vdwlem2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > vdwlem2		Structured version Visualization version GIF version

Theorem vdwlem2 17016

Description: Lemma for vdw 17028. (Contributed by Mario Carneiro, 12-Sep-2014.)

**Hypotheses**
Ref	Expression
vdwlem2.r	⊢ (𝜑 → 𝑅 ∈ Fin)
vdwlem2.k	⊢ (𝜑 → 𝐾 ∈ ℕ₀)
vdwlem2.w	⊢ (𝜑 → 𝑊 ∈ ℕ)
vdwlem2.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
vdwlem2.f	⊢ (𝜑 → 𝐹:(1...𝑀)⟶𝑅)
vdwlem2.m	⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘(𝑊 + 𝑁)))
vdwlem2.g	⊢ 𝐺 = (𝑥 ∈ (1...𝑊) ↦ (𝐹‘(𝑥 + 𝑁)))

**Assertion**
Ref	Expression
vdwlem2	⊢ (𝜑 → (𝐾 MonoAP 𝐺 → 𝐾 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.

Step	Hyp	Ref	Expression
1		id 22	. . . . . 6 ⊢ (𝑎 ∈ ℕ → 𝑎 ∈ ℕ)
2		vdwlem2.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
3		nnaddcl 12287	. . . . . 6 ⊢ ((𝑎 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑎 + 𝑁) ∈ ℕ)
4	1, 2, 3	syl2anr 597	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → (𝑎 + 𝑁) ∈ ℕ)
5		simpllr 776	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑎 ∈ ℕ)
6	5	nncnd 12280	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑎 ∈ ℂ)
7	2	ad3antrrr 730	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑁 ∈ ℕ)
8	7	nncnd 12280	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑁 ∈ ℂ)
9		elfznn0 13657	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (0...(𝐾 − 1)) → 𝑚 ∈ ℕ₀)
10	9	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℕ₀)
11	10	nn0cnd 12587	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℂ)
12		simplrl 777	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑑 ∈ ℕ)
13	12	nncnd 12280	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑑 ∈ ℂ)
14	11, 13	mulcld 11279	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · 𝑑) ∈ ℂ)
15	6, 8, 14	add32d 11487	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ 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))
17	16	eleq1d 2824	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑎 + (𝑚 · 𝑑)) → ((𝑥 + 𝑁) ∈ (1...𝑀) ↔ ((𝑎 + (𝑚 · 𝑑)) + 𝑁) ∈ (1...𝑀)))
18		elfznn 13590	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (1...𝑊) → 𝑥 ∈ ℕ)
19		nnaddcl 12287	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑥 + 𝑁) ∈ ℕ)
20	18, 2, 19	syl2anr 597	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑊)) → (𝑥 + 𝑁) ∈ ℕ)
21		nnuz 12919	. . . . . . . . . . . . . . . . . 18 ⊢ ℕ = (ℤ_≥‘1)
22	20, 21	eleqtrdi 2849	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑊)) → (𝑥 + 𝑁) ∈ (ℤ_≥‘1))
23		vdwlem2.m	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘(𝑊 + 𝑁)))
24		elfzuz3 13558	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (1...𝑊) → 𝑊 ∈ (ℤ_≥‘𝑥))
25	2	nnzd 12638	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℤ)
26		eluzadd 12905	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑊 ∈ (ℤ_≥‘𝑥) ∧ 𝑁 ∈ ℤ) → (𝑊 + 𝑁) ∈ (ℤ_≥‘(𝑥 + 𝑁)))
27	24, 25, 26	syl2anr 597	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑊)) → (𝑊 + 𝑁) ∈ (ℤ_≥‘(𝑥 + 𝑁)))
28		uztrn 12894	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ (ℤ_≥‘(𝑊 + 𝑁)) ∧ (𝑊 + 𝑁) ∈ (ℤ_≥‘(𝑥 + 𝑁))) → 𝑀 ∈ (ℤ_≥‘(𝑥 + 𝑁)))
29	23, 27, 28	syl2an2r 685	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑊)) → 𝑀 ∈ (ℤ_≥‘(𝑥 + 𝑁)))
30		elfzuzb 13555	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 + 𝑁) ∈ (1...𝑀) ↔ ((𝑥 + 𝑁) ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ (ℤ_≥‘(𝑥 + 𝑁))))
31	22, 29, 30	sylanbrc 583	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑊)) → (𝑥 + 𝑁) ∈ (1...𝑀))
32	31	ralrimiva 3144	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ (1...𝑊)(𝑥 + 𝑁) ∈ (1...𝑀))
33	32	ad3antrrr 730	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ∀𝑥 ∈ (1...𝑊)(𝑥 + 𝑁) ∈ (1...𝑀))
34		simplrr 778	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))
35		eqid 2735	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑚 · 𝑑))
