quartlem3 - Metamath Proof Explorer

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Theorem quartlem3 25153

Description: Closure lemmas for quart 25155. (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 11513	. . . . . . . . . . 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 25148	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ))
12	11	simp1d 1123	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℂ)
13		mulcl 10417	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ 𝑃 ∈ ℂ) → (2 · 𝑃) ∈ ℂ)
14	3, 12, 13	sylancr 579	. . . . . . . . . 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 25152	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑉 ∈ ℂ ∧ 𝑊 ∈ ℂ))
21	20	simp2d 1124	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ∈ ℂ)
22	20	simp3d 1125	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑊 ∈ ℂ)
23	21, 22	addcld 10457	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑉 + 𝑊) ∈ ℂ)
24	23	halfcld 11690	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑉 + 𝑊) / 2) ∈ ℂ)
25		3nn 11517	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℕ
26		nnrecre 11480	. . . . . . . . . . . . . 14 ⊢ (3 ∈ ℕ → (1 / 3) ∈ ℝ)
27	25, 26	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (1 / 3) ∈ ℝ
28	27	recni 10452	. . . . . . . . . . . 12 ⊢ (1 / 3) ∈ ℂ
29		cxpcl 24973	. . . . . . . . . . . 12 ⊢ ((((𝑉 + 𝑊) / 2) ∈ ℂ ∧ (1 / 3) ∈ ℂ) → (((𝑉 + 𝑊) / 2)↑_𝑐(1 / 3)) ∈ ℂ)
30	24, 28, 29	sylancl 578	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑉 + 𝑊) / 2)↑_𝑐(1 / 3)) ∈ ℂ)
31	15, 30	eqeltrd 2859	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ∈ ℂ)
32	14, 31	addcld 10457	. . . . . . . . 9 ⊢ (𝜑 → ((2 · 𝑃) + 𝑇) ∈ ℂ)
33	20	simp1d 1123	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ ℂ)
34		quart.t0	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ≠ 0)
35	33, 31, 34	divcld 11215	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 / 𝑇) ∈ ℂ)
36	32, 35	addcld 10457	. . . . . . . 8 ⊢ (𝜑 → (((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) ∈ ℂ)
37		3cn 11519	. . . . . . . . 9 ⊢ 3 ∈ ℂ
38	37	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 3 ∈ ℂ)
39		3ne0 11551	. . . . . . . . 9 ⊢ 3 ≠ 0
40	39	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 3 ≠ 0)
41	36, 38, 40	divcld 11215	. . . . . . 7 ⊢ (𝜑 → ((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3) ∈ ℂ)
42	41	negcld 10783	. . . . . 6 ⊢ (𝜑 → -((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3) ∈ ℂ)
43	2, 42	eqeltrd 2859	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℂ)
44	43	sqrtcld 14656	. . . 4 ⊢ (𝜑 → (√‘𝑀) ∈ ℂ)
45	44	halfcld 11690	. . 3 ⊢ (𝜑 → ((√‘𝑀) / 2) ∈ ℂ)
46	1, 45	eqeltrd 2859	. 2 ⊢ (𝜑 → 𝑆 ∈ ℂ)
47	46, 43, 31	3jca 1109	1 ⊢ (𝜑 → (𝑆 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑇 ∈ ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ≠ wne 2960 ‘cfv 6185 (class class class)co 6974 ℂcc 10331 ℝcr 10332 0cc0 10333 1c1 10334 + caddc 10336 · cmul 10338 − cmin 10668 -cneg 10669 / cdiv 11096 ℕcn 11437 2c2 11493 3c3 11494 4c4 11495 5c5 11496 6c6 11497 7c7 11498 8c8 11499 cdc 11909 ↑cexp 13242 √csqrt 14451 ↑_𝑐ccxp 24855

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-inf2 8896 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-pre-sup 10411 ax-addf 10412 ax-mulf 10413

This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-iin 4791 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-of 7225 df-om 7395 df-1st 7499 df-2nd 7500 df-supp 7632 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-2o 7904 df-oadd 7907 df-er 8087 df-map 8206 df-pm 8207 df-ixp 8258 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-fsupp 8627 df-fi 8668 df-sup 8699 df-inf 8700 df-oi 8767 df-card 9160 df-cda 9386 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-z 11792 df-dec 11910 df-uz 12057 df-q 12161 df-rp 12203 df-xneg 12322 df-xadd 12323 df-xmul 12324 df-ioo 12556 df-ioc 12557 df-ico 12558 df-icc 12559 df-fz 12707 df-fzo 12848 df-fl 12975 df-mod 13051 df-seq 13183 df-exp 13243 df-fac 13447 df-bc 13476 df-hash 13504 df-shft 14285 df-cj 14317 df-re 14318 df-im 14319 df-sqrt 14453 df-abs 14454 df-limsup 14687 df-clim 14704 df-rlim 14705 df-sum 14902 df-ef 15279 df-sin 15281 df-cos 15282 df-pi 15284 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-mulr 16433 df-starv 16434 df-sca 16435 df-vsca 16436 df-ip 16437 df-tset 16438 df-ple 16439 df-ds 16441 df-unif 16442 df-hom 16443 df-cco 16444 df-rest 16550 df-topn 16551 df-0g 16569 df-gsum 16570 df-topgen 16571 df-pt 16572 df-prds 16575 df-xrs 16629 df-qtop 16634 df-imas 16635 df-xps 16637 df-mre 16727 df-mrc 16728 df-acs 16730 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-submnd 17816 df-mulg 18024 df-cntz 18230 df-cmn 18680 df-psmet 20254 df-xmet 20255 df-met 20256 df-bl 20257 df-mopn 20258 df-fbas 20259 df-fg 20260 df-cnfld 20263 df-top 21221 df-topon 21238 df-topsp 21260 df-bases 21273 df-cld 21346 df-ntr 21347 df-cls 21348 df-nei 21425 df-lp 21463 df-perf 21464 df-cn 21554 df-cnp 21555 df-haus 21642 df-tx 21889 df-hmeo 22082 df-fil 22173 df-fm 22265 df-flim 22266 df-flf 22267 df-xms 22648 df-ms 22649 df-tms 22650 df-cncf 23204 df-limc 24182 df-dv 24183 df-log 24856 df-cxp 24857

This theorem is referenced by: quartlem4 25154 quart 25155

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