quartlem2 - Metamath Proof Explorer

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Theorem quartlem2 26835

Description: Closure lemmas for quart 26838. (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 26831	. . . . . 6 ⊢ (𝜑 → (𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ))
10	9	simp1d 1139	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ)
11	10	sqcld 14144	. . . 4 ⊢ (𝜑 → (𝑃↑2) ∈ ℂ)
12		1nn0 12521	. . . . . . 7 ⊢ 1 ∈ ℕ₀
13		2nn 12318	. . . . . . 7 ⊢ 2 ∈ ℕ
14	12, 13	decnncl 12730	. . . . . 6 ⊢ ;12 ∈ ℕ
15	14	nncni 12255	. . . . 5 ⊢ ;12 ∈ ℂ
16	9	simp3d 1141	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℂ)
17		mulcl 11224	. . . . 5 ⊢ ((;12 ∈ ℂ ∧ 𝑅 ∈ ℂ) → (;12 · 𝑅) ∈ ℂ)
18	15, 16, 17	sylancr 585	. . . 4 ⊢ (𝜑 → (;12 · 𝑅) ∈ ℂ)
19	11, 18	addcld 11265	. . 3 ⊢ (𝜑 → ((𝑃↑2) + (;12 · 𝑅)) ∈ ℂ)
20	1, 19	eqeltrd 2825	. 2 ⊢ (𝜑 → 𝑈 ∈ ℂ)
21		quart.v	. . 3 ⊢ (𝜑 → 𝑉 = ((-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅))))
22		2cn 12320	. . . . . . 7 ⊢ 2 ∈ ℂ
23		3nn0 12523	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
24		expcl 14080	. . . . . . . 8 ⊢ ((𝑃 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝑃↑3) ∈ ℂ)
25	10, 23, 24	sylancl 584	. . . . . . 7 ⊢ (𝜑 → (𝑃↑3) ∈ ℂ)
26		mulcl 11224	. . . . . . 7 ⊢ ((2 ∈ ℂ ∧ (𝑃↑3) ∈ ℂ) → (2 · (𝑃↑3)) ∈ ℂ)
27	22, 25, 26	sylancr 585	. . . . . 6 ⊢ (𝜑 → (2 · (𝑃↑3)) ∈ ℂ)
28	27	negcld 11590	. . . . 5 ⊢ (𝜑 → -(2 · (𝑃↑3)) ∈ ℂ)
29		2nn0 12522	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
30		7nn 12337	. . . . . . . 8 ⊢ 7 ∈ ℕ
31	29, 30	decnncl 12730	. . . . . . 7 ⊢ ;27 ∈ ℕ
32	31	nncni 12255	. . . . . 6 ⊢ ;27 ∈ ℂ
33	9	simp2d 1140	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℂ)
34	33	sqcld 14144	. . . . . 6 ⊢ (𝜑 → (𝑄↑2) ∈ ℂ)
35		mulcl 11224	. . . . . 6 ⊢ ((;27 ∈ ℂ ∧ (𝑄↑2) ∈ ℂ) → (;27 · (𝑄↑2)) ∈ ℂ)
36	32, 34, 35	sylancr 585	. . . . 5 ⊢ (𝜑 → (;27 · (𝑄↑2)) ∈ ℂ)
37	28, 36	subcld 11603	. . . 4 ⊢ (𝜑 → (-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) ∈ ℂ)
38		7nn0 12527	. . . . . . 7 ⊢ 7 ∈ ℕ₀
39	38, 13	decnncl 12730	. . . . . 6 ⊢ ;72 ∈ ℕ
40	39	nncni 12255	. . . . 5 ⊢ ;72 ∈ ℂ
41	10, 16	mulcld 11266	. . . . 5 ⊢ (𝜑 → (𝑃 · 𝑅) ∈ ℂ)
42		mulcl 11224	. . . . 5 ⊢ ((;72 ∈ ℂ ∧ (𝑃 · 𝑅) ∈ ℂ) → (;72 · (𝑃 · 𝑅)) ∈ ℂ)
43	40, 41, 42	sylancr 585	. . . 4 ⊢ (𝜑 → (;72 · (𝑃 · 𝑅)) ∈ ℂ)
44	37, 43	addcld 11265	. . 3 ⊢ (𝜑 → ((-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅))) ∈ ℂ)
45	21, 44	eqeltrd 2825	. 2 ⊢ (𝜑 → 𝑉 ∈ ℂ)
46		quart.w	. . 3 ⊢ (𝜑 → 𝑊 = (√‘((𝑉↑2) − (4 · (𝑈↑3)))))
47	45	sqcld 14144	. . . . 5 ⊢ (𝜑 → (𝑉↑2) ∈ ℂ)
48		4cn 12330	. . . . . 6 ⊢ 4 ∈ ℂ
49		expcl 14080	. . . . . . 7 ⊢ ((𝑈 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝑈↑3) ∈ ℂ)
50	20, 23, 49	sylancl 584	. . . . . 6 ⊢ (𝜑 → (𝑈↑3) ∈ ℂ)
51		mulcl 11224	. . . . . 6 ⊢ ((4 ∈ ℂ ∧ (𝑈↑3) ∈ ℂ) → (4 · (𝑈↑3)) ∈ ℂ)
52	48, 50, 51	sylancr 585	. . . . 5 ⊢ (𝜑 → (4 · (𝑈↑3)) ∈ ℂ)
53	47, 52	subcld 11603	. . . 4 ⊢ (𝜑 → ((𝑉↑2) − (4 · (𝑈↑3))) ∈ ℂ)
54	53	sqrtcld 15420	. . 3 ⊢ (𝜑 → (√‘((𝑉↑2) − (4 · (𝑈↑3)))) ∈ ℂ)
55	46, 54	eqeltrd 2825	. 2 ⊢ (𝜑 → 𝑊 ∈ ℂ)
56	20, 45, 55	3jca 1125	1 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑉 ∈ ℂ ∧ 𝑊 ∈ ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 1c1 11141 + caddc 11143 · cmul 11145 − cmin 11476 -cneg 11477 / cdiv 11903 2c2 12300 3c3 12301 4c4 12302 5c5 12303 6c6 12304 7c7 12305 8c8 12306 ℕ₀cn0 12505 cdc 12710 ↑cexp 14062 √csqrt 15216

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 ax-pre-sup 11218

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-rmo 3363 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-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-sup 9467 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-rp 13010 df-seq 14003 df-exp 14063 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219

This theorem is referenced by: quartlem3 26836 quart 26838

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