quartlem2 - Metamath Proof Explorer

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Theorem quartlem2 24585

Description: Closure lemmas for quart 24588. (Contributed by Mario Carneiro, 7-May-2015.)

**Hypotheses**
Ref	Expression
quart.a	⊢ (𝜑 → 𝐴 ∈ ℂ)
quart.b	⊢ (𝜑 → 𝐵 ∈ ℂ)
quart.c	⊢ (𝜑 → 𝐶 ∈ ℂ)
quart.d	⊢ (𝜑 → 𝐷 ∈ ℂ)
quart.x	⊢ (𝜑 → 𝑋 ∈ ℂ)
quart.e	⊢ (𝜑 → 𝐸 = -(𝐴 / 4))
quart.p	⊢ (𝜑 → 𝑃 = (𝐵 − ((3 / 8) · (𝐴↑2))))
quart.q	⊢ (𝜑 → 𝑄 = ((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8)))
quart.r	⊢ (𝜑 → 𝑅 = ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))
quart.u	⊢ (𝜑 → 𝑈 = ((𝑃↑2) + (;12 · 𝑅)))
quart.v	⊢ (𝜑 → 𝑉 = ((-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅))))
quart.w	⊢ (𝜑 → 𝑊 = (√‘((𝑉↑2) − (4 · (𝑈↑3)))))

