quartlem3 - Metamath Proof Explorer

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Theorem quartlem3 26917

Description: Closure lemmas for quart 26919. (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 12339	. . . . . . . . . . 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 26912	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ))
12	11	simp1d 1141	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℂ)
13		mulcl 11237	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ 𝑃 ∈ ℂ) → (2 · 𝑃) ∈ ℂ)
14	3, 12, 13	sylancr 587	. . . . . . . . . 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 26916	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑉 ∈ ℂ ∧ 𝑊 ∈ ℂ))
21	20	simp2d 1142	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ∈ ℂ)
22	20	simp3d 1143	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑊 ∈ ℂ)
23	21, 22	addcld 11278	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑉 + 𝑊) ∈ ℂ)
24	23	halfcld 12509	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑉 + 𝑊) / 2) ∈ ℂ)
25		3nn 12343	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℕ
26		nnrecre 12306	. . . . . . . . . . . . . 14 ⊢ (3 ∈ ℕ → (1 / 3) ∈ ℝ)
27	25, 26	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (1 / 3) ∈ ℝ
28	27	recni 11273	. . . . . . . . . . . 12 ⊢ (1 / 3) ∈ ℂ
29		cxpcl 26731	. . . . . . . . . . . 12 ⊢ ((((𝑉 + 𝑊) / 2) ∈ ℂ ∧ (1 / 3) ∈ ℂ) → (((𝑉 + 𝑊) / 2)↑_𝑐(1 / 3)) ∈ ℂ)
30	24, 28, 29	sylancl 586	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑉 + 𝑊) / 2)↑_𝑐(1 / 3)) ∈ ℂ)
31	15, 30	eqeltrd 2839	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ∈ ℂ)
32	14, 31	addcld 11278	. . . . . . . . 9 ⊢ (𝜑 → ((2 · 𝑃) + 𝑇) ∈ ℂ)
33	20	simp1d 1141	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ ℂ)
34		quart.t0	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ≠ 0)
35	33, 31, 34	divcld 12041	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 / 𝑇) ∈ ℂ)
36	32, 35	addcld 11278	. . . . . . . 8 ⊢ (𝜑 → (((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) ∈ ℂ)
37		3cn 12345	. . . . . . . . 9 ⊢ 3 ∈ ℂ
38	37	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 3 ∈ ℂ)
39		3ne0 12370	. . . . . . . . 9 ⊢ 3 ≠ 0
40	39	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 3 ≠ 0)
41	36, 38, 40	divcld 12041	. . . . . . 7 ⊢ (𝜑 → ((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3) ∈ ℂ)
42	41	negcld 11605	. . . . . 6 ⊢ (𝜑 → -((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3) ∈ ℂ)
43	2, 42	eqeltrd 2839	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℂ)
44	43	sqrtcld 15473	. . . 4 ⊢ (𝜑 → (√‘𝑀) ∈ ℂ)
45	44	halfcld 12509	. . 3 ⊢ (𝜑 → ((√‘𝑀) / 2) ∈ ℂ)
46	1, 45	eqeltrd 2839	. 2 ⊢ (𝜑 → 𝑆 ∈ ℂ)
47	46, 43, 31	3jca 1127	1 ⊢ (𝜑 → (𝑆 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑇 ∈ ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 − cmin 11490 -cneg 11491 / cdiv 11918 ℕcn 12264 2c2 12319 3c3 12320 4c4 12321 5c5 12322 6c6 12323 7c7 12324 8c8 12325 cdc 12731 ↑cexp 14099 √csqrt 15269 ↑_𝑐ccxp 26612

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917 df-log 26613 df-cxp 26614

This theorem is referenced by: quartlem4 26918 quart 26919

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