![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > quartlem3 | Structured version Visualization version GIF version |
Description: Closure lemmas for quart 26708. (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 12283 | . . . . . . . . . . 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 26701 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ)) |
12 | 11 | simp1d 1139 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
13 | mulcl 11189 | . . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ 𝑃 ∈ ℂ) → (2 · 𝑃) ∈ ℂ) | |
14 | 3, 12, 13 | sylancr 586 | . . . . . . . . . 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 26705 | . . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑉 ∈ ℂ ∧ 𝑊 ∈ ℂ)) |
21 | 20 | simp2d 1140 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ∈ ℂ) |
22 | 20 | simp3d 1141 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑊 ∈ ℂ) |
23 | 21, 22 | addcld 11229 | . . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑉 + 𝑊) ∈ ℂ) |
24 | 23 | halfcld 12453 | . . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑉 + 𝑊) / 2) ∈ ℂ) |
25 | 3nn 12287 | . . . . . . . . . . . . . 14 ⊢ 3 ∈ ℕ | |
26 | nnrecre 12250 | . . . . . . . . . . . . . 14 ⊢ (3 ∈ ℕ → (1 / 3) ∈ ℝ) | |
27 | 25, 26 | ax-mp 5 | . . . . . . . . . . . . 13 ⊢ (1 / 3) ∈ ℝ |
28 | 27 | recni 11224 | . . . . . . . . . . . 12 ⊢ (1 / 3) ∈ ℂ |
29 | cxpcl 26523 | . . . . . . . . . . . 12 ⊢ ((((𝑉 + 𝑊) / 2) ∈ ℂ ∧ (1 / 3) ∈ ℂ) → (((𝑉 + 𝑊) / 2)↑𝑐(1 / 3)) ∈ ℂ) | |
30 | 24, 28, 29 | sylancl 585 | . . . . . . . . . . 11 ⊢ (𝜑 → (((𝑉 + 𝑊) / 2)↑𝑐(1 / 3)) ∈ ℂ) |
31 | 15, 30 | eqeltrd 2825 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ∈ ℂ) |
32 | 14, 31 | addcld 11229 | . . . . . . . . 9 ⊢ (𝜑 → ((2 · 𝑃) + 𝑇) ∈ ℂ) |
33 | 20 | simp1d 1139 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ ℂ) |
34 | quart.t0 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ≠ 0) | |
35 | 33, 31, 34 | divcld 11986 | . . . . . . . . 9 ⊢ (𝜑 → (𝑈 / 𝑇) ∈ ℂ) |
36 | 32, 35 | addcld 11229 | . . . . . . . 8 ⊢ (𝜑 → (((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) ∈ ℂ) |
37 | 3cn 12289 | . . . . . . . . 9 ⊢ 3 ∈ ℂ | |
38 | 37 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 3 ∈ ℂ) |
39 | 3ne0 12314 | . . . . . . . . 9 ⊢ 3 ≠ 0 | |
40 | 39 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 3 ≠ 0) |
41 | 36, 38, 40 | divcld 11986 | . . . . . . 7 ⊢ (𝜑 → ((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3) ∈ ℂ) |
42 | 41 | negcld 11554 | . . . . . 6 ⊢ (𝜑 → -((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3) ∈ ℂ) |
43 | 2, 42 | eqeltrd 2825 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
44 | 43 | sqrtcld 15380 | . . . 4 ⊢ (𝜑 → (√‘𝑀) ∈ ℂ) |
45 | 44 | halfcld 12453 | . . 3 ⊢ (𝜑 → ((√‘𝑀) / 2) ∈ ℂ) |
46 | 1, 45 | eqeltrd 2825 | . 2 ⊢ (𝜑 → 𝑆 ∈ ℂ) |
47 | 46, 43, 31 | 3jca 1125 | 1 ⊢ (𝜑 → (𝑆 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑇 ∈ ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ‘cfv 6533 (class class class)co 7401 ℂcc 11103 ℝcr 11104 0cc0 11105 1c1 11106 + caddc 11108 · cmul 11110 − cmin 11440 -cneg 11441 / cdiv 11867 ℕcn 12208 2c2 12263 3c3 12264 4c4 12265 5c5 12266 6c6 12267 7c7 12268 8c8 12269 ;cdc 12673 ↑cexp 14023 √csqrt 15176 ↑𝑐ccxp 26405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9631 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 ax-addf 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8698 df-map 8817 df-pm 8818 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-fi 9401 df-sup 9432 df-inf 9433 df-oi 9500 df-card 9929 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17143 df-ress 17172 df-plusg 17208 df-mulr 17209 df-starv 17210 df-sca 17211 df-vsca 17212 df-ip 17213 df-tset 17214 df-ple 17215 df-ds 17217 df-unif 17218 df-hom 17219 df-cco 17220 df-rest 17366 df-topn 17367 df-0g 17385 df-gsum 17386 df-topgen 17387 df-pt 17388 df-prds 17391 df-xrs 17446 df-qtop 17451 df-imas 17452 df-xps 17454 df-mre 17528 df-mrc 17529 df-acs 17531 df-mgm 18562 df-sgrp 18641 df-mnd 18657 df-submnd 18703 df-mulg 18985 df-cntz 19222 df-cmn 19691 df-psmet 21219 df-xmet 21220 df-met 21221 df-bl 21222 df-mopn 21223 df-fbas 21224 df-fg 21225 df-cnfld 21228 df-top 22717 df-topon 22734 df-topsp 22756 df-bases 22770 df-cld 22844 df-ntr 22845 df-cls 22846 df-nei 22923 df-lp 22961 df-perf 22962 df-cn 23052 df-cnp 23053 df-haus 23140 df-tx 23387 df-hmeo 23580 df-fil 23671 df-fm 23763 df-flim 23764 df-flf 23765 df-xms 24147 df-ms 24148 df-tms 24149 df-cncf 24719 df-limc 25716 df-dv 25717 df-log 26406 df-cxp 26407 |
This theorem is referenced by: quartlem4 26707 quart 26708 |
Copyright terms: Public domain | W3C validator |