quartlem2 - Metamath Proof Explorer

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Theorem quartlem2 24385

Description: Closure lemmas for quart 24388. (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 24381	. . . . . 6 ⊢ (𝜑 → (𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ))
10	9	simp1d 1066	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ)
11	10	sqcld 12868	. . . 4 ⊢ (𝜑 → (𝑃↑2) ∈ ℂ)
12		1nn0 11185	. . . . . . 7 ⊢ 1 ∈ ℕ₀
13		2nn 11062	. . . . . . 7 ⊢ 2 ∈ ℕ
14	12, 13	decnncl 11394	. . . . . 6 ⊢ ;12 ∈ ℕ
15	14	nncni 10907	. . . . 5 ⊢ ;12 ∈ ℂ
16	9	simp3d 1068	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℂ)
17		mulcl 9899	. . . . 5 ⊢ ((;12 ∈ ℂ ∧ 𝑅 ∈ ℂ) → (;12 · 𝑅) ∈ ℂ)
18	15, 16, 17	sylancr 694	. . . 4 ⊢ (𝜑 → (;12 · 𝑅) ∈ ℂ)
19	11, 18	addcld 9938	. . 3 ⊢ (𝜑 → ((𝑃↑2) + (;12 · 𝑅)) ∈ ℂ)
20	1, 19	eqeltrd 2688	. 2 ⊢ (𝜑 → 𝑈 ∈ ℂ)
21		quart.v	. . 3 ⊢ (𝜑 → 𝑉 = ((-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅))))
22		2cn 10968	. . . . . . 7 ⊢ 2 ∈ ℂ
23		3nn0 11187	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
24		expcl 12740	. . . . . . . 8 ⊢ ((𝑃 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝑃↑3) ∈ ℂ)
25	10, 23, 24	sylancl 693	. . . . . . 7 ⊢ (𝜑 → (𝑃↑3) ∈ ℂ)
26		mulcl 9899	. . . . . . 7 ⊢ ((2 ∈ ℂ ∧ (𝑃↑3) ∈ ℂ) → (2 · (𝑃↑3)) ∈ ℂ)
27	22, 25, 26	sylancr 694	. . . . . 6 ⊢ (𝜑 → (2 · (𝑃↑3)) ∈ ℂ)
28	27	negcld 10258	. . . . 5 ⊢ (𝜑 → -(2 · (𝑃↑3)) ∈ ℂ)
29		2nn0 11186	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
30		7nn 11067	. . . . . . . 8 ⊢ 7 ∈ ℕ
31	29, 30	decnncl 11394	. . . . . . 7 ⊢ ;27 ∈ ℕ
32	31	nncni 10907	. . . . . 6 ⊢ ;27 ∈ ℂ
33	9	simp2d 1067	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℂ)
34	33	sqcld 12868	. . . . . 6 ⊢ (𝜑 → (𝑄↑2) ∈ ℂ)
35		mulcl 9899	. . . . . 6 ⊢ ((;27 ∈ ℂ ∧ (𝑄↑2) ∈ ℂ) → (;27 · (𝑄↑2)) ∈ ℂ)
36	32, 34, 35	sylancr 694	. . . . 5 ⊢ (𝜑 → (;27 · (𝑄↑2)) ∈ ℂ)
37	28, 36	subcld 10271	. . . 4 ⊢ (𝜑 → (-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) ∈ ℂ)
38		7nn0 11191	. . . . . . 7 ⊢ 7 ∈ ℕ₀
39	38, 13	decnncl 11394	. . . . . 6 ⊢ ;72 ∈ ℕ
40	39	nncni 10907	. . . . 5 ⊢ ;72 ∈ ℂ
41	10, 16	mulcld 9939	. . . . 5 ⊢ (𝜑 → (𝑃 · 𝑅) ∈ ℂ)
42		mulcl 9899	. . . . 5 ⊢ ((;72 ∈ ℂ ∧ (𝑃 · 𝑅) ∈ ℂ) → (;72 · (𝑃 · 𝑅)) ∈ ℂ)
43	40, 41, 42	sylancr 694	. . . 4 ⊢ (𝜑 → (;72 · (𝑃 · 𝑅)) ∈ ℂ)
44	37, 43	addcld 9938	. . 3 ⊢ (𝜑 → ((-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅))) ∈ ℂ)
45	21, 44	eqeltrd 2688	. 2 ⊢ (𝜑 → 𝑉 ∈ ℂ)
46		quart.w	. . 3 ⊢ (𝜑 → 𝑊 = (√‘((𝑉↑2) − (4 · (𝑈↑3)))))
47	45	sqcld 12868	. . . . 5 ⊢ (𝜑 → (𝑉↑2) ∈ ℂ)
48		4cn 10975	. . . . . 6 ⊢ 4 ∈ ℂ
49		expcl 12740	. . . . . . 7 ⊢ ((𝑈 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝑈↑3) ∈ ℂ)
50	20, 23, 49	sylancl 693	. . . . . 6 ⊢ (𝜑 → (𝑈↑3) ∈ ℂ)
51		mulcl 9899	. . . . . 6 ⊢ ((4 ∈ ℂ ∧ (𝑈↑3) ∈ ℂ) → (4 · (𝑈↑3)) ∈ ℂ)
52	48, 50, 51	sylancr 694	. . . . 5 ⊢ (𝜑 → (4 · (𝑈↑3)) ∈ ℂ)
53	47, 52	subcld 10271	. . . 4 ⊢ (𝜑 → ((𝑉↑2) − (4 · (𝑈↑3))) ∈ ℂ)
54	53	sqrtcld 14024	. . 3 ⊢ (𝜑 → (√‘((𝑉↑2) − (4 · (𝑈↑3)))) ∈ ℂ)
55	46, 54	eqeltrd 2688	. 2 ⊢ (𝜑 → 𝑊 ∈ ℂ)
56	20, 45, 55	3jca 1235	1 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑉 ∈ ℂ ∧ 𝑊 ∈ ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 1c1 9816 + caddc 9818 · cmul 9820 − cmin 10145 -cneg 10146 / cdiv 10563 2c2 10947 3c3 10948 4c4 10949 5c5 10950 6c6 10951 7c7 10952 8c8 10953 ℕ₀cn0 11169 cdc 11369 ↑cexp 12722 √csqrt 13821

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824

This theorem is referenced by: quartlem3 24386 quart 24388

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