quartlem4 - Metamath Proof Explorer

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Theorem quartlem4 26983

Description: Closure lemmas for quart 26984. (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)
quart.m0	⊢ (𝜑 → 𝑀 ≠ 0)
quart.i	⊢ (𝜑 → 𝐼 = (√‘((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆))))
quart.j	⊢ (𝜑 → 𝐽 = (√‘((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆))))

**Assertion**
Ref	Expression
quartlem4	⊢ (𝜑 → (𝑆 ≠ 0 ∧ 𝐼 ∈ ℂ ∧ 𝐽 ∈ ℂ))

**Proof of Theorem quartlem4**
Step	Hyp	Ref	Expression
1		quart.s	. . 3 ⊢ (𝜑 → 𝑆 = ((√‘𝑀) / 2))
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.e	. . . . . . 7 ⊢ (𝜑 → 𝐸 = -(𝐴 / 4))
7		quart.p	. . . . . . 7 ⊢ (𝜑 → 𝑃 = (𝐵 − ((3 / 8) · (𝐴↑2))))
8		quart.q	. . . . . . 7 ⊢ (𝜑 → 𝑄 = ((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8)))
9		quart.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 = ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4)))))
10		quart.u	. . . . . . 7 ⊢ (𝜑 → 𝑈 = ((𝑃↑2) + (;12 · 𝑅)))
11		quart.v	. . . . . . 7 ⊢ (𝜑 → 𝑉 = ((-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅))))
12		quart.w	. . . . . . 7 ⊢ (𝜑 → 𝑊 = (√‘((𝑉↑2) − (4 · (𝑈↑3)))))
13		quart.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 = -((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3))
14		quart.t	. . . . . . 7 ⊢ (𝜑 → 𝑇 = (((𝑉 + 𝑊) / 2)↑_𝑐(1 / 3)))
15		quart.t0	. . . . . . 7 ⊢ (𝜑 → 𝑇 ≠ 0)
16	2, 3, 4, 5, 2, 6, 7, 8, 9, 10, 11, 12, 1, 13, 14, 15	quartlem3 26982	. . . . . 6 ⊢ (𝜑 → (𝑆 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑇 ∈ ℂ))
17	16	simp2d 1159	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℂ)
18	17	sqrtcld 15481	. . . 4 ⊢ (𝜑 → (√‘𝑀) ∈ ℂ)
19		2cnd 12310	. . . 4 ⊢ (𝜑 → 2 ∈ ℂ)
20	17	sqsqrtd 15483	. . . . . 6 ⊢ (𝜑 → ((√‘𝑀)↑2) = 𝑀)
21		quart.m0	. . . . . 6 ⊢ (𝜑 → 𝑀 ≠ 0)
22	20, 21	eqnetrd 3027	. . . . 5 ⊢ (𝜑 → ((√‘𝑀)↑2) ≠ 0)
23		sqne0 14150	. . . . . 6 ⊢ ((√‘𝑀) ∈ ℂ → (((√‘𝑀)↑2) ≠ 0 ↔ (√‘𝑀) ≠ 0))
24	18, 23	syl 18	. . . . 5 ⊢ (𝜑 → (((√‘𝑀)↑2) ≠ 0 ↔ (√‘𝑀) ≠ 0))
25	22, 24	mpbid 235	. . . 4 ⊢ (𝜑 → (√‘𝑀) ≠ 0)
26		2ne0 12338	. . . . 5 ⊢ 2 ≠ 0
27	26	a1i 11	. . . 4 ⊢ (𝜑 → 2 ≠ 0)
28	18, 19, 25, 27	divne0d 11998	. . 3 ⊢ (𝜑 → ((√‘𝑀) / 2) ≠ 0)
29	1, 28	eqnetrd 3027	. 2 ⊢ (𝜑 → 𝑆 ≠ 0)
30		quart.i	. . 3 ⊢ (𝜑 → 𝐼 = (√‘((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆))))
31	16	simp1d 1158	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℂ)
32	31	sqcld 14171	. . . . . . 7 ⊢ (𝜑 → (𝑆↑2) ∈ ℂ)
33	32	negcld 11544	. . . . . 6 ⊢ (𝜑 → -(𝑆↑2) ∈ ℂ)
34	2, 3, 4, 5, 7, 8, 9	quart1cl 26977	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ))
35	34	simp1d 1158	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℂ)
36	35	halfcld 12480	. . . . . 6 ⊢ (𝜑 → (𝑃 / 2) ∈ ℂ)
37	33, 36	subcld 11557	. . . . 5 ⊢ (𝜑 → (-(𝑆↑2) − (𝑃 / 2)) ∈ ℂ)
38	34	simp2d 1159	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℂ)
39		4cn 12317	. . . . . . . 8 ⊢ 4 ∈ ℂ
40	39	a1i 11	. . . . . . 7 ⊢ (𝜑 → 4 ∈ ℂ)
41		4ne0 12343	. . . . . . . 8 ⊢ 4 ≠ 0
42	41	a1i 11	. . . . . . 7 ⊢ (𝜑 → 4 ≠ 0)
43	38, 40, 42	divcld 11982	. . . . . 6 ⊢ (𝜑 → (𝑄 / 4) ∈ ℂ)
44	43, 31, 29	divcld 11982	. . . . 5 ⊢ (𝜑 → ((𝑄 / 4) / 𝑆) ∈ ℂ)
45	37, 44	addcld 11216	. . . 4 ⊢ (𝜑 → ((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)) ∈ ℂ)
46	45	sqrtcld 15481	. . 3 ⊢ (𝜑 → (√‘((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆))) ∈ ℂ)
47	30, 46	eqeltrd 2865	. 2 ⊢ (𝜑 → 𝐼 ∈ ℂ)
48		quart.j	. . 3 ⊢ (𝜑 → 𝐽 = (√‘((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆))))
49	37, 44	subcld 11557	. . . 4 ⊢ (𝜑 → ((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)) ∈ ℂ)
50	49	sqrtcld 15481	. . 3 ⊢ (𝜑 → (√‘((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆))) ∈ ℂ)
51	48, 50	eqeltrd 2865	. 2 ⊢ (𝜑 → 𝐽 ∈ ℂ)
52	29, 47, 51	3jca 1144	1 ⊢ (𝜑 → (𝑆 ≠ 0 ∧ 𝐼 ∈ ℂ ∧ 𝐽 ∈ ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1101 = wceq 1563 ∈ wcel 2145 ≠ wne 2960 ‘cfv 6525 (class class class)co 7400 ℂcc 11086 0cc0 11088 1c1 11089 + caddc 11091 · cmul 11093 − cmin 11429 -cneg 11430 / cdiv 11859 2c2 12286 3c3 12287 4c4 12288 5c5 12289 6c6 12290 7c7 12291 8c8 12292 cdc 12702 ↑cexp 14088 √csqrt 15274 ↑_𝑐ccxp 26678

