quartlem4 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > quartlem4		Structured version Visualization version GIF version

Theorem quartlem4 27006

Description: Closure lemmas for quart 27007. (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 27005	. . . . . 6 ⊢ (𝜑 → (𝑆 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑇 ∈ ℂ))
17	16	simp2d 1161	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℂ)
18	17	sqrtcld 15493	. . . 4 ⊢ (𝜑 → (√‘𝑀) ∈ ℂ)
19		2cnd 12320	. . . 4 ⊢ (𝜑 → 2 ∈ ℂ)
20	17	sqsqrtd 15495	. . . . . 6 ⊢ (𝜑 → ((√‘𝑀)↑2) = 𝑀)
21		quart.m0	. . . . . 6 ⊢ (𝜑 → 𝑀 ≠ 0)
22	20, 21	eqnetrd 3025	. . . . 5 ⊢ (𝜑 → ((√‘𝑀)↑2) ≠ 0)
23		sqne0 14161	. . . . . 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 12348	. . . . 5 ⊢ 2 ≠ 0
27	26	a1i 11	. . . 4 ⊢ (𝜑 → 2 ≠ 0)
28	18, 19, 25, 27	divne0d 12008	. . 3 ⊢ (𝜑 → ((√‘𝑀) / 2) ≠ 0)
29	1, 28	eqnetrd 3025	. 2 ⊢ (𝜑 → 𝑆 ≠ 0)
30		quart.i	. . 3 ⊢ (𝜑 → 𝐼 = (√‘((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆))))
31	16	simp1d 1160	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℂ)
32	31	sqcld 14182	. . . . . . 7 ⊢ (𝜑 → (𝑆↑2) ∈ ℂ)
33	32	negcld 11557	. . . . . 6 ⊢ (𝜑 → -(𝑆↑2) ∈ ℂ)
34	2, 3, 4, 5, 7, 8, 9	quart1cl 27000	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ))
35	34	simp1d 1160	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℂ)
36	35	halfcld 12490	. . . . . 6 ⊢ (𝜑 → (𝑃 / 2) ∈ ℂ)
37	33, 36	subcld 11570	. . . . 5 ⊢ (𝜑 → (-(𝑆↑2) − (𝑃 / 2)) ∈ ℂ)
38	34	simp2d 1161	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℂ)
39		4cn 12327	. . . . . . . 8 ⊢ 4 ∈ ℂ
40	39	a1i 11	. . . . . . 7 ⊢ (𝜑 → 4 ∈ ℂ)
41		4ne0 12353	. . . . . . . 8 ⊢ 4 ≠ 0
42	41	a1i 11	. . . . . . 7 ⊢ (𝜑 → 4 ≠ 0)
43	38, 40, 42	divcld 11992	. . . . . 6 ⊢ (𝜑 → (𝑄 / 4) ∈ ℂ)
44	43, 31, 29	divcld 11992	. . . . 5 ⊢ (𝜑 → ((𝑄 / 4) / 𝑆) ∈ ℂ)
45	37, 44	addcld 11229	. . . 4 ⊢ (𝜑 → ((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)) ∈ ℂ)
46	45	sqrtcld 15493	. . 3 ⊢ (𝜑 → (√‘((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆))) ∈ ℂ)
47	30, 46	eqeltrd 2863	. 2 ⊢ (𝜑 → 𝐼 ∈ ℂ)
48		quart.j	. . 3 ⊢ (𝜑 → 𝐽 = (√‘((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆))))
49	37, 44	subcld 11570	. . . 4 ⊢ (𝜑 → ((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)) ∈ ℂ)
50	49	sqrtcld 15493	. . 3 ⊢ (𝜑 → (√‘((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆))) ∈ ℂ)
51	48, 50	eqeltrd 2863	. 2 ⊢ (𝜑 → 𝐽 ∈ ℂ)
52	29, 47, 51	3jca 1146	1 ⊢ (𝜑 → (𝑆 ≠ 0 ∧ 𝐼 ∈ ℂ ∧ 𝐽 ∈ ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ‘cfv 6538 (class class class)co 7412 ℂcc 11099 0cc0 11101 1c1 11102 + caddc 11104 · cmul 11106 − cmin 11442 -cneg 11443 / cdiv 11872 2c2 12296 3c3 12297 4c4 12298 5c5 12299 6c6 12300 7c7 12301 8c8 12302 cdc 12712 ↑cexp 14099 √csqrt 15286 ↑_𝑐ccxp 26701

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179 ax-addf 11180

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 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 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8158 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-2o 8455 df-er 8695 df-map 8827 df-pm 8828 df-ixp 8897 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-fsupp 9323 df-fi 9372 df-sup 9403 df-inf 9404 df-oi 9473 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-4 12306 df-5 12307 df-6 12308 df-7 12309 df-8 12310 df-9 12311 df-n0 12506 df-z 12593 df-dec 12713 df-uz 12864 df-q 12974 df-rp 13018 df-xneg 13138 df-xadd 13139 df-xmul 13140 df-ioo 13377 df-ioc 13378 df-ico 13379 df-icc 13380 df-fz 13537 df-fzo 13685 df-fl 13827 df-mod 13905 df-seq 14040 df-exp 14100 df-fac 14312 df-bc 14341 df-hash 14369 df-shft 15106 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-limsup 15524 df-clim 15541 df-rlim 15542 df-sum 15740 df-ef 16122 df-sin 16124 df-cos 16125 df-pi 16127 df-struct 17208 df-sets 17225 df-slot 17243 df-ndx 17255 df-base 17271 df-ress 17292 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17476 df-topn 17477 df-0g 17495 df-gsum 17496 df-topgen 17497 df-pt 17498 df-prds 17501 df-xrs 17557 df-qtop 17562 df-imas 17563 df-xps 17565 df-mre 17639 df-mrc 17640 df-acs 17642 df-mgm 18699 df-sgrp 18778 df-mnd 18794 df-submnd 18843 df-mulg 19135 df-cntz 19388 df-cmn 19853 df-psmet 21495 df-xmet 21496 df-met 21497 df-bl 21498 df-mopn 21499 df-fbas 21500 df-fg 21501 df-cnfld 21504 df-top 23032 df-topon 23049 df-topsp 23071 df-bases 23084 df-cld 23157 df-ntr 23158 df-cls 23159 df-nei 23236 df-lp 23274 df-perf 23275 df-cn 23365 df-cnp 23366 df-haus 23453 df-tx 23700 df-hmeo 23893 df-fil 23984 df-fm 24076 df-flim 24077 df-flf 24078 df-xms 24458 df-ms 24459 df-tms 24460 df-cncf 25018 df-limc 26006 df-dv 26007 df-log 26702 df-cxp 26703

This theorem is referenced by: quart 27007

W3C validator