vdwlem3 - Metamath Proof Explorer

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Theorem vdwlem3 16955

Description: Lemma for vdw 16966. (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 13565	. . . . . 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 13565	. . . . . . . . 9 ⊢ (𝐴 ∈ (1...𝑉) → 𝐴 ∈ ℕ)
7	5, 6	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℕ)
8		nnm1nn0 12546	. . . . . . . 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 12536	. . . . . . 7 ⊢ (((𝐴 − 1) ∈ ℕ₀ ∧ 𝑉 ∈ ℕ) → ((𝐴 − 1) + 𝑉) ∈ ℕ)
12	9, 10, 11	syl2anc 582	. . . . . 6 ⊢ (𝜑 → ((𝐴 − 1) + 𝑉) ∈ ℕ)
13	4, 12	nnmulcld 12298	. . . . 5 ⊢ (𝜑 → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℕ)
14	3, 13	nnaddcld 12297	. . . 4 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ ℕ)
15	14	nnred 12260	. . 3 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ ℝ)
16	7, 10	nnaddcld 12297	. . . . 5 ⊢ (𝜑 → (𝐴 + 𝑉) ∈ ℕ)
17	4, 16	nnmulcld 12298	. . . 4 ⊢ (𝜑 → (𝑊 · (𝐴 + 𝑉)) ∈ ℕ)
18	17	nnred 12260	. . 3 ⊢ (𝜑 → (𝑊 · (𝐴 + 𝑉)) ∈ ℝ)
19		2nn 12318	. . . . . 6 ⊢ 2 ∈ ℕ
20		nnmulcl 12269	. . . . . 6 ⊢ ((2 ∈ ℕ ∧ 𝑉 ∈ ℕ) → (2 · 𝑉) ∈ ℕ)
21	19, 10, 20	sylancr 585	. . . . 5 ⊢ (𝜑 → (2 · 𝑉) ∈ ℕ)
22	4, 21	nnmulcld 12298	. . . 4 ⊢ (𝜑 → (𝑊 · (2 · 𝑉)) ∈ ℕ)
23	22	nnred 12260	. . 3 ⊢ (𝜑 → (𝑊 · (2 · 𝑉)) ∈ ℝ)
24		elfzle2 13540	. . . . . 6 ⊢ (𝐵 ∈ (1...𝑊) → 𝐵 ≤ 𝑊)
25	1, 24	syl 17	. . . . 5 ⊢ (𝜑 → 𝐵 ≤ 𝑊)
26		nnre 12252	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
27		nnre 12252	. . . . . . 7 ⊢ (𝑊 ∈ ℕ → 𝑊 ∈ ℝ)
28		nnre 12252	. . . . . . 7 ⊢ ((𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℕ → (𝑊 · ((𝐴 − 1) + 𝑉)) ∈ ℝ)
29		leadd1 11714	. . . . . . 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 1157	. . . . . 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 1368	. . . . 5 ⊢ (𝜑 → (𝐵 ≤ 𝑊 ↔ (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ≤ (𝑊 + (𝑊 · ((𝐴 − 1) + 𝑉)))))
32	25, 31	mpbid 231	. . . 4 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ≤ (𝑊 + (𝑊 · ((𝐴 − 1) + 𝑉))))
33	4	nncnd 12261	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℂ)
34		1cnd 11241	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
35	12	nncnd 12261	. . . . . 6 ⊢ (𝜑 → ((𝐴 − 1) + 𝑉) ∈ ℂ)
36	33, 34, 35	adddid 11270	. . . . 5 ⊢ (𝜑 → (𝑊 · (1 + ((𝐴 − 1) + 𝑉))) = ((𝑊 · 1) + (𝑊 · ((𝐴 − 1) + 𝑉))))
37	9	nn0cnd 12567	. . . . . . . 8 ⊢ (𝜑 → (𝐴 − 1) ∈ ℂ)
38	10	nncnd 12261	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ ℂ)
39	34, 37, 38	addassd 11268	. . . . . . 7 ⊢ (𝜑 → ((1 + (𝐴 − 1)) + 𝑉) = (1 + ((𝐴 − 1) + 𝑉)))
40		ax-1cn 11198	. . . . . . . . 9 ⊢ 1 ∈ ℂ
41	7	nncnd 12261	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