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-inf2 9598 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166 ax-addf 11167

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-iin 4955 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-se 5606 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7664 df-om 7851 df-1st 7974 df-2nd 7975 df-supp 8145 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-er 8682 df-map 8814 df-pm 8815 df-ixp 8884 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-9 12301 df-n0 12496 df-z 12583 df-dec 12703 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13367 df-ioc 13368 df-ico 13369 df-icc 13370 df-fz 13527 df-fzo 13674 df-fl 13816 df-mod 13894 df-seq 14029 df-exp 14089 df-fac 14301 df-bc 14330 df-hash 14358 df-shft 15094 df-cj 15140 df-re 15141 df-im 15142 df-sqrt 15276 df-abs 15277 df-limsup 15512 df-clim 15529 df-rlim 15530 df-sum 15728 df-ef 16111 df-sin 16113 df-cos 16114 df-pi 16116 df-struct 17197 df-sets 17214 df-slot 17232 df-ndx 17244 df-base 17260 df-ress 17281 df-plusg 17313 df-mulr 17314 df-starv 17315 df-sca 17316 df-vsca 17317 df-ip 17318 df-tset 17319 df-ple 17320 df-ds 17322 df-unif 17323 df-hom 17324 df-cco 17325 df-rest 17465 df-topn 17466 df-0g 17484 df-gsum 17485 df-topgen 17486 df-pt 17487 df-prds 17490 df-xrs 17546 df-qtop 17551 df-imas 17552 df-xps 17554 df-mre 17628 df-mrc 17629 df-acs 17631 df-mgm 18688 df-sgrp 18767 df-mnd 18783 df-submnd 18832 df-mulg 19125 df-cntz 19378 df-cmn 19843 df-psmet 21474 df-xmet 21475 df-met 21476 df-bl 21477 df-mopn 21478 df-fbas 21479 df-fg 21480 df-cnfld 21483 df-top 23012 df-topon 23029 df-topsp 23051 df-bases 23064 df-cld 23137 df-ntr 23138 df-cls 23139 df-nei 23216 df-lp 23254 df-perf 23255 df-cn 23345 df-cnp 23346 df-haus 23433 df-tx 23680 df-hmeo 23873 df-fil 23964 df-fm 24056 df-flim 24057 df-flf 24058 df-xms 24438 df-ms 24439 df-tms 24440 df-cncf 24998 df-limc 25986 df-dv 25987 df-log 26679 df-cxp 26680

This theorem is referenced by: quart 26984

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