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Mirrors > Home > MPE Home > Th. List > quartlem2 | Structured version Visualization version GIF version |
Description: Closure lemmas for quart 25366. (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))))) |
Ref | Expression |
---|---|
quartlem2 | ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑉 ∈ ℂ ∧ 𝑊 ∈ ℂ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | quart.u | . . 3 ⊢ (𝜑 → 𝑈 = ((𝑃↑2) + (;12 · 𝑅))) | |
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.p | . . . . . . 7 ⊢ (𝜑 → 𝑃 = (𝐵 − ((3 / 8) · (𝐴↑2)))) | |
7 | quart.q | . . . . . . 7 ⊢ (𝜑 → 𝑄 = ((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))) | |
8 | quart.r | . . . . . . 7 ⊢ (𝜑 → 𝑅 = ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))) | |
9 | 2, 3, 4, 5, 6, 7, 8 | quart1cl 25359 | . . . . . 6 ⊢ (𝜑 → (𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ)) |
10 | 9 | simp1d 1134 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
11 | 10 | sqcld 13496 | . . . 4 ⊢ (𝜑 → (𝑃↑2) ∈ ℂ) |
12 | 1nn0 11901 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
13 | 2nn 11698 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
14 | 12, 13 | decnncl 12106 | . . . . . 6 ⊢ ;12 ∈ ℕ |
15 | 14 | nncni 11636 | . . . . 5 ⊢ ;12 ∈ ℂ |
16 | 9 | simp3d 1136 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℂ) |
17 | mulcl 10609 | . . . . 5 ⊢ ((;12 ∈ ℂ ∧ 𝑅 ∈ ℂ) → (;12 · 𝑅) ∈ ℂ) | |
18 | 15, 16, 17 | sylancr 587 | . . . 4 ⊢ (𝜑 → (;12 · 𝑅) ∈ ℂ) |
19 | 11, 18 | addcld 10648 | . . 3 ⊢ (𝜑 → ((𝑃↑2) + (;12 · 𝑅)) ∈ ℂ) |
20 | 1, 19 | eqeltrd 2910 | . 2 ⊢ (𝜑 → 𝑈 ∈ ℂ) |
21 | quart.v | . . 3 ⊢ (𝜑 → 𝑉 = ((-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅)))) | |
22 | 2cn 11700 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
23 | 3nn0 11903 | . . . . . . . 8 ⊢ 3 ∈ ℕ0 | |
24 | expcl 13435 | . . . . . . . 8 ⊢ ((𝑃 ∈ ℂ ∧ 3 ∈ ℕ0) → (𝑃↑3) ∈ ℂ) | |
25 | 10, 23, 24 | sylancl 586 | . . . . . . 7 ⊢ (𝜑 → (𝑃↑3) ∈ ℂ) |
26 | mulcl 10609 | . . . . . . 7 ⊢ ((2 ∈ ℂ ∧ (𝑃↑3) ∈ ℂ) → (2 · (𝑃↑3)) ∈ ℂ) | |
27 | 22, 25, 26 | sylancr 587 | . . . . . 6 ⊢ (𝜑 → (2 · (𝑃↑3)) ∈ ℂ) |
28 | 27 | negcld 10972 | . . . . 5 ⊢ (𝜑 → -(2 · (𝑃↑3)) ∈ ℂ) |
29 | 2nn0 11902 | . . . . . . . 8 ⊢ 2 ∈ ℕ0 | |
