vdwlem3 - Metamath Proof Explorer

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Theorem vdwlem3 17030

Description: Lemma for vdw 17041. (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 13613	. . . . . 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 13613	. . . . . . . . 9 ⊢ (𝐴 ∈ (1...𝑉) → 𝐴 ∈ ℕ)
7	5, 6	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℕ)
8		nnm1nn0 12594	. . . . . . . 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 12584	. . . . . . 7 ⊢ (((𝐴 − 1) ∈ ℕ₀ ∧ 𝑉 ∈ ℕ) → ((𝐴 − 1) + 𝑉) ∈ ℕ)
12	9, 10, 11	syl2anc 583	. . . . . 6 ⊢ (𝜑 → ((𝐴 − 1) + 𝑉) ∈ ℕ)
13	4, 12	nnmulcld 12346	. . . . 5 ⊢ (𝜑 → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℕ)
14	3, 13	nnaddcld 12345	. . . 4 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ ℕ)
15	14	nnred 12308	. . 3 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ ℝ)
16	7, 10	nnaddcld 12345	. . . . 5 ⊢ (𝜑 → (𝐴 + 𝑉) ∈ ℕ)
17	4, 16	nnmulcld 12346	. . . 4 ⊢ (𝜑 → (𝑊 · (𝐴 + 𝑉)) ∈ ℕ)
18	17	nnred 12308	. . 3 ⊢ (𝜑 → (𝑊 · (𝐴 + 𝑉)) ∈ ℝ)
19		2nn 12366	. . . . . 6 ⊢ 2 ∈ ℕ
20		nnmulcl 12317	. . . . . 6 ⊢ ((2 ∈ ℕ ∧ 𝑉 ∈ ℕ) → (2 · 𝑉) ∈ ℕ)
21	19, 10, 20	sylancr 586	. . . . 5 ⊢ (𝜑 → (2 · 𝑉) ∈ ℕ)
22	4, 21	nnmulcld 12346	. . . 4 ⊢ (𝜑 → (𝑊 · (2 · 𝑉)) ∈ ℕ)
23	22	nnred 12308	. . 3 ⊢ (𝜑 → (𝑊 · (2 · 𝑉)) ∈ ℝ)
24		elfzle2 13588	. . . . . 6 ⊢ (𝐵 ∈ (1...𝑊) → 𝐵 ≤ 𝑊)
25	1, 24	syl 17	. . . . 5 ⊢ (𝜑 → 𝐵 ≤ 𝑊)
26		nnre 12300	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
27		nnre 12300	. . . . . . 7 ⊢ (𝑊 ∈ ℕ → 𝑊 ∈ ℝ)
28		nnre 12300	. . . . . . 7 ⊢ ((𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℕ → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℝ)
29		leadd1 11758	. . . . . . 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 1160	. . . . . 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 1371	. . . . 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 12309	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℂ)
34		1cnd 11285	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
35	12	nncnd 12309	. . . . . 6 ⊢ (𝜑 → ((𝐴 − 1) + 𝑉) ∈ ℂ)
36	33, 34, 35	adddid 11314	. . . . 5 ⊢ (𝜑 → (𝑊 · (1 + ((𝐴 − 1) + 𝑉))) = ((𝑊 · 1) + (𝑊 · ((𝐴 − 1) + 𝑉))))
37	9	nn0cnd 12615	. . . . . . . 8 ⊢ (𝜑 → (𝐴 − 1) ∈ ℂ)
38	10	nncnd 12309	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ ℂ)
39	34, 37, 38	addassd 11312	. . . . . . 7 ⊢ (𝜑 → ((1 + (𝐴 − 1)) + 𝑉) = (1 + ((𝐴 − 1) + 𝑉)))
40		ax-1cn 11242	. . . . . . . . 9 ⊢ 1 ∈ ℂ
41	7	nncnd 12309	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
42		pncan3 11544	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 + (𝐴 − 1)) = 𝐴)
43	40, 41, 42	sylancr 586	. . . . . . . 8 ⊢ (𝜑 → (1 + (𝐴 − 1)) = 𝐴)
44	43	oveq1d 7463	. . . . . . 7 ⊢ (𝜑 → ((1 + (𝐴 − 1)) + 𝑉) = (𝐴 + 𝑉))
45	39, 44	eqtr3d 2782	. . . . . 6 ⊢ (𝜑 → (1 + ((𝐴 − 1) + 𝑉)) = (𝐴 + 𝑉))
46	45	oveq2d 7464	. . . . 5 ⊢ (𝜑 → (𝑊 · (1 + ((𝐴 − 1) + 𝑉))) = (𝑊 · (𝐴 + 𝑉)))
47	33	mulridd 11307	. . . . . 6 ⊢ (𝜑 → (𝑊 · 1) = 𝑊)
48	47	oveq1d 7463	. . . . 5 ⊢ (𝜑 → ((𝑊 · 1) + (𝑊 · ((𝐴 − 1) + 𝑉))) = (𝑊 + (𝑊 · ((𝐴 − 1) + 𝑉))))
49	36, 46, 48	3eqtr3d 2788	. . . 4 ⊢ (𝜑 → (𝑊 · (𝐴 + 𝑉)) = (𝑊 + (𝑊 · ((𝐴 − 1) + 𝑉))))
50	32, 49	breqtrrd 5194	. . 3 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ≤ (𝑊 · (𝐴 + 𝑉)))
51	7	nnred 12308	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
52	10	nnred 12308	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ ℝ)
53		elfzle2 13588	. . . . . . 7 ⊢ (𝐴 ∈ (1...𝑉) → 𝐴 ≤ 𝑉)
54	5, 53	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐴 ≤ 𝑉)
55	51, 52, 52, 54	leadd1dd 11904	. . . . 5 ⊢ (𝜑 → (𝐴 + 𝑉) ≤ (𝑉 + 𝑉))
56	38	2timesd 12536	. . . . 5 ⊢ (𝜑 → (2 · 𝑉) = (𝑉 + 𝑉))
57	55, 56	breqtrrd 5194	. . . 4 ⊢ (𝜑 → (𝐴 + 𝑉) ≤ (2 · 𝑉))
58	16	nnred 12308	. . . . 5 ⊢ (𝜑 → (𝐴 + 𝑉) ∈ ℝ)
59	21	nnred 12308	. . . . 5 ⊢ (𝜑 → (2 · 𝑉) ∈ ℝ)
60	4	nnred 12308	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ ℝ)
61	4	nngt0d 12342	. . . . 5 ⊢ (𝜑 → 0 < 𝑊)
62		lemul2 12147	. . . . 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 1374	. . . 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 11447	. 2 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ≤ (𝑊 · (2 · 𝑉)))
66		nnuz 12946	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
67	14, 66	eleqtrdi 2854	. . 3 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ (ℤ_≥‘1))
68	22	nnzd 12666	. . 3 ⊢ (𝜑 → (𝑊 · (2 · 𝑉)) ∈ ℤ)
69		elfz5 13576	. . 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 583	. 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 1537 ∈ wcel 2108 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 < clt 11324 ≤ cle 11325 − cmin 11520 ℕcn 12293 2c2 12348 ℕ₀cn0 12553 ℤcz 12639 ℤ_≥cuz 12903 ...cfz 13567

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-n0 12554 df-z 12640 df-uz 12904 df-fz 13568

This theorem is referenced by: vdwlem4 17031 vdwlem6 17033

W3C validator