quartlem2 - Metamath Proof Explorer

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Theorem quartlem2 26768

Description: Closure lemmas for quart 26771. (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 26764	. . . . . 6 ⊢ (𝜑 → (𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ))
10	9	simp1d 1142	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ)
11	10	sqcld 14109	. . . 4 ⊢ (𝜑 → (𝑃↑2) ∈ ℂ)
12		1nn0 12458	. . . . . . 7 ⊢ 1 ∈ ℕ₀
13		2nn 12259	. . . . . . 7 ⊢ 2 ∈ ℕ
14	12, 13	decnncl 12669	. . . . . 6 ⊢ ;12 ∈ ℕ
15	14	nncni 12196	. . . . 5 ⊢ ;12 ∈ ℂ
16	9	simp3d 1144	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℂ)
17		mulcl 11152	. . . . 5 ⊢ ((;12 ∈ ℂ ∧ 𝑅 ∈ ℂ) → (;12 · 𝑅) ∈ ℂ)
18	15, 16, 17	sylancr 587	. . . 4 ⊢ (𝜑 → (;12 · 𝑅) ∈ ℂ)
19	11, 18	addcld 11193	. . 3 ⊢ (𝜑 → ((𝑃↑2) + (;12 · 𝑅)) ∈ ℂ)
20	1, 19	eqeltrd 2828	. 2 ⊢ (𝜑 → 𝑈 ∈ ℂ)
21		quart.v	. . 3 ⊢ (𝜑 → 𝑉 = ((-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅))))
22		2cn 12261	. . . . . . 7 ⊢ 2 ∈ ℂ
23		3nn0 12460	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
24		expcl 14044	. . . . . . . 8 ⊢ ((𝑃 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝑃↑3) ∈ ℂ)
25	10, 23, 24	sylancl 586	. . . . . . 7 ⊢ (𝜑 → (𝑃↑3) ∈ ℂ)
26		mulcl 11152	. . . . . . 7 ⊢ ((2 ∈ ℂ ∧ (𝑃↑3) ∈ ℂ) → (2 · (𝑃↑3)) ∈ ℂ)
27	22, 25, 26	sylancr 587	. . . . . 6 ⊢ (𝜑 → (2 · (𝑃↑3)) ∈ ℂ)
28	27	negcld 11520	. . . . 5 ⊢ (𝜑 → -(2 · (𝑃↑3)) ∈ ℂ)
29		2nn0 12459	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
30		7nn 12278	. . . . . . . 8 ⊢ 7 ∈ ℕ
31	29, 30	decnncl 12669	. . . . . . 7 ⊢ ;27 ∈ ℕ
32	31	nncni 12196	. . . . . 6 ⊢ ;27 ∈ ℂ
33	9	simp2d 1143	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℂ)
34	33	sqcld 14109	. . . . . 6 ⊢ (𝜑 → (𝑄↑2) ∈ ℂ)
35		mulcl 11152	. . . . . 6 ⊢ ((;27 ∈ ℂ ∧ (𝑄↑2) ∈ ℂ) → (;27 · (𝑄↑2)) ∈ ℂ)
36	32, 34, 35	sylancr 587	. . . . 5 ⊢ (𝜑 → (;27 · (𝑄↑2)) ∈ ℂ)
37	28, 36	subcld 11533	. . . 4 ⊢ (𝜑 → (-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) ∈ ℂ)
38		7nn0 12464	. . . . . . 7 ⊢ 7 ∈ ℕ₀
39	38, 13	decnncl 12669	. . . . . 6 ⊢ ;72 ∈ ℕ
40	39	nncni 12196	. . . . 5 ⊢ ;72 ∈ ℂ
41	10, 16	mulcld 11194	. . . . 5 ⊢ (𝜑 → (𝑃 · 𝑅) ∈ ℂ)
42		mulcl 11152	. . . . 5 ⊢ ((;72 ∈ ℂ ∧ (𝑃 · 𝑅) ∈ ℂ) → (;72 · (𝑃 · 𝑅)) ∈ ℂ)
43	40, 41, 42	sylancr 587	. . . 4 ⊢ (𝜑 → (;72 · (𝑃 · 𝑅)) ∈ ℂ)
44	37, 43	addcld 11193	. . 3 ⊢ (𝜑 → ((-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅))) ∈ ℂ)
45	21, 44	eqeltrd 2828	. 2 ⊢ (𝜑 → 𝑉 ∈ ℂ)
46		quart.w	. . 3 ⊢ (𝜑 → 𝑊 = (√‘((𝑉↑2) − (4 · (𝑈↑3)))))
47	45	sqcld 14109	. . . . 5 ⊢ (𝜑 → (𝑉↑2) ∈ ℂ)
48		4cn 12271	. . . . . 6 ⊢ 4 ∈ ℂ
49		expcl 14044	. . . . . . 7 ⊢ ((𝑈 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝑈↑3) ∈ ℂ)
50	20, 23, 49	sylancl 586	. . . . . 6 ⊢ (𝜑 → (𝑈↑3) ∈ ℂ)
51		mulcl 11152	. . . . . 6 ⊢ ((4 ∈ ℂ ∧ (𝑈↑3) ∈ ℂ) → (4 · (𝑈↑3)) ∈ ℂ)
52	48, 50, 51	sylancr 587	. . . . 5 ⊢ (𝜑 → (4 · (𝑈↑3)) ∈ ℂ)
53	47, 52	subcld 11533	. . . 4 ⊢ (𝜑 → ((𝑉↑2) − (4 · (𝑈↑3))) ∈ ℂ)
54	53	sqrtcld 15406	. . 3 ⊢ (𝜑 → (√‘((𝑉↑2) − (4 · (𝑈↑3)))) ∈ ℂ)
55	46, 54	eqeltrd 2828	. 2 ⊢ (𝜑 → 𝑊 ∈ ℂ)
56	20, 45, 55	3jca 1128	1 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑉 ∈ ℂ ∧ 𝑊 ∈ ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 1c1 11069 + caddc 11071 · cmul 11073 − cmin 11405 -cneg 11406 / cdiv 11835 2c2 12241 3c3 12242 4c4 12243 5c5 12244 6c6 12245 7c7 12246 8c8 12247 ℕ₀cn0 12442 cdc 12649 ↑cexp 14026 √csqrt 15199

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-sup 9393 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-rp 12952 df-seq 13967 df-exp 14027 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202

This theorem is referenced by: quartlem3 26769 quart 26771

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