vdwlem3 - Metamath Proof Explorer

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Theorem vdwlem3 17042

Description: Lemma for vdw 17053. (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 13580	. . . . . 6 ⊢ (𝐵 ∈ (1...𝑊) → 𝐵 ∈ ℕ)
3	1, 2	syl 18	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℕ)
4		vdwlem3.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℕ)
5		vdwlem3.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (1...𝑉))
6		elfznn 13580	. . . . . . . . 9 ⊢ (𝐴 ∈ (1...𝑉) → 𝐴 ∈ ℕ)
7	5, 6	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℕ)
8		nnm1nn0 12544	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (𝐴 − 1) ∈ ℕ₀)
9	7, 8	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐴 − 1) ∈ ℕ₀)
10		vdwlem3.v	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ ℕ)
11		nn0nnaddcl 12534	. . . . . . 7 ⊢ (((𝐴 − 1) ∈ ℕ₀ ∧ 𝑉 ∈ ℕ) → ((𝐴 − 1) + 𝑉) ∈ ℕ)
12	9, 10, 11	syl2anc 595	. . . . . 6 ⊢ (𝜑 → ((𝐴 − 1) + 𝑉) ∈ ℕ)
13	4, 12	nnmulcld 12288	. . . . 5 ⊢ (𝜑 → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℕ)
14	3, 13	nnaddcld 12287	. . . 4 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ ℕ)
15	14	nnred 12247	. . 3 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ ℝ)
16	7, 10	nnaddcld 12287	. . . . 5 ⊢ (𝜑 → (𝐴 + 𝑉) ∈ ℕ)
17	4, 16	nnmulcld 12288	. . . 4 ⊢ (𝜑 → (𝑊 · (𝐴 + 𝑉)) ∈ ℕ)
18	17	nnred 12247	. . 3 ⊢ (𝜑 → (𝑊 · (𝐴 + 𝑉)) ∈ ℝ)
19		2nn 12313	. . . . . 6 ⊢ 2 ∈ ℕ
20		nnmulcl 12256	. . . . . 6 ⊢ ((2 ∈ ℕ ∧ 𝑉 ∈ ℕ) → (2 · 𝑉) ∈ ℕ)
21	19, 10, 20	sylancr 598	. . . . 5 ⊢ (𝜑 → (2 · 𝑉) ∈ ℕ)
22	4, 21	nnmulcld 12288	. . . 4 ⊢ (𝜑 → (𝑊 · (2 · 𝑉)) ∈ ℕ)
23	22	nnred 12247	. . 3 ⊢ (𝜑 → (𝑊 · (2 · 𝑉)) ∈ ℝ)
24		elfzle2 13555	. . . . . 6 ⊢ (𝐵 ∈ (1...𝑊) → 𝐵 ≤ 𝑊)
25	1, 24	syl 18	. . . . 5 ⊢ (𝜑 → 𝐵 ≤ 𝑊)
26		nnre 12239	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
27		nnre 12239	. . . . . . 7 ⊢ (𝑊 ∈ ℕ → 𝑊 ∈ ℝ)
28		nnre 12239	. . . . . . 7 ⊢ ((𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℕ → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℝ)
29		leadd1 11681	. . . . . . 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 1176	. . . . . 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 1396	. . . . 5 ⊢ (𝜑 → (𝐵 ≤ 𝑊 ↔ (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ≤ (𝑊 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
32	25, 31	mpbid 235	. . . 4 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ≤ (𝑊 + (𝑊 · ((𝐴 − 1) + 𝑉))))
33	4	nncnd 12248	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℂ)
34		1cnd 11201	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
35	12	nncnd 12248	. . . . . 6 ⊢ (𝜑 → ((𝐴 − 1) + 𝑉) ∈ ℂ)
36	33, 34, 35	adddid 11232	. . . . 5 ⊢ (𝜑 → (𝑊 · (1 + ((𝐴 − 1) + 𝑉))) = ((𝑊 · 1) + (𝑊 · ((𝐴 − 1) + 𝑉))))
37	9	nn0cnd 12566	. . . . . . . 8 ⊢ (𝜑 → (𝐴 − 1) ∈ ℂ)
38	10	nncnd 12248	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ ℂ)
39	34, 37, 38	addassd 11230	. . . . . . 7 ⊢ (𝜑 → ((1 + (𝐴 − 1)) + 𝑉) = (1 + ((𝐴 − 1) + 𝑉)))