42		pncan3 11500	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 + (𝐴 − 1)) = 𝐴)
43	40, 41, 42	sylancr 585	. . . . . . . 8 ⊢ (𝜑 → (1 + (𝐴 − 1)) = 𝐴)
44	43	oveq1d 7434	. . . . . . 7 ⊢ (𝜑 → ((1 + (𝐴 − 1)) + 𝑉) = (𝐴 + 𝑉))
45	39, 44	eqtr3d 2767	. . . . . 6 ⊢ (𝜑 → (1 + ((𝐴 − 1) + 𝑉)) = (𝐴 + 𝑉))
46	45	oveq2d 7435	. . . . 5 ⊢ (𝜑 → (𝑊 · (1 + ((𝐴 − 1) + 𝑉))) = (𝑊 · (𝐴 + 𝑉)))
47	33	mulridd 11263	. . . . . 6 ⊢ (𝜑 → (𝑊 · 1) = 𝑊)
48	47	oveq1d 7434	. . . . 5 ⊢ (𝜑 → ((𝑊 · 1) + (𝑊 · ((𝐴 − 1) + 𝑉))) = (𝑊 + (𝑊 · ((𝐴 − 1) + 𝑉))))
49	36, 46, 48	3eqtr3d 2773	. . . 4 ⊢ (𝜑 → (𝑊 · (𝐴 + 𝑉)) = (𝑊 + (𝑊 · ((𝐴 − 1) + 𝑉))))
50	32, 49	breqtrrd 5177	. . 3 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ≤ (𝑊 · (𝐴 + 𝑉)))
51	7	nnred 12260	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
52	10	nnred 12260	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ ℝ)
53		elfzle2 13540	. . . . . . 7 ⊢ (𝐴 ∈ (1...𝑉) → 𝐴 ≤ 𝑉)
54	5, 53	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐴 ≤ 𝑉)
55	51, 52, 52, 54	leadd1dd 11860	. . . . 5 ⊢ (𝜑 → (𝐴 + 𝑉) ≤ (𝑉 + 𝑉))
56	38	2timesd 12488	. . . . 5 ⊢ (𝜑 → (2 · 𝑉) = (𝑉 + 𝑉))
57	55, 56	breqtrrd 5177	. . . 4 ⊢ (𝜑 → (𝐴 + 𝑉) ≤ (2 · 𝑉))
58	16	nnred 12260	. . . . 5 ⊢ (𝜑 → (𝐴 + 𝑉) ∈ ℝ)
59	21	nnred 12260	. . . . 5 ⊢ (𝜑 → (2 · 𝑉) ∈ ℝ)
60	4	nnred 12260	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ ℝ)
61	4	nngt0d 12294	. . . . 5 ⊢ (𝜑 → 0 < 𝑊)
62		lemul2 12100	. . . . 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 1371	. . . 4 ⊢ (𝜑 → ((𝐴 + 𝑉) ≤ (2 · 𝑉) ↔ (𝑊 · (𝐴 + 𝑉)) ≤ (𝑊 · (2 · 𝑉))))
64	57, 63	mpbid 231	. . 3 ⊢ (𝜑 → (𝑊 · (𝐴 + 𝑉)) ≤ (𝑊 · (2 · 𝑉)))
65	15, 18, 23, 50, 64	letrd 11403	. 2 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ≤ (𝑊 · (2 · 𝑉)))
66		nnuz 12898	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
67	14, 66	eleqtrdi 2835	. . 3 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ (ℤ_≥‘1))
68	22	nnzd 12618	. . 3 ⊢ (𝜑 → (𝑊 · (2 · 𝑉)) ∈ ℤ)
69		elfz5 13528	. . 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 582	. 2 ⊢ (𝜑 → ((𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ (1...(𝑊 · (2 · 𝑉))) ↔ (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ≤ (𝑊 · (2 · 𝑉))))
71	65, 70	mpbird 256	1 ⊢ (𝜑 → (𝐵 + (𝑊 · ((𝐴 − 1) + 𝑉))) ∈ (1...(𝑊 · (2 · 𝑉))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 ℝcr 11139 0cc0 11140 1c1 11141 + caddc 11143 · cmul 11145 < clt 11280 ≤ cle 11281 − cmin 11476 ℕcn 12245 2c2 12300 ℕ₀cn0 12505 ℤcz 12591 ℤ_≥cuz 12855 ...cfz 13519

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520

This theorem is referenced by: vdwlem4 16956 vdwlem6 16958

W3C validator