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Mirrors > Home > MPE Home > Th. List > quartlem2 | Structured version Visualization version GIF version |
Description: Closure lemmas for quart 25156. (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 25149 | . . . . . 6 ⊢ (𝜑 → (𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ)) |
10 | 9 | simp1d 1123 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
11 | 10 | sqcld 13322 | . . . 4 ⊢ (𝜑 → (𝑃↑2) ∈ ℂ) |
12 | 1nn0 11724 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
13 | 2nn 11512 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
14 | 12, 13 | decnncl 11931 | . . . . . 6 ⊢ ;12 ∈ ℕ |
15 | 14 | nncni 11449 | . . . . 5 ⊢ ;12 ∈ ℂ |
16 | 9 | simp3d 1125 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℂ) |
17 | mulcl 10418 | . . . . 5 ⊢ ((;12 ∈ ℂ ∧ 𝑅 ∈ ℂ) → (;12 · 𝑅) ∈ ℂ) | |
18 | 15, 16, 17 | sylancr 579 | . . . 4 ⊢ (𝜑 → (;12 · 𝑅) ∈ ℂ) |
19 | 11, 18 | addcld 10458 | . . 3 ⊢ (𝜑 → ((𝑃↑2) + (;12 · 𝑅)) ∈ ℂ) |
20 | 1, 19 | eqeltrd 2861 | . 2 ⊢ (𝜑 → 𝑈 ∈ ℂ) |
21 | quart.v | . . 3 ⊢ (𝜑 → 𝑉 = ((-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅)))) | |
22 | 2cn 11514 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
23 | 3nn0 11726 | . . . . . . . 8 ⊢ 3 ∈ ℕ0 | |
24 | expcl 13261 | . . . . . . . 8 ⊢ ((𝑃 ∈ ℂ ∧ 3 ∈ ℕ0) → (𝑃↑3) ∈ ℂ) | |
25 | 10, 23, 24 | sylancl 578 | . . . . . . 7 ⊢ (𝜑 → (𝑃↑3) ∈ ℂ) |
26 | mulcl 10418 | . . . . . . 7 ⊢ ((2 ∈ ℂ ∧ (𝑃↑3) ∈ ℂ) → (2 · (𝑃↑3)) ∈ ℂ) | |
27 | 22, 25, 26 | sylancr 579 | . . . . . 6 ⊢ (𝜑 → (2 · (𝑃↑3)) ∈ ℂ) |
28 | 27 | negcld 10784 | . . . . 5 ⊢ (𝜑 → -(2 · (𝑃↑3)) ∈ ℂ) |
29 | 2nn0 11725 | . . . . . . . 8 ⊢ 2 ∈ ℕ0 | |
30 | 7nn 11535 | . . . . . . . 8 ⊢ 7 ∈ ℕ | |
31 | 29, 30 | decnncl 11931 | . . . . . . 7 ⊢ ;27 ∈ ℕ |
32 | 31 | nncni 11449 | . . . . . 6 ⊢ ;27 ∈ ℂ |
33 | 9 | simp2d 1124 | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℂ) |
34 | 33 | sqcld 13322 | . . . . . 6 ⊢ (𝜑 → (𝑄↑2) ∈ ℂ) |
35 | mulcl 10418 | . . . . . 6 ⊢ ((;27 ∈ ℂ ∧ (𝑄↑2) ∈ ℂ) → (;27 · (𝑄↑2)) ∈ ℂ) | |
36 | 32, 34, 35 | sylancr 579 | . . . . 5 ⊢ (𝜑 → (;27 · (𝑄↑2)) ∈ ℂ) |
37 | 28, 36 | subcld 10797 | . . . 4 ⊢ (𝜑 → (-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) ∈ ℂ) |
38 | 7nn0 11730 | . . . . . . 7 ⊢ 7 ∈ ℕ0 | |
39 | 38, 13 | decnncl 11931 | . . . . . 6 ⊢ ;72 ∈ ℕ |
40 | 39 | nncni 11449 | . . . . 5 ⊢ ;72 ∈ ℂ |
41 | 10, 16 | mulcld 10459 | . . . . 5 ⊢ (𝜑 → (𝑃 · 𝑅) ∈ ℂ) |
42 | mulcl 10418 | . . . . 5 ⊢ ((;72 ∈ ℂ ∧ (𝑃 · 𝑅) ∈ ℂ) → (;72 · (𝑃 · 𝑅)) ∈ ℂ) | |
43 | 40, 41, 42 | sylancr 579 | . . . 4 ⊢ (𝜑 → (;72 · (𝑃 · 𝑅)) ∈ ℂ) |
44 | 37, 43 | addcld 10458 | . . 3 ⊢ (𝜑 → ((-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅))) ∈ ℂ) |
45 | 21, 44 | eqeltrd 2861 | . 2 ⊢ (𝜑 → 𝑉 ∈ ℂ) |
46 | quart.w | . . 3 ⊢ (𝜑 → 𝑊 = (√‘((𝑉↑2) − (4 · (𝑈↑3))))) | |
47 | 45 | sqcld 13322 | . . . . 5 ⊢ (𝜑 → (𝑉↑2) ∈ ℂ) |
48 | 4cn 11525 | . . . . . 6 ⊢ 4 ∈ ℂ | |
49 | expcl 13261 | . . . . . . 7 ⊢ ((𝑈 ∈ ℂ ∧ 3 ∈ ℕ0) → (𝑈↑3) ∈ ℂ) | |
50 | 20, 23, 49 | sylancl 578 | . . . . . 6 ⊢ (𝜑 → (𝑈↑3) ∈ ℂ) |
51 | mulcl 10418 | . . . . . 6 ⊢ ((4 ∈ ℂ ∧ (𝑈↑3) ∈ ℂ) → (4 · (𝑈↑3)) ∈ ℂ) | |
52 | 48, 50, 51 | sylancr 579 | . . . . 5 ⊢ (𝜑 → (4 · (𝑈↑3)) ∈ ℂ) |
53 | 47, 52 | subcld 10797 | . . . 4 ⊢ (𝜑 → ((𝑉↑2) − (4 · (𝑈↑3))) ∈ ℂ) |
54 | 53 | sqrtcld 14657 | . . 3 ⊢ (𝜑 → (√‘((𝑉↑2) − (4 · (𝑈↑3)))) ∈ ℂ) |
55 | 46, 54 | eqeltrd 2861 | . 2 ⊢ (𝜑 → 𝑊 ∈ ℂ) |
56 | 20, 45, 55 | 3jca 1109 | 1 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑉 ∈ ℂ ∧ 𝑊 ∈ ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ‘cfv 6186 (class class class)co 6975 ℂcc 10332 1c1 10335 + caddc 10337 · cmul 10339 − cmin 10669 -cneg 10670 / cdiv 11097 2c2 11494 3c3 11495 4c4 11496 5c5 11497 6c6 11498 7c7 11499 8c8 11500 ℕ0cn0 11706 ;cdc 11910 ↑cexp 13243 √csqrt 14452 |
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 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 ax-pre-sup 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-2nd 7501 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-sup 8700 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-div 11098 df-nn 11439 df-2 11502 df-3 11503 df-4 11504 df-5 11505 df-6 11506 df-7 11507 df-8 11508 df-9 11509 df-n0 11707 df-z 11793 df-dec 11911 df-uz 12058 df-rp 12204 df-seq 13184 df-exp 13244 df-cj 14318 df-re 14319 df-im 14320 df-sqrt 14454 df-abs 14455 |
This theorem is referenced by: quartlem3 25154 quart 25156 |
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