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Mirrors > Home > MPE Home > Th. List > quartlem2 | Structured version Visualization version GIF version |
Description: Closure lemmas for quart 25716. (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 25709 | . . . . . 6 ⊢ (𝜑 → (𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ)) |
10 | 9 | simp1d 1144 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
11 | 10 | sqcld 13697 | . . . 4 ⊢ (𝜑 → (𝑃↑2) ∈ ℂ) |
12 | 1nn0 12089 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
13 | 2nn 11886 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
14 | 12, 13 | decnncl 12296 | . . . . . 6 ⊢ ;12 ∈ ℕ |
15 | 14 | nncni 11823 | . . . . 5 ⊢ ;12 ∈ ℂ |
16 | 9 | simp3d 1146 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℂ) |
17 | mulcl 10796 | . . . . 5 ⊢ ((;12 ∈ ℂ ∧ 𝑅 ∈ ℂ) → (;12 · 𝑅) ∈ ℂ) | |
18 | 15, 16, 17 | sylancr 590 | . . . 4 ⊢ (𝜑 → (;12 · 𝑅) ∈ ℂ) |
19 | 11, 18 | addcld 10835 | . . 3 ⊢ (𝜑 → ((𝑃↑2) + (;12 · 𝑅)) ∈ ℂ) |
20 | 1, 19 | eqeltrd 2834 | . 2 ⊢ (𝜑 → 𝑈 ∈ ℂ) |
21 | quart.v | . . 3 ⊢ (𝜑 → 𝑉 = ((-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅)))) | |
22 | 2cn 11888 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
23 | 3nn0 12091 | . . . . . . . 8 ⊢ 3 ∈ ℕ0 | |
24 | expcl 13636 | . . . . . . . 8 ⊢ ((𝑃 ∈ ℂ ∧ 3 ∈ ℕ0) → (𝑃↑3) ∈ ℂ) | |
25 | 10, 23, 24 | sylancl 589 | . . . . . . 7 ⊢ (𝜑 → (𝑃↑3) ∈ ℂ) |
26 | mulcl 10796 | . . . . . . 7 ⊢ ((2 ∈ ℂ ∧ (𝑃↑3) ∈ ℂ) → (2 · (𝑃↑3)) ∈ ℂ) | |
27 | 22, 25, 26 | sylancr 590 | . . . . . 6 ⊢ (𝜑 → (2 · (𝑃↑3)) ∈ ℂ) |
28 | 27 | negcld 11159 | . . . . 5 ⊢ (𝜑 → -(2 · (𝑃↑3)) ∈ ℂ) |
29 | 2nn0 12090 | . . . . . . . 8 ⊢ 2 ∈ ℕ0 | |
30 | 7nn 11905 | . . . . . . . 8 ⊢ 7 ∈ ℕ | |
31 | 29, 30 | decnncl 12296 | . . . . . . 7 ⊢ ;27 ∈ ℕ |
32 | 31 | nncni 11823 | . . . . . 6 ⊢ ;27 ∈ ℂ |
33 | 9 | simp2d 1145 | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℂ) |
34 | 33 | sqcld 13697 | . . . . . 6 ⊢ (𝜑 → (𝑄↑2) ∈ ℂ) |
35 | mulcl 10796 | . . . . . 6 ⊢ ((;27 ∈ ℂ ∧ (𝑄↑2) ∈ ℂ) → (;27 · (𝑄↑2)) ∈ ℂ) | |
36 | 32, 34, 35 | sylancr 590 | . . . . 5 ⊢ (𝜑 → (;27 · (𝑄↑2)) ∈ ℂ) |
37 | 28, 36 | subcld 11172 | . . . 4 ⊢ (𝜑 → (-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) ∈ ℂ) |
38 | 7nn0 12095 | . . . . . . 7 ⊢ 7 ∈ ℕ0 | |
39 | 38, 13 | decnncl 12296 | . . . . . 6 ⊢ ;72 ∈ ℕ |
40 | 39 | nncni 11823 | . . . . 5 ⊢ ;72 ∈ ℂ |
41 | 10, 16 | mulcld 10836 | . . . . 5 ⊢ (𝜑 → (𝑃 · 𝑅) ∈ ℂ) |
42 | mulcl 10796 | . . . . 5 ⊢ ((;72 ∈ ℂ ∧ (𝑃 · 𝑅) ∈ ℂ) → (;72 · (𝑃 · 𝑅)) ∈ ℂ) | |
43 | 40, 41, 42 | sylancr 590 | . . . 4 ⊢ (𝜑 → (;72 · (𝑃 · 𝑅)) ∈ ℂ) |
44 | 37, 43 | addcld 10835 | . . 3 ⊢ (𝜑 → ((-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅))) ∈ ℂ) |
45 | 21, 44 | eqeltrd 2834 | . 2 ⊢ (𝜑 → 𝑉 ∈ ℂ) |
46 | quart.w | . . 3 ⊢ (𝜑 → 𝑊 = (√‘((𝑉↑2) − (4 · (𝑈↑3))))) | |
47 | 45 | sqcld 13697 | . . . . 5 ⊢ (𝜑 → (𝑉↑2) ∈ ℂ) |
48 | 4cn 11898 | . . . . . 6 ⊢ 4 ∈ ℂ | |
49 | expcl 13636 | . . . . . . 7 ⊢ ((𝑈 ∈ ℂ ∧ 3 ∈ ℕ0) → (𝑈↑3) ∈ ℂ) | |
50 | 20, 23, 49 | sylancl 589 | . . . . . 6 ⊢ (𝜑 → (𝑈↑3) ∈ ℂ) |
51 | mulcl 10796 | . . . . . 6 ⊢ ((4 ∈ ℂ ∧ (𝑈↑3) ∈ ℂ) → (4 · (𝑈↑3)) ∈ ℂ) | |
52 | 48, 50, 51 | sylancr 590 | . . . . 5 ⊢ (𝜑 → (4 · (𝑈↑3)) ∈ ℂ) |
53 | 47, 52 | subcld 11172 | . . . 4 ⊢ (𝜑 → ((𝑉↑2) − (4 · (𝑈↑3))) ∈ ℂ) |
54 | 53 | sqrtcld 14984 | . . 3 ⊢ (𝜑 → (√‘((𝑉↑2) − (4 · (𝑈↑3)))) ∈ ℂ) |
55 | 46, 54 | eqeltrd 2834 | . 2 ⊢ (𝜑 → 𝑊 ∈ ℂ) |
56 | 20, 45, 55 | 3jca 1130 | 1 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑉 ∈ ℂ ∧ 𝑊 ∈ ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ‘cfv 6369 (class class class)co 7202 ℂcc 10710 1c1 10713 + caddc 10715 · cmul 10717 − cmin 11045 -cneg 11046 / cdiv 11472 2c2 11868 3c3 11869 4c4 11870 5c5 11871 6c6 11872 7c7 11873 8c8 11874 ℕ0cn0 12073 ;cdc 12276 ↑cexp 13618 √csqrt 14779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-sup 9047 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-rp 12570 df-seq 13558 df-exp 13619 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 |
This theorem is referenced by: quartlem3 25714 quart 25716 |
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