**Assertion**
Ref	Expression
quartlem2	⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑉 ∈ ℂ ∧ 𝑊 ∈ ℂ))

**Proof of Theorem quartlem2**
Step	Hyp	Ref	Expression
1		quart.u	. . 3 ⊢ (𝜑 → 𝑈 = ((𝑃↑2) + (;12 · 𝑅)))
2		quart.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
3		quart.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
4		quart.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ)
5		quart.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ℂ)
6		quart.p	. . . . . . 7 ⊢ (𝜑 → 𝑃 = (𝐵 − ((3 / 8) · (𝐴↑2))))
7		quart.q	. . . . . . 7 ⊢ (𝜑 → 𝑄 = ((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8)))
8		quart.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 = ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))
9	2, 3, 4, 5, 6, 7, 8	quart1cl 24581	. . . . . 6 ⊢ (𝜑 → (𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ))
10	9	simp1d 1073	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ)
11	10	sqcld 13006	. . . 4 ⊢ (𝜑 → (𝑃↑2) ∈ ℂ)
12		1nn0 11308	. . . . . . 7 ⊢ 1 ∈ ℕ₀
13		2nn 11185	. . . . . . 7 ⊢ 2 ∈ ℕ
14	12, 13	decnncl 11518	. . . . . 6 ⊢ ;12 ∈ ℕ
15	14	nncni 11030	. . . . 5 ⊢ ;12 ∈ ℂ
16	9	simp3d 1075	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℂ)
17		mulcl 10020	. . . . 5 ⊢ ((;12 ∈ ℂ ∧ 𝑅 ∈ ℂ) → (;12 · 𝑅) ∈ ℂ)
18	15, 16, 17	sylancr 695	. . . 4 ⊢ (𝜑 → (;12 · 𝑅) ∈ ℂ)
19	11, 18	addcld 10059	. . 3 ⊢ (𝜑 → ((𝑃↑2) + (;12 · 𝑅)) ∈ ℂ)
20	1, 19	eqeltrd 2701	. 2 ⊢ (𝜑 → 𝑈 ∈ ℂ)
21		quart.v	. . 3 ⊢ (𝜑 → 𝑉 = ((-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅))))
22		2cn 11091	. . . . . . 7 ⊢ 2 ∈ ℂ
23		3nn0 11310	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
24		expcl 12878	. . . . . . . 8 ⊢ ((𝑃 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝑃↑3) ∈ ℂ)
25	10, 23, 24	sylancl 694	. . . . . . 7 ⊢ (𝜑 → (𝑃↑3) ∈ ℂ)
26		mulcl 10020	. . . . . . 7 ⊢ ((2 ∈ ℂ ∧ (𝑃↑3) ∈ ℂ) → (2 · (𝑃↑3)) ∈ ℂ)
27	22, 25, 26	sylancr 695	. . . . . 6 ⊢ (𝜑 → (2 · (𝑃↑3)) ∈ ℂ)
28	27	negcld 10379	. . . . 5 ⊢ (𝜑 → -(2 · (𝑃↑3)) ∈ ℂ)
29		2nn0 11309	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
30		7nn 11190	. . . . . . . 8 ⊢ 7 ∈ ℕ
31	29, 30	decnncl 11518	. . . . . . 7 ⊢ ;27 ∈ ℕ
32	31	nncni 11030	. . . . . 6 ⊢ ;27 ∈ ℂ
33	9	simp2d 1074	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℂ)
34	33	sqcld 13006	. . . . . 6 ⊢ (𝜑 → (𝑄↑2) ∈ ℂ)
35		mulcl 10020	. . . . . 6 ⊢ ((;27 ∈ ℂ ∧ (𝑄↑2) ∈ ℂ) → (;27 · (𝑄↑2)) ∈ ℂ)
36	32, 34, 35	sylancr 695	. . . . 5 ⊢ (𝜑 → (;27 · (𝑄↑2)) ∈ ℂ)
37	28, 36	subcld 10392	. . . 4 ⊢ (𝜑 → (-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) ∈ ℂ)
38		7nn0 11314	. . . . . . 7 ⊢ 7 ∈ ℕ₀
39	38, 13	decnncl 11518	. . . . . 6 ⊢ ;72 ∈ ℕ
40	39	nncni 11030	. . . . 5 ⊢ ;72 ∈ ℂ
41	10, 16	mulcld 10060	. . . . 5 ⊢ (𝜑 → (𝑃 · 𝑅) ∈ ℂ)
42		mulcl 10020	. . . . 5 ⊢ ((;72 ∈ ℂ ∧ (𝑃 · 𝑅) ∈ ℂ) → (;72 · (𝑃 · 𝑅)) ∈ ℂ)
43	40, 41, 42	sylancr 695	. . . 4 ⊢ (𝜑 → (;72 · (𝑃 · 𝑅)) ∈ ℂ)
44	37, 43	addcld 10059	. . 3 ⊢ (𝜑 → ((-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅))) ∈ ℂ)
45	21, 44	eqeltrd 2701	. 2 ⊢ (𝜑 → 𝑉 ∈ ℂ)
46		quart.w	. . 3 ⊢ (𝜑 → 𝑊 = (√‘((𝑉↑2) − (4 · (𝑈↑3)))))
47	45	sqcld 13006	. . . . 5 ⊢ (𝜑 → (𝑉↑2) ∈ ℂ)
48		4cn 11098	. . . . . 6 ⊢ 4 ∈ ℂ
49		expcl 12878	. . . . . . 7 ⊢ ((𝑈 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝑈↑3) ∈ ℂ)
50	20, 23, 49	sylancl 694	. . . . . 6 ⊢ (𝜑 → (𝑈↑3) ∈ ℂ)
51		mulcl 10020	. . . . . 6 ⊢ ((4 ∈ ℂ ∧ (𝑈↑3) ∈ ℂ) → (4 · (𝑈↑3)) ∈ ℂ)
52	48, 50, 51	sylancr 695	. . . . 5 ⊢ (𝜑 → (4 · (𝑈↑3)) ∈ ℂ)
53	47, 52	subcld 10392	. . . 4 ⊢ (𝜑 → ((𝑉↑2) − (4 · (𝑈↑3))) ∈ ℂ)
54	53	sqrtcld 14176	. . 3 ⊢ (𝜑 → (√‘((𝑉↑2) − (4 · (𝑈↑3)))) ∈ ℂ)
55	46, 54	eqeltrd 2701	. 2 ⊢ (𝜑 → 𝑊 ∈ ℂ)
56	20, 45, 55	3jca 1242	1 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑉 ∈ ℂ ∧ 𝑊 ∈ ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 1c1 9937 + caddc 9939 · cmul 9941 − cmin 10266 -cneg 10267 / cdiv 10684 2c2 11070 3c3 11071 4c4 11072 5c5 11073 6c6 11074 7c7 11075 8c8 11076 ℕ₀cn0 11292 cdc 11493 ↑cexp 12860 √csqrt 13973

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976

This theorem is referenced by: quartlem3 24586 quart 24588

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