quartlem2 - Metamath Proof Explorer

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Theorem quartlem2 27216

Description: Closure lemmas for quart 27219. (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 27212	. . . . . 6 ⊢ (𝜑 → (𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ))
10	9	simp1d 1160	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ)
11	10	sqcld 14314	. . . 4 ⊢ (𝜑 → (𝑃↑2) ∈ ℂ)
12		1nn0 12649	. . . . . . 7 ⊢ 1 ∈ ℕ₀
13		2nn 12443	. . . . . . 7 ⊢ 2 ∈ ℕ
14	12, 13	decnncl 12865	. . . . . 6 ⊢ ;12 ∈ ℕ
15	14	nncni 12372	. . . . 5 ⊢ ;12 ∈ ℂ
16	9	simp3d 1162	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℂ)
17		mulcl 11311	. . . . 5 ⊢ ((;12 ∈ ℂ ∧ 𝑅 ∈ ℂ) → (;12 · 𝑅) ∈ ℂ)
18	15, 16, 17	sylancr 599	. . . 4 ⊢ (𝜑 → (;12 · 𝑅) ∈ ℂ)
19	11, 18	addcld 11355	. . 3 ⊢ (𝜑 → ((𝑃↑2) + (;12 · 𝑅)) ∈ ℂ)
20	1, 19	eqeltrd 2864	. 2 ⊢ (𝜑 → 𝑈 ∈ ℂ)
21		quart.v	. . 3 ⊢ (𝜑 → 𝑉 = (( -(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅))))
22		2cn 12445	. . . . . . 7 ⊢ 2 ∈ ℂ
23		3nn0 12651	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
24		expcl 14249	. . . . . . . 8 ⊢ ((𝑃 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝑃↑3) ∈ ℂ)
25	10, 23, 24	sylancl 598	. . . . . . 7 ⊢ (𝜑 → (𝑃↑3) ∈ ℂ)
26		mulcl 11311	. . . . . . 7 ⊢ ((2 ∈ ℂ ∧ (𝑃↑3) ∈ ℂ) → (2 · (𝑃↑3)) ∈ ℂ)
27	22, 25, 26	sylancr 599	. . . . . 6 ⊢ (𝜑 → (2 · (𝑃↑3)) ∈ ℂ)
28	27	negcld 11683	. . . . 5 ⊢ (𝜑 → -(2 · (𝑃↑3)) ∈ ℂ)
29		2nn0 12650	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
30		7nn 12462	. . . . . . . 8 ⊢ 7 ∈ ℕ
31	29, 30	decnncl 12865	. . . . . . 7 ⊢ ;27 ∈ ℕ
32	31	nncni 12372	. . . . . 6 ⊢ ;27 ∈ ℂ
33	9	simp2d 1161	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℂ)
34	33	sqcld 14314	. . . . . 6 ⊢ (𝜑 → (𝑄↑2) ∈ ℂ)
35		mulcl 11311	. . . . . 6 ⊢ ((;27 ∈ ℂ ∧ (𝑄↑2) ∈ ℂ) → (;27 · (𝑄↑2)) ∈ ℂ)
36	32, 34, 35	sylancr 599	. . . . 5 ⊢ (𝜑 → (;27 · (𝑄↑2)) ∈ ℂ)
37	28, 36	subcld 11696	. . . 4 ⊢ (𝜑 → ( -(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) ∈ ℂ)
38		7nn0 12655	. . . . . . 7 ⊢ 7 ∈ ℕ₀
39	38, 13	decnncl 12865	. . . . . 6 ⊢ ;72 ∈ ℕ
40	39	nncni 12372	. . . . 5 ⊢ ;72 ∈ ℂ
41	10, 16	mulcld 11356	. . . . 5 ⊢ (𝜑 → (𝑃 · 𝑅) ∈ ℂ)
42		mulcl 11311	. . . . 5 ⊢ ((;72 ∈ ℂ ∧ (𝑃 · 𝑅) ∈ ℂ) → (;72 · (𝑃 · 𝑅)) ∈ ℂ)
43	40, 41, 42	sylancr 599	. . . 4 ⊢ (𝜑 → (;72 · (𝑃 · 𝑅)) ∈ ℂ)
44	37, 43	addcld 11355	. . 3 ⊢ (𝜑 → (( -(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅))) ∈ ℂ)
45	21, 44	eqeltrd 2864	. 2 ⊢ (𝜑 → 𝑉 ∈ ℂ)
46		quart.w	. . 3 ⊢ (𝜑 → 𝑊 = (√‘((𝑉↑2) − (4 · (𝑈↑3)))))
47	45	sqcld 14314	. . . . 5 ⊢ (𝜑 → (𝑉↑2) ∈ ℂ)
48		4cn 12455	. . . . . 6 ⊢ 4 ∈ ℂ
49		expcl 14249	. . . . . . 7 ⊢ ((𝑈 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝑈↑3) ∈ ℂ)
50	20, 23, 49	sylancl 598	. . . . . 6 ⊢ (𝜑 → (𝑈↑3) ∈ ℂ)
51		mulcl 11311	. . . . . 6 ⊢ ((4 ∈ ℂ ∧ (𝑈↑3) ∈ ℂ) → (4 · (𝑈↑3)) ∈ ℂ)
52	48, 50, 51	sylancr 599	. . . . 5 ⊢ (𝜑 → (4 · (𝑈↑3)) ∈ ℂ)
53	47, 52	subcld 11696	. . . 4 ⊢ (𝜑 → ((𝑉↑2) − (4 · (𝑈↑3))) ∈ ℂ)
54	53	sqrtcld 15636	. . 3 ⊢ (𝜑 → (√‘((𝑉↑2) − (4 · (𝑈↑3)))) ∈ ℂ)
55	46, 54	eqeltrd 2864	. 2 ⊢ (𝜑 → 𝑊 ∈ ℂ)
56	20, 45, 55	3jca 1146	1 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑉 ∈ ℂ ∧ 𝑊 ∈ ℂ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ‘cfv 6558 (class class class)co 7443 ℂcc 11225 1c1 11228 + caddc 11230 · cmul 11232 − cmin 11568 -cneg 11569 / cdiv 12000 2c2 12424 3c3 12425 4c4 12426 5c5 12427 6c6 12428 7c7 12429 8c8 12430 ℕ₀cn0 12633 cdc 12841 ↑cexp 14231 √csqrt 15429

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-pre-sup 11305

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-om 7902 df-2nd 8026 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-er 8745 df-en 9002 df-dom 9003 df-sdom 9004 df-sup 9461 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-div 12001 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-5 12435 df-6 12436 df-7 12437 df-8 12438 df-9 12439 df-n0 12634 df-z 12721 df-dec 12842 df-uz 12993 df-rp 13148 df-seq 14172 df-exp 14232 df-cj 15295 df-re 15296 df-im 15297 df-sqrt 15431 df-abs 15432

This theorem is used by: quartlem3 27217 quart 27219

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