30 | 7nn 11717 | . . . . . . . 8 ⊢ 7 ∈ ℕ | |
31 | 29, 30 | decnncl 12106 | . . . . . . 7 ⊢ ;27 ∈ ℕ |
32 | 31 | nncni 11636 | . . . . . 6 ⊢ ;27 ∈ ℂ |
33 | 9 | simp2d 1135 | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℂ) |
34 | 33 | sqcld 13496 | . . . . . 6 ⊢ (𝜑 → (𝑄↑2) ∈ ℂ) |
35 | mulcl 10609 | . . . . . 6 ⊢ ((;27 ∈ ℂ ∧ (𝑄↑2) ∈ ℂ) → (;27 · (𝑄↑2)) ∈ ℂ) | |
36 | 32, 34, 35 | sylancr 587 | . . . . 5 ⊢ (𝜑 → (;27 · (𝑄↑2)) ∈ ℂ) |
37 | 28, 36 | subcld 10985 | . . . 4 ⊢ (𝜑 → (-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) ∈ ℂ) |
38 | 7nn0 11907 | . . . . . . 7 ⊢ 7 ∈ ℕ0 | |
39 | 38, 13 | decnncl 12106 | . . . . . 6 ⊢ ;72 ∈ ℕ |
40 | 39 | nncni 11636 | . . . . 5 ⊢ ;72 ∈ ℂ |
41 | 10, 16 | mulcld 10649 | . . . . 5 ⊢ (𝜑 → (𝑃 · 𝑅) ∈ ℂ) |
42 | mulcl 10609 | . . . . 5 ⊢ ((;72 ∈ ℂ ∧ (𝑃 · 𝑅) ∈ ℂ) → (;72 · (𝑃 · 𝑅)) ∈ ℂ) | |
43 | 40, 41, 42 | sylancr 587 | . . . 4 ⊢ (𝜑 → (;72 · (𝑃 · 𝑅)) ∈ ℂ) |
44 | 37, 43 | addcld 10648 | . . 3 ⊢ (𝜑 → ((-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅))) ∈ ℂ) |
45 | 21, 44 | eqeltrd 2910 | . 2 ⊢ (𝜑 → 𝑉 ∈ ℂ) |
46 | quart.w | . . 3 ⊢ (𝜑 → 𝑊 = (√‘((𝑉↑2) − (4 · (𝑈↑3))))) | |
47 | 45 | sqcld 13496 | . . . . 5 ⊢ (𝜑 → (𝑉↑2) ∈ ℂ) |
48 | 4cn 11710 | . . . . . 6 ⊢ 4 ∈ ℂ | |
49 | expcl 13435 | . . . . . . 7 ⊢ ((𝑈 ∈ ℂ ∧ 3 ∈ ℕ0) → (𝑈↑3) ∈ ℂ) | |
50 | 20, 23, 49 | sylancl 586 | . . . . . 6 ⊢ (𝜑 → (𝑈↑3) ∈ ℂ) |
51 | mulcl 10609 | . . . . . 6 ⊢ ((4 ∈ ℂ ∧ (𝑈↑3) ∈ ℂ) → (4 · (𝑈↑3)) ∈ ℂ) | |
52 | 48, 50, 51 | sylancr 587 | . . . . 5 ⊢ (𝜑 → (4 · (𝑈↑3)) ∈ ℂ) |
53 | 47, 52 | subcld 10985 | . . . 4 ⊢ (𝜑 → ((𝑉↑2) − (4 · (𝑈↑3))) ∈ ℂ) |
54 | 53 | sqrtcld 14785 | . . 3 ⊢ (𝜑 → (√‘((𝑉↑2) − (4 · (𝑈↑3)))) ∈ ℂ) |
55 | 46, 54 | eqeltrd 2910 | . 2 ⊢ (𝜑 → 𝑊 ∈ ℂ) |
56 | 20, 45, 55 | 3jca 1120 | 1 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑉 ∈ ℂ ∧ 𝑊 ∈ ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 1c1 10526 + caddc 10528 · cmul 10530 − cmin 10858 -cneg 10859 / cdiv 11285 2c2 11680 3c3 11681 4c4 11682 5c5 11683 6c6 11684 7c7 11685 8c8 11686 ℕ0cn0 11885 ;cdc 12086 ↑cexp 13417 √csqrt 14580 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 |
This theorem is referenced by: quartlem3 25364 quart 25366 |
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