . . 15 (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)) ∈ V
6866, 49, 67fvmpt 6239 . . . . . . . . . . . . . 14 ((𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊) → (𝐺‘(𝑎 + (𝑚 · 𝑑))) = (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)))
6957, 68syl 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)))
7056simprd 479 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐺‘(𝑎 + (𝑚 · 𝑑))) = 𝑐)
7165, 69, 703eqtr2d 2661 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = 𝑐)
7264, 71jca 554 . . . . . . . . . . 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))
73 eleq1 2686 . . . . . . . . . . . 12 (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → (𝑥 ∈ (1...𝑀) ↔ ((𝑎 + 𝑁) + (𝑚 · 𝑑)) ∈ (1...𝑀)))
74 fveq2 6148 . . . . . . . . . . . . 13 (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → (𝐹𝑥) = (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))))
7574eqeq1d 2623 . . . . . . . . . . . 12 (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → ((𝐹𝑥) = 𝑐 ↔ (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = 𝑐))
7673, 75anbi12d 746 . . . . . . . . . . 11 (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → ((𝑥 ∈ (1...𝑀) ∧ (𝐹𝑥) = 𝑐) ↔ (((𝑎 + 𝑁) + (𝑚 · 𝑑)) ∈ (1...𝑀) ∧ (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = 𝑐)))
7772, 76syl5ibrcom 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)))
7877rexlimdva 3024 . . . . . . . . 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)))
794adantr 481 . . . . . . . . . 10 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → (𝑎 + 𝑁) ∈ ℕ)
80 simprl 793 . . . . . . . . . 10 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → 𝑑 ∈ ℕ)
81 vdwapval 15601 . . . . . . . . . 10 ((𝐾 ∈ ℕ0 ∧ (𝑎 + 𝑁) ∈ ℕ ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑))))
8225, 79, 80, 81syl3anc 1323 . . . . . . . . 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))))
83 ffn 6002 . . . . . . . . . . . 12 (𝐹:(1...𝑀)⟶𝑅𝐹 Fn (1...𝑀))
8446, 83syl 17 . . . . . . . . . . 11 (𝜑𝐹 Fn (1...𝑀))
8584ad2antrr 761 . . . . . . . . . 10 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → 𝐹 Fn (1...𝑀))
86 fniniseg 6294 . . . . . . . . . 10 (𝐹 Fn (1...𝑀) → (𝑥 ∈ (𝐹 “ {𝑐}) ↔ (𝑥 ∈ (1...𝑀) ∧ (𝐹𝑥) = 𝑐)))
8785, 86syl 17 . . . . . . . . 9 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → (𝑥 ∈ (𝐹 “ {𝑐}) ↔ (𝑥 ∈ (1...𝑀) ∧ (𝐹𝑥) = 𝑐)))
8878, 82, 873imtr4d 283 . . . . . . . 8 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → (𝑥 ∈ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) → 𝑥 ∈ (𝐹 “ {𝑐})))
8988ssrdv 3589 . . . . . . 7 (((𝜑𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}))) → ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))
9089expr 642 . . . . . 6 (((𝜑𝑎 ∈ ℕ) ∧ 𝑑 ∈ ℕ) → ((𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}) → ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
9190reximdva 3011 . . . . 5 ((𝜑𝑎 ∈ ℕ) → (∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}) → ∃𝑑 ∈ ℕ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
92 oveq1 6611 . . . . . . . 8 (𝑏 = (𝑎 + 𝑁) → (𝑏(AP‘𝐾)𝑑) = ((𝑎 + 𝑁)(AP‘𝐾)𝑑))
9392sseq1d 3611 . . . . . . 7 (𝑏 = (𝑎 + 𝑁) → ((𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
9493rexbidv 3045 . . . . . 6 (𝑏 = (𝑎 + 𝑁) → (∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ ∃𝑑 ∈ ℕ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
9594rspcev 3295 . . . . 5 (((𝑎 + 𝑁) ∈ ℕ ∧ ∃𝑑 ∈ ℕ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})) → ∃𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐}))
964, 91, 95syl6an 567 . . . 4 ((𝜑𝑎 ∈ ℕ) → (∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}) → ∃𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
9796rexlimdva 3024 . . 3 (𝜑 → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}) → ∃𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
9897eximdv 1843 . 2 (𝜑 → (∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐}) → ∃𝑐𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
99 ovex 6632 . . 3 (1...𝑊) ∈ V
10099, 24, 50vdwmc 15606 . 2 (𝜑 → (𝐾 MonoAP 𝐺 ↔ ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐺 “ {𝑐})))
101 ovex 6632 . . 3 (1...𝑀) ∈ V
102101, 24, 46vdwmc 15606 . 2 (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑐})))
10398, 100, 1023imtr4d 283 1 (𝜑 → (𝐾 MonoAP 𝐺𝐾 MonoAP 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  wral 2907  wrex 2908  wss 3555  {csn 4148   class class class wbr 4613  cmpt 4673  ccnv 5073  cima 5077   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  Fincfn 7899  0cc0 9880  1c1 9881   + caddc 9883   · cmul 9885  cmin 10210  cn 10964  0cn0 11236  cz 11321  cuz 11631  ...cfz 12268  APcvdwa 15593   MonoAP cvdwm 15594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-vdwap 15596  df-vdwmc 15597
This theorem is referenced by:  vdwlem9  15617
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