40		ax-1cn 11157	. . . . . . . . 9 ⊢ 1 ∈ ℂ
41	7	nncnd 12248	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
42		pncan3 11464	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 + (𝐴 − 1)) = 𝐴)
43	40, 41, 42	sylancr 598	. . . . . . . 8 ⊢ (𝜑 → (1 + (𝐴 − 1)) = 𝐴)
44	43	oveq1d 7426	. . . . . . 7 ⊢ (𝜑 → ((1 + (𝐴 − 1)) + 𝑉) = (𝐴 + 𝑉))
45	39, 44	eqtr3d 2806	. . . . . 6 ⊢ (𝜑 → (1 + ((𝐴 − 1) + 𝑉)) = (𝐴 + 𝑉))
46	45	oveq2d 7427	. . . . 5 ⊢ (𝜑 → (𝑊 · (1 + ((𝐴 − 1) + 𝑉))) = (𝑊 · (𝐴 + 𝑉)))
47	33	mulridd 11225	. . . . . 6 ⊢ (𝜑 → (𝑊 · 1) = 𝑊)
48	47	oveq1d 7426	. . . . 5 ⊢ (𝜑 → ((𝑊 · 1) + (𝑊 · ((𝐴 − 1) + 𝑉))) = (𝑊 + (𝑊 · ((𝐴 − 1) + 𝑉))))
49	36, 46, 48	3eqtr3d 2812	. . . 4 ⊢ (𝜑 → (𝑊 · (𝐴 + 𝑉)) = (𝑊 + (𝑊 · ((𝐴 − 1) + 𝑉))))
50	32, 49	breqtrrd 5143	. . 3 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ≤ (𝑊 · (𝐴 + 𝑉)))
51	7	nnred 12247	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
52	10	nnred 12247	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ ℝ)
53		elfzle2 13555	. . . . . . 7 ⊢ (𝐴 ∈ (1...𝑉) → 𝐴 ≤ 𝑉)
54	5, 53	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐴 ≤ 𝑉)
55	51, 52, 52, 54	leadd1dd 11827	. . . . 5 ⊢ (𝜑 → (𝐴 + 𝑉) ≤ (𝑉 + 𝑉))
56	38	2timesd 12486	. . . . 5 ⊢ (𝜑 → (2 · 𝑉) = (𝑉 + 𝑉))
57	55, 56	breqtrrd 5143	. . . 4 ⊢ (𝜑 → (𝐴 + 𝑉) ≤ (2 · 𝑉))
58	16	nnred 12247	. . . . 5 ⊢ (𝜑 → (𝐴 + 𝑉) ∈ ℝ)
59	21	nnred 12247	. . . . 5 ⊢ (𝜑 → (2 · 𝑉) ∈ ℝ)
60	4	nnred 12247	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ ℝ)
61	4	nngt0d 12284	. . . . 5 ⊢ (𝜑 → 0 < 𝑊)
62		lemul2 12067	. . . . 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 1399	. . . 4 ⊢ (𝜑 → ((𝐴 + 𝑉) ≤ (2 · 𝑉) ↔ (𝑊 · (𝐴 + 𝑉)) ≤ (𝑊 · (2 · 𝑉))))
64	57, 63	mpbid 235	. . 3 ⊢ (𝜑 → (𝑊 · (𝐴 + 𝑉)) ≤ (𝑊 · (2 · 𝑉)))
65	15, 18, 23, 50, 64	letrd 11366	. 2 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ≤ (𝑊 · (2 · 𝑉)))
66		nnuz 12900	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
67	14, 66	eleqtrdi 2879	. . 3 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ (ℤ_≥‘1))
68	22	nnzd 12616	. . 3 ⊢ (𝜑 → (𝑊 · (2 · 𝑉)) ∈ ℤ)
69		elfz5 13543	. . 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 595	. 2 ⊢ (𝜑 → ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ (1...(𝑊 · (2 · 𝑉))) ↔ (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ≤ (𝑊 · (2 · 𝑉))))
71	65, 70	mpbird 260	1 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ (1...(𝑊 · (2 · 𝑉))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 = wceq 1567 ∈ wcel 2149 class class class wbr 5113 ‘cfv 6537 (class class class)co 7411 ℂcc 11097 ℝcr 11098 0cc0 11099 1c1 11100 + caddc 11102 · cmul 11104 < clt 11242 ≤ cle 11243 − cmin 11440 ℕcn 12232 2c2 12294 ℕ₀cn0 12503 ℤcz 12590 ℤ_≥cuz 12861 ...cfz 13534

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-er 8693 df-en 8943 df-dom 8944 df-sdom 8945 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-nn 12233 df-2 12302 df-n0 12504 df-z 12591 df-uz 12862 df-fz 13535

This theorem is referenced by: vdwlem4 17043 vdwlem6 17045

W3C validator