vdwlem3 - Metamath Proof Explorer

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Theorem vdwlem3 17021

Description: Lemma for vdw 17032. (Contributed by Mario Carneiro, 13-Sep-2014.)

**Hypotheses**
Ref	Expression
vdwlem3.v	⊢ (𝜑 → 𝑉 ∈ ℕ)
vdwlem3.w	⊢ (𝜑 → 𝑊 ∈ ℕ)
vdwlem3.a	⊢ (𝜑 → 𝐴 ∈ (1...𝑉))
vdwlem3.b	⊢ (𝜑 → 𝐵 ∈ (1...𝑊))

**Assertion**
Ref	Expression
vdwlem3	⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ (1...(𝑊 · (2 · 𝑉))))

**Proof of Theorem vdwlem3**
Step	Hyp	Ref	Expression
1		vdwlem3.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (1...𝑊))
2		elfznn 13593	. . . . . 6 ⊢ (𝐵 ∈ (1...𝑊) → 𝐵 ∈ ℕ)
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℕ)
4		vdwlem3.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℕ)
5		vdwlem3.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (1...𝑉))
6		elfznn 13593	. . . . . . . . 9 ⊢ (𝐴 ∈ (1...𝑉) → 𝐴 ∈ ℕ)
7	5, 6	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℕ)
8		nnm1nn0 12567	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (𝐴 − 1) ∈ ℕ₀)
9	7, 8	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐴 − 1) ∈ ℕ₀)
10		vdwlem3.v	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ ℕ)
11		nn0nnaddcl 12557	. . . . . . 7 ⊢ (((𝐴 − 1) ∈ ℕ₀ ∧ 𝑉 ∈ ℕ) → ((𝐴 − 1) + 𝑉) ∈ ℕ)
12	9, 10, 11	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((𝐴 − 1) + 𝑉) ∈ ℕ)
13	4, 12	nnmulcld 12319	. . . . 5 ⊢ (𝜑 → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℕ)
14	3, 13	nnaddcld 12318	. . . 4 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ ℕ)
15	14	nnred 12281	. . 3 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ ℝ)
16	7, 10	nnaddcld 12318	. . . . 5 ⊢ (𝜑 → (𝐴 + 𝑉) ∈ ℕ)
17	4, 16	nnmulcld 12319	. . . 4 ⊢ (𝜑 → (𝑊 · (𝐴 + 𝑉)) ∈ ℕ)
18	17	nnred 12281	. . 3 ⊢ (𝜑 → (𝑊 · (𝐴 + 𝑉)) ∈ ℝ)
19		2nn 12339	. . . . . 6 ⊢ 2 ∈ ℕ
20		nnmulcl 12290	. . . . . 6 ⊢ ((2 ∈ ℕ ∧ 𝑉 ∈ ℕ) → (2 · 𝑉) ∈ ℕ)
21	19, 10, 20	sylancr 587	. . . . 5 ⊢ (𝜑 → (2 · 𝑉) ∈ ℕ)
22	4, 21	nnmulcld 12319	. . . 4 ⊢ (𝜑 → (𝑊 · (2 · 𝑉)) ∈ ℕ)
23	22	nnred 12281	. . 3 ⊢ (𝜑 → (𝑊 · (2 · 𝑉)) ∈ ℝ)
24		elfzle2 13568	. . . . . 6 ⊢ (𝐵 ∈ (1...𝑊) → 𝐵 ≤ 𝑊)
25	1, 24	syl 17	. . . . 5 ⊢ (𝜑 → 𝐵 ≤ 𝑊)
26		nnre 12273	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
27		nnre 12273	. . . . . . 7 ⊢ (𝑊 ∈ ℕ → 𝑊 ∈ ℝ)
28		nnre 12273	. . . . . . 7 ⊢ ((𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℕ → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℝ)
29		leadd1 11731	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝑊 ∈ ℝ ∧ (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℝ) → (𝐵 ≤ 𝑊 ↔ (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ≤ (𝑊 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
30	26, 27, 28, 29	syl3an 1161	. . . . . 6 ⊢ ((𝐵 ∈ ℕ ∧ 𝑊 ∈ ℕ ∧ (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℕ) → (𝐵 ≤ 𝑊 ↔ (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ≤ (𝑊 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
31	3, 4, 13, 30	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝐵 ≤ 𝑊 ↔ (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ≤ (𝑊 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
32	25, 31	mpbid 232	. . . 4 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ≤ (𝑊 + (𝑊 · ((𝐴 − 1) + 𝑉))))
33	4	nncnd 12282	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℂ)
34		1cnd 11256	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
35	12	nncnd 12282	. . . . . 6 ⊢ (𝜑 → ((𝐴 − 1) + 𝑉) ∈ ℂ)
