quartlem3 - Metamath Proof Explorer

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Theorem quartlem3 26785

Description: Closure lemmas for quart 26787. (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 12221	. . . . . . . . . . 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 26780	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ))
12	11	simp1d 1142	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℂ)
13		mulcl 11112	. . . . . . . . . . 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 26784	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑉 ∈ ℂ ∧ 𝑊 ∈ ℂ))
21	20	simp2d 1143	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ∈ ℂ)
22	20	simp3d 1144	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑊 ∈ ℂ)
23	21, 22	addcld 11153	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑉 + 𝑊) ∈ ℂ)
24	23	halfcld 12387	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑉 + 𝑊) / 2) ∈ ℂ)
25		3nn 12225	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℕ
26		nnrecre 12188	. . . . . . . . . . . . . 14 ⊢ (3 ∈ ℕ → (1 / 3) ∈ ℝ)
27	25, 26	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (1 / 3) ∈ ℝ
28	27	recni 11148	. . . . . . . . . . . 12 ⊢ (1 / 3) ∈ ℂ
29		cxpcl 26599	. . . . . . . . . . . 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 2828	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ∈ ℂ)
32	14, 31	addcld 11153	. . . . . . . . 9 ⊢ (𝜑 → ((2 · 𝑃) + 𝑇) ∈ ℂ)
33	20	simp1d 1142	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ ℂ)
34		quart.t0	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ≠ 0)
35	33, 31, 34	divcld 11918	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 / 𝑇) ∈ ℂ)
36	32, 35	addcld 11153	. . . . . . . 8 ⊢ (𝜑 → (((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) ∈ ℂ)
37		3cn 12227	. . . . . . . . 9 ⊢ 3 ∈ ℂ
38	37	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 3 ∈ ℂ)
39		3ne0 12252	. . . . . . . . 9 ⊢ 3 ≠ 0
40	39	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 3 ≠ 0)
41	36, 38, 40	divcld 11918	. . . . . . 7 ⊢ (𝜑 → ((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3) ∈ ℂ)
42	41	negcld 11480	. . . . . 6 ⊢ (𝜑 → -((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3) ∈ ℂ)
43	2, 42	eqeltrd 2828	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℂ)
44	43	sqrtcld 15365	. . . 4 ⊢ (𝜑 → (√‘𝑀) ∈ ℂ)
45	44	halfcld 12387	. . 3 ⊢ (𝜑 → ((√‘𝑀) / 2) ∈ ℂ)
46	1, 45	eqeltrd 2828	. 2 ⊢ (𝜑 → 𝑆 ∈ ℂ)
47	46, 43, 31	3jca 1128	1 ⊢ (𝜑 → (𝑆 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑇 ∈ ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ‘cfv 6486 (class class class)co 7353 ℂcc 11026 ℝcr 11027 0cc0 11028 1c1 11029 + caddc 11031 · cmul 11033 − cmin 11365 -cneg 11366 / cdiv 11795 ℕcn 12146 2c2 12201 3c3 12202 4c4 12203 5c5 12204 6c6 12205 7c7 12206 8c8 12207 cdc 12609 ↑cexp 13986 √csqrt 15158 ↑_𝑐ccxp 26480

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8632 df-map 8762 df-pm 8763 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-fi 9320 df-sup 9351 df-inf 9352 df-oi 9421 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-q 12868 df-rp 12912 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-ioo 13270 df-ioc 13271 df-ico 13272 df-icc 13273 df-fz 13429 df-fzo 13576 df-fl 13714 df-mod 13792 df-seq 13927 df-exp 13987 df-fac 14199 df-bc 14228 df-hash 14256 df-shft 14992 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-limsup 15396 df-clim 15413 df-rlim 15414 df-sum 15612 df-ef 15992 df-sin 15994 df-cos 15995 df-pi 15997 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17344 df-topn 17345 df-0g 17363 df-gsum 17364 df-topgen 17365 df-pt 17366 df-prds 17369 df-xrs 17424 df-qtop 17429 df-imas 17430 df-xps 17432 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-submnd 18676 df-mulg 18965 df-cntz 19214 df-cmn 19679 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-fbas 21276 df-fg 21277 df-cnfld 21280 df-top 22797 df-topon 22814 df-topsp 22836 df-bases 22849 df-cld 22922 df-ntr 22923 df-cls 22924 df-nei 23001 df-lp 23039 df-perf 23040 df-cn 23130 df-cnp 23131 df-haus 23218 df-tx 23465 df-hmeo 23658 df-fil 23749 df-fm 23841 df-flim 23842 df-flf 23843 df-xms 24224 df-ms 24225 df-tms 24226 df-cncf 24787 df-limc 25783 df-dv 25784 df-log 26481 df-cxp 26482

This theorem is referenced by: quartlem4 26786 quart 26787

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