vdwlem3 - Metamath Proof Explorer

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Theorem vdwlem3 16909

Description: Lemma for vdw 16920. (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 13467	. . . . . 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 13467	. . . . . . . . 9 ⊢ (𝐴 ∈ (1...𝑉) → 𝐴 ∈ ℕ)
7	5, 6	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℕ)
8		nnm1nn0 12440	. . . . . . . 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 12430	. . . . . . 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 12196	. . . . 5 ⊢ (𝜑 → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℕ)
14	3, 13	nnaddcld 12195	. . . 4 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ ℕ)
15	14	nnred 12158	. . 3 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ ℝ)
16	7, 10	nnaddcld 12195	. . . . 5 ⊢ (𝜑 → (𝐴 + 𝑉) ∈ ℕ)
17	4, 16	nnmulcld 12196	. . . 4 ⊢ (𝜑 → (𝑊 · (𝐴 + 𝑉)) ∈ ℕ)
18	17	nnred 12158	. . 3 ⊢ (𝜑 → (𝑊 · (𝐴 + 𝑉)) ∈ ℝ)
19		2nn 12216	. . . . . 6 ⊢ 2 ∈ ℕ
20		nnmulcl 12167	. . . . . 6 ⊢ ((2 ∈ ℕ ∧ 𝑉 ∈ ℕ) → (2 · 𝑉) ∈ ℕ)
21	19, 10, 20	sylancr 587	. . . . 5 ⊢ (𝜑 → (2 · 𝑉) ∈ ℕ)
22	4, 21	nnmulcld 12196	. . . 4 ⊢ (𝜑 → (𝑊 · (2 · 𝑉)) ∈ ℕ)
23	22	nnred 12158	. . 3 ⊢ (𝜑 → (𝑊 · (2 · 𝑉)) ∈ ℝ)
24		elfzle2 13442	. . . . . 6 ⊢ (𝐵 ∈ (1...𝑊) → 𝐵 ≤ 𝑊)
25	1, 24	syl 17	. . . . 5 ⊢ (𝜑 → 𝐵 ≤ 𝑊)
26		nnre 12150	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
27		nnre 12150	. . . . . . 7 ⊢ (𝑊 ∈ ℕ → 𝑊 ∈ ℝ)
28		nnre 12150	. . . . . . 7 ⊢ ((𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℕ → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℝ)
29		leadd1 11603	. . . . . . 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 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 12159	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℂ)
34		1cnd 11125	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
35	12	nncnd 12159	. . . . . 6 ⊢ (𝜑 → ((𝐴 − 1) + 𝑉) ∈ ℂ)
36	33, 34, 35	adddid 11154	. . . . 5 ⊢ (𝜑 → (𝑊 · (1 + ((𝐴 − 1) + 𝑉))) = ((𝑊 · 1) + (𝑊 · ((𝐴 − 1) + 𝑉))))
37	9	nn0cnd 12462	. . . . . . . 8 ⊢ (𝜑 → (𝐴 − 1) ∈ ℂ)
38	10	nncnd 12159	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ ℂ)
39	34, 37, 38	addassd 11152	. . . . . . 7 ⊢ (𝜑 → ((1 + (𝐴 − 1)) + 𝑉) = (1 + ((𝐴 − 1) + 𝑉)))
40		ax-1cn 11082	. . . . . . . . 9 ⊢ 1 ∈ ℂ
41	7	nncnd 12159	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
42		pncan3 11386	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 + (𝐴 − 1)) = 𝐴)
43	40, 41, 42	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (1 + (𝐴 − 1)) = 𝐴)
44	43	oveq1d 7371	. . . . . . 7 ⊢ (𝜑 → ((1 + (𝐴 − 1)) + 𝑉) = (𝐴 + 𝑉))
45	39, 44	eqtr3d 2771	. . . . . 6 ⊢ (𝜑 → (1 + ((𝐴 − 1) + 𝑉)) = (𝐴 + 𝑉))
46	45	oveq2d 7372	. . . . 5 ⊢ (𝜑 → (𝑊 · (1 + ((𝐴 − 1) + 𝑉))) = (𝑊 · (𝐴 + 𝑉)))
47	33	mulridd 11147	. . . . . 6 ⊢ (𝜑 → (𝑊 · 1) = 𝑊)
48	47	oveq1d 7371	. . . . 5 ⊢ (𝜑 → ((𝑊 · 1) + (𝑊 · ((𝐴 − 1) + 𝑉))) = (𝑊 + (𝑊 · ((𝐴 − 1) + 𝑉))))
49	36, 46, 48	3eqtr3d 2777	. . . 4 ⊢ (𝜑 → (𝑊 · (𝐴 + 𝑉)) = (𝑊 + (𝑊 · ((𝐴 − 1) + 𝑉))))
50	32, 49	breqtrrd 5124	. . 3 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ≤ (𝑊 · (𝐴 + 𝑉)))
51	7	nnred 12158	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
52	10	nnred 12158	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ ℝ)
53		elfzle2 13442	. . . . . . 7 ⊢ (𝐴 ∈ (1...𝑉) → 𝐴 ≤ 𝑉)
54	5, 53	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐴 ≤ 𝑉)
55	51, 52, 52, 54	leadd1dd 11749	. . . . 5 ⊢ (𝜑 → (𝐴 + 𝑉) ≤ (𝑉 + 𝑉))
56	38	2timesd 12382	. . . . 5 ⊢ (𝜑 → (2 · 𝑉) = (𝑉 + 𝑉))
57	55, 56	breqtrrd 5124	. . . 4 ⊢ (𝜑 → (𝐴 + 𝑉) ≤ (2 · 𝑉))
58	16	nnred 12158	. . . . 5 ⊢ (𝜑 → (𝐴 + 𝑉) ∈ ℝ)
59	21	nnred 12158	. . . . 5 ⊢ (𝜑 → (2 · 𝑉) ∈ ℝ)
60	4	nnred 12158	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ ℝ)
61	4	nngt0d 12192	. . . . 5 ⊢ (𝜑 → 0 < 𝑊)
62		lemul2 11992	. . . . 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 11288	. 2 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ≤ (𝑊 · (2 · 𝑉)))
66		nnuz 12788	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
67	14, 66	eleqtrdi 2844	. . 3 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ (ℤ_≥‘1))
68	22	nnzd 12512	. . 3 ⊢ (𝜑 → (𝑊 · (2 · 𝑉)) ∈ ℤ)
69		elfz5 13430	. . 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 1541 ∈ wcel 2113 class class class wbr 5096 ‘cfv 6490 (class class class)co 7356 ℂcc 11022 ℝcr 11023 0cc0 11024 1c1 11025 + caddc 11027 · cmul 11029 < clt 11164 ≤ cle 11165 − cmin 11362 ℕcn 12143 2c2 12198 ℕ₀cn0 12399 ℤcz 12486 ℤ_≥cuz 12749 ...cfz 13421

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8633 df-en 8882 df-dom 8883 df-sdom 8884 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-nn 12144 df-2 12206 df-n0 12400 df-z 12487 df-uz 12750 df-fz 13422

This theorem is referenced by: vdwlem4 16910 vdwlem6 16912

W3C validator