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Mirrors > Home > MPE Home > Th. List > quartlem3 | Structured version Visualization version GIF version |
Description: Closure lemmas for quart 25155. (Contributed by Mario Carneiro, 7-May-2015.) |
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) |
Ref | Expression |
---|---|
quartlem3 | ⊢ (𝜑 → (𝑆 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑇 ∈ ℂ)) |
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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