36		oveq1 7438	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑚 → (𝑛 · 𝑑) = (𝑚 · 𝑑))
37	36	oveq2d 7447	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑚 → (𝑎 + (𝑛 · 𝑑)) = (𝑎 + (𝑚 · 𝑑)))
38	37	rspceeqv 3645	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ (0...(𝐾 − 1)) ∧ (𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑚 · 𝑑))) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑)))
39	35, 38	mpan2 691	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ (0...(𝐾 − 1)) → ∃𝑛 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑)))
40	39	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 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 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
42	41	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) → 𝐾 ∈ ℕ₀)
43	42	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐾 ∈ ℕ₀)
44		vdwapval 17007	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → ((𝑎 + (𝑚 · 𝑑)) ∈ (𝑎(AP‘𝐾)𝑑) ↔ ∃𝑛 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) = (𝑎 + (𝑛 · 𝑑))))
45	43, 5, 12, 44	syl3anc 1370	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 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))))
46	40, 45	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎 + (𝑚 · 𝑑)) ∈ (𝑎(AP‘𝐾)𝑑))
47	34, 46	sseldd 3996	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎 + (𝑚 · 𝑑)) ∈ (^◡𝐺 “ {𝑐}))
48		vdwlem2.f	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐹:(1...𝑀)⟶𝑅)
49	48	ffvelcdmda 7104	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑥 + 𝑁) ∈ (1...𝑀)) → (𝐹‘(𝑥 + 𝑁)) ∈ 𝑅)
50	31, 49	syldan 591	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑊)) → (𝐹‘(𝑥 + 𝑁)) ∈ 𝑅)
51		vdwlem2.g	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐺 = (𝑥 ∈ (1...𝑊) ↦ (𝐹‘(𝑥 + 𝑁)))
52	50, 51	fmptd 7134	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐺:(1...𝑊)⟶𝑅)
53	52	ffnd 6738	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐺 Fn (1...𝑊))
54	53	ad3antrrr 730	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ 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)))
56	54, 55	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ 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)))
57	47, 56	mpbid 232	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊) ∧ (𝐺‘(𝑎 + (𝑚 · 𝑑))) = 𝑐))
58	57	simpld 494	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊))
59	17, 33, 58	rspcdva 3623	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + (𝑚 · 𝑑)) + 𝑁) ∈ (1...𝑀))
60	15, 59	eqeltrd 2839	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑎 + 𝑁) + (𝑚 · 𝑑)) ∈ (1...𝑀))
61	15	fveq2d 6911	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)))
62	16	fveq2d 6911	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑎 + (𝑚 · 𝑑)) → (𝐹‘(𝑥 + 𝑁)) = (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)))
63		fvex 6920	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)) ∈ V
64	62, 51, 63	fvmpt 7016	. . . . . . . . . . . . . 14 ⊢ ((𝑎 + (𝑚 · 𝑑)) ∈ (1...𝑊) → (𝐺‘(𝑎 + (𝑚 · 𝑑))) = (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)))
65	58, 64	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐺‘(𝑎 + (𝑚 · 𝑑))) = (𝐹‘((𝑎 + (𝑚 · 𝑑)) + 𝑁)))
66	57	simprd 495	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐺‘(𝑎 + (𝑚 · 𝑑))) = 𝑐)
67	61, 65, 66	3eqtr2d 2781	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = 𝑐)
68	60, 67	jca 511	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ 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 2827	. . . . . . . . . . . 12 ⊢ (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → (𝑥 ∈ (1...𝑀) ↔ ((𝑎 + 𝑁) + (𝑚 · 𝑑)) ∈ (1...𝑀)))
70		fveqeq2 6916	. . . . . . . . . . . 12 ⊢ (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → ((𝐹‘𝑥) = 𝑐 ↔ (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = 𝑐))