36	33, 34, 35	adddid 11285	. . . . 5 ⊢ (𝜑 → (𝑊 · (1 + ((𝐴 − 1) + 𝑉))) = ((𝑊 · 1) + (𝑊 · ((𝐴 − 1) + 𝑉))))
37	9	nn0cnd 12589	. . . . . . . 8 ⊢ (𝜑 → (𝐴 − 1) ∈ ℂ)
38	10	nncnd 12282	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ ℂ)
39	34, 37, 38	addassd 11283	. . . . . . 7 ⊢ (𝜑 → ((1 + (𝐴 − 1)) + 𝑉) = (1 + ((𝐴 − 1) + 𝑉)))
40		ax-1cn 11213	. . . . . . . . 9 ⊢ 1 ∈ ℂ
41	7	nncnd 12282	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
42		pncan3 11516	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 + (𝐴 − 1)) = 𝐴)
43	40, 41, 42	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (1 + (𝐴 − 1)) = 𝐴)
44	43	oveq1d 7446	. . . . . . 7 ⊢ (𝜑 → ((1 + (𝐴 − 1)) + 𝑉) = (𝐴 + 𝑉))
45	39, 44	eqtr3d 2779	. . . . . 6 ⊢ (𝜑 → (1 + ((𝐴 − 1) + 𝑉)) = (𝐴 + 𝑉))
46	45	oveq2d 7447	. . . . 5 ⊢ (𝜑 → (𝑊 · (1 + ((𝐴 − 1) + 𝑉))) = (𝑊 · (𝐴 + 𝑉)))
47	33	mulridd 11278	. . . . . 6 ⊢ (𝜑 → (𝑊 · 1) = 𝑊)
48	47	oveq1d 7446	. . . . 5 ⊢ (𝜑 → ((𝑊 · 1) + (𝑊 · ((𝐴 − 1) + 𝑉))) = (𝑊 + (𝑊 · ((𝐴 − 1) + 𝑉))))
49	36, 46, 48	3eqtr3d 2785	. . . 4 ⊢ (𝜑 → (𝑊 · (𝐴 + 𝑉)) = (𝑊 + (𝑊 · ((𝐴 − 1) + 𝑉))))
50	32, 49	breqtrrd 5171	. . 3 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ≤ (𝑊 · (𝐴 + 𝑉)))
51	7	nnred 12281	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
52	10	nnred 12281	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ ℝ)
53		elfzle2 13568	. . . . . . 7 ⊢ (𝐴 ∈ (1...𝑉) → 𝐴 ≤ 𝑉)
54	5, 53	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐴 ≤ 𝑉)
55	51, 52, 52, 54	leadd1dd 11877	. . . . 5 ⊢ (𝜑 → (𝐴 + 𝑉) ≤ (𝑉 + 𝑉))
56	38	2timesd 12509	. . . . 5 ⊢ (𝜑 → (2 · 𝑉) = (𝑉 + 𝑉))
57	55, 56	breqtrrd 5171	. . . 4 ⊢ (𝜑 → (𝐴 + 𝑉) ≤ (2 · 𝑉))
58	16	nnred 12281	. . . . 5 ⊢ (𝜑 → (𝐴 + 𝑉) ∈ ℝ)
59	21	nnred 12281	. . . . 5 ⊢ (𝜑 → (2 · 𝑉) ∈ ℝ)
60	4	nnred 12281	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ ℝ)
61	4	nngt0d 12315	. . . . 5 ⊢ (𝜑 → 0 < 𝑊)
62		lemul2 12120	. . . . 5 ⊢ (((𝐴 + 𝑉) ∈ ℝ ∧ (2 · 𝑉) ∈ ℝ ∧ (𝑊 ∈ ℝ ∧ 0 < 𝑊)) → ((𝐴 + 𝑉) ≤ (2 · 𝑉) ↔ (𝑊 · (𝐴 + 𝑉)) ≤ (𝑊 · (2 · 𝑉))))
63	58, 59, 60, 61, 62	syl112anc 1376	. . . 4 ⊢ (𝜑 → ((𝐴 + 𝑉) ≤ (2 · 𝑉) ↔ (𝑊 · (𝐴 + 𝑉)) ≤ (𝑊 · (2 · 𝑉))))
64	57, 63	mpbid 232	. . 3 ⊢ (𝜑 → (𝑊 · (𝐴 + 𝑉)) ≤ (𝑊 · (2 · 𝑉)))
65	15, 18, 23, 50, 64	letrd 11418	. 2 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ≤ (𝑊 · (2 · 𝑉)))
66		nnuz 12921	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
67	14, 66	eleqtrdi 2851	. . 3 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ (ℤ_≥‘1))
68	22	nnzd 12640	. . 3 ⊢ (𝜑 → (𝑊 · (2 · 𝑉)) ∈ ℤ)
69		elfz5 13556	. . 3 ⊢ (((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ (ℤ_≥‘1) ∧ (𝑊 · (2 · 𝑉)) ∈ ℤ) → ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ (1...(𝑊 · (2 · 𝑉))) ↔ (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ≤ (𝑊 · (2 · 𝑉))))
70	67, 68, 69	syl2anc 584	. 2 ⊢ (𝜑 → ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ (1...(𝑊 · (2 · 𝑉))) ↔ (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ≤ (𝑊 · (2 · 𝑉))))
71	65, 70	mpbird 257	1 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ (1...(𝑊 · (2 · 𝑉))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 ≤ cle 11296 − cmin 11492 ℕcn 12266 2c2 12321 ℕ₀cn0 12526 ℤcz 12613 ℤ_≥cuz 12878 ...cfz 13547

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-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-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548

This theorem is referenced by: vdwlem4 17022 vdwlem6 17024

W3C validator