quartlem3 - Metamath Proof Explorer

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Theorem quartlem3 27192

Description: Closure lemmas for quart 27194. (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)))))
quart.s	⊢ (𝜑 → 𝑆 = ((√‘𝑀) / 2))
quart.m	⊢ (𝜑 → 𝑀 = -((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3))
quart.t	⊢ (𝜑 → 𝑇 = (((𝑉 + 𝑊) / 2)↑_𝑐(1 / 3)))
quart.t0	⊢ (𝜑 → 𝑇 ≠ 0)

**Assertion**
Ref	Expression
quartlem3	⊢ (𝜑 → (𝑆 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑇 ∈ ℂ))

**Proof of Theorem quartlem3**
Step	Hyp	Ref	Expression
1		quart.s	. . 3 ⊢ (𝜑 → 𝑆 = ((√‘𝑀) / 2))
2		quart.m	. . . . . 6 ⊢ (𝜑 → 𝑀 = -((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3))
3		2cn 12421	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
4		quart.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℂ)
5		quart.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℂ)
6		quart.c	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ ℂ)
7		quart.d	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ∈ ℂ)
8		quart.p	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 = (𝐵 − ((3 / 8) · (𝐴↑2))))
9		quart.q	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 = ((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8)))
10		quart.r	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 = ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))
11	4, 5, 6, 7, 8, 9, 10	quart1cl 27187	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ))
12	11	simp1d 1160	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℂ)
13		mulcl 11287	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ 𝑃 ∈ ℂ) → (2 · 𝑃) ∈ ℂ)
14	3, 12, 13	sylancr 599	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 𝑃) ∈ ℂ)
15		quart.t	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 = (((𝑉 + 𝑊) / 2)↑_𝑐(1 / 3)))
16		quart.e	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐸 = -(𝐴 / 4))
17		quart.u	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑈 = ((𝑃↑2) + (;12 · 𝑅)))
18		quart.v	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑉 = (( -(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅))))
19		quart.w	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑊 = (√‘((𝑉↑2) − (4 · (𝑈↑3)))))
20	4, 5, 6, 7, 4, 16, 8, 9, 10, 17, 18, 19	quartlem2 27191	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑉 ∈ ℂ ∧ 𝑊 ∈ ℂ))
21	20	simp2d 1161	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ∈ ℂ)
22	20	simp3d 1162	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑊 ∈ ℂ)
23	21, 22	addcld 11331	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑉 + 𝑊) ∈ ℂ)
24	23	halfcld 12594	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑉 + 𝑊) / 2) ∈ ℂ)
25		3nn 12425	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℕ
26		nnrecre 12383	. . . . . . . . . . . . . 14 ⊢ (3 ∈ ℕ → (1 / 3) ∈ ℝ)
27	25, 26	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (1 / 3) ∈ ℝ
28	27	recni 11326	. . . . . . . . . . . 12 ⊢ (1 / 3) ∈ ℂ
29		cxpcl 27007	. . . . . . . . . . . 12 ⊢ ((((𝑉 + 𝑊) / 2) ∈ ℂ ∧ (1 / 3) ∈ ℂ) → (((𝑉 + 𝑊) / 2)↑_𝑐(1 / 3)) ∈ ℂ)
30	24, 28, 29	sylancl 598	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑉 + 𝑊) / 2)↑_𝑐(1 / 3)) ∈ ℂ)
31	15, 30	eqeltrd 2860	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ∈ ℂ)
32	14, 31	addcld 11331	. . . . . . . . 9 ⊢ (𝜑 → ((2 · 𝑃) + 𝑇) ∈ ℂ)
33	20	simp1d 1160	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ ℂ)
34		quart.t0	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ≠ 0)
35	33, 31, 34	divcld 12096	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 / 𝑇) ∈ ℂ)
36	32, 35	addcld 11331	. . . . . . . 8 ⊢ (𝜑 → (((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) ∈ ℂ)
37		3cn 12427	. . . . . . . . 9 ⊢ 3 ∈ ℂ
38	37	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 3 ∈ ℂ)
39		3ne0 12455	. . . . . . . . 9 ⊢ 3 ≠ 0
40	39	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 3 ≠ 0)
41	36, 38, 40	divcld 12096	. . . . . . 7 ⊢ (𝜑 → ((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3) ∈ ℂ)
42	41	negcld 11659	. . . . . 6 ⊢ (𝜑 → -((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3) ∈ ℂ)
43	2, 42	eqeltrd 2860	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℂ)
44	43	sqrtcld 15612	. . . 4 ⊢ (𝜑 → (√‘𝑀) ∈ ℂ)
45	44	halfcld 12594	. . 3 ⊢ (𝜑 → ((√‘𝑀) / 2) ∈ ℂ)
46	1, 45	eqeltrd 2860	. 2 ⊢ (𝜑 → 𝑆 ∈ ℂ)
47	46, 43, 31	3jca 1146	1 ⊢ (𝜑 → (𝑆 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑇 ∈ ℂ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ‘cfv 6538 (class class class)co 7420 ℂcc 11201 ℝcr 11202 0cc0 11203 1c1 11204 + caddc 11206 · cmul 11208 − cmin 11544 -cneg 11545 / cdiv 11976 ℕcn 12338 2c2 12400 3c3 12401 4c4 12402 5c5 12403 6c6 12404 7c7 12405 8c8 12406 cdc 12817 ↑cexp 14207 √csqrt 15405 ↑_𝑐ccxp 26888