71	69, 70	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → ((𝑥 ∈ (1...𝑀) ∧ (𝐹‘𝑥) = 𝑐) ↔ (((𝑎 + 𝑁) + (𝑚 · 𝑑)) ∈ (1...𝑀) ∧ (𝐹‘((𝑎 + 𝑁) + (𝑚 · 𝑑))) = 𝑐)))
72	68, 71	syl5ibrcom 247	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → (𝑥 ∈ (1...𝑀) ∧ (𝐹‘𝑥) = 𝑐)))
73	72	rexlimdva 3153	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) → (∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑)) → (𝑥 ∈ (1...𝑀) ∧ (𝐹‘𝑥) = 𝑐)))
74	4	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) → (𝑎 + 𝑁) ∈ ℕ)
75		simprl 771	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) → 𝑑 ∈ ℕ)
76		vdwapval 17007	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℕ₀ ∧ (𝑎 + 𝑁) ∈ ℕ ∧ 𝑑 ∈ ℕ) → (𝑥 ∈ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑))))
77	42, 74, 75, 76	syl3anc 1370	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) → (𝑥 ∈ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝑎 + 𝑁) + (𝑚 · 𝑑))))
78	48	ffnd 6738	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn (1...𝑀))
79	78	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 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) ∧ (𝐹‘𝑥) = 𝑐)))
81	79, 80	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) → (𝑥 ∈ (^◡𝐹 “ {𝑐}) ↔ (𝑥 ∈ (1...𝑀) ∧ (𝐹‘𝑥) = 𝑐)))
82	73, 77, 81	3imtr4d 294	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) → (𝑥 ∈ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) → 𝑥 ∈ (^◡𝐹 “ {𝑐})))
83	82	ssrdv 4001	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ (𝑑 ∈ ℕ ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}))) → ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐}))
84	83	expr 456	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℕ) ∧ 𝑑 ∈ ℕ) → ((𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}) → ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐})))
85	84	reximdva 3166	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 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))
87	86	sseq1d 4027	. . . . . . 7 ⊢ (𝑏 = (𝑎 + 𝑁) → ((𝑏(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐}) ↔ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐})))
88	87	rexbidv 3177	. . . . . 6 ⊢ (𝑏 = (𝑎 + 𝑁) → (∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐}) ↔ ∃𝑑 ∈ ℕ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐})))
89	88	rspcev 3622	. . . . 5 ⊢ (((𝑎 + 𝑁) ∈ ℕ ∧ ∃𝑑 ∈ ℕ ((𝑎 + 𝑁)(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐})) → ∃𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐}))
90	4, 85, 89	syl6an 684	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → (∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}) → ∃𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐})))
91	90	rexlimdva 3153	. . 3 ⊢ (𝜑 → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}) → ∃𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐})))
92	91	eximdv 1915	. 2 ⊢ (𝜑 → (∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐}) → ∃𝑐∃𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐})))
93		ovex 7464	. . 3 ⊢ (1...𝑊) ∈ V
94	93, 41, 52	vdwmc 17012	. 2 ⊢ (𝜑 → (𝐾 MonoAP 𝐺 ↔ ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐺 “ {𝑐})))
95		ovex 7464	. . 3 ⊢ (1...𝑀) ∈ V
96	95, 41, 48	vdwmc 17012	. 2 ⊢ (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑐∃𝑏 ∈ ℕ ∃𝑑 ∈ ℕ (𝑏(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑐})))
97	92, 94, 96	3imtr4d 294	1 ⊢ (𝜑 → (𝐾 MonoAP 𝐺 → 𝐾 MonoAP 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 {csn 4631 class class class wbr 5148 ↦ cmpt 5231 ^◡ccnv 5688 “ cima 5692 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 − cmin 11490 ℕcn 12264 ℕ₀cn0 12524 ℤcz 12611 ℤ_≥cuz 12876 ...cfz 13544 APcvdwa 16999 MonoAP cvdwm 17000

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-vdwap 17002 df-vdwmc 17003

This theorem is referenced by: vdwlem9 17023

W3C validator