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9645 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280 ax-pre-sup 11281 ax-addf 11282

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8180 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-1o 8479 df-2o 8480 df-er 8720 df-map 8852 df-pm 8853 df-ixp 8929 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-fsupp 9357 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9507 df-card 10023 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-div 11977 df-nn 12339 df-2 12408 df-3 12409 df-4 12410 df-5 12411 df-6 12412 df-7 12413 df-8 12414 df-9 12415 df-n0 12610 df-z 12697 df-dec 12818 df-uz 12969 df-q 13079 df-rp 13124 df-xneg 13244 df-xadd 13245 df-xmul 13246 df-ioo 13483 df-ioc 13484 df-ico 13485 df-icc 13486 df-fz 13643 df-fzo 13792 df-fl 13935 df-mod 14013 df-seq 14148 df-exp 14208 df-fac 14421 df-bc 14450 df-hash 14478 df-shft 15225 df-cj 15271 df-re 15272 df-im 15273 df-sqrt 15407 df-abs 15408 df-limsup 15643 df-clim 15660 df-rlim 15661 df-sum 15859 df-ef 16238 df-sin 16240 df-cos 16241 df-pi 16243 df-struct 17330 df-sets 17347 df-slot 17365 df-ndx 17377 df-base 17393 df-ress 17414 df-plusg 17446 df-mulr 17447 df-starv 17448 df-sca 17449 df-vsca 17450 df-ip 17451 df-tset 17452 df-ple 17453 df-ds 17455 df-unif 17456 df-hom 17457 df-cco 17458 df-rest 17598 df-topn 17599 df-0g 17617 df-gsum 17618 df-topgen 17619 df-pt 17620 df-prds 17623 df-xrs 17679 df-qtop 17684 df-imas 17685 df-xps 17687 df-mre 17761 df-mrc 17762 df-acs 17764 df-mgm 18821 df-sgrp 18913 df-mnd 18929 df-submnd 18984 df-mulg 19283 df-cntz 19536 df-cmn 20001 df-psmet 21675 df-xmet 21676 df-met 21677 df-bl 21678 df-mopn 21679 df-fbas 21680 df-fg 21681 df-cnfld 21684 df-top 23217 df-topon 23234 df-topsp 23256 df-bases 23269 df-cld 23342 df-ntr 23343 df-cls 23344 df-nei 23421 df-lp 23459 df-perf 23460 df-cn 23550 df-cnp 23551 df-haus 23638 df-tx 23886 df-hmeo 24079 df-fil 24170 df-fm 24262 df-flim 24263 df-flf 24264 df-xms 24644 df-ms 24645 df-tms 24646 df-cncf 25204 df-limc 26191 df-dv 26192 df-log 26889 df-cxp 26890

This theorem is used by: quartlem4 27193 quart 27194

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