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Mirrors > Home > MPE Home > Th. List > quartlem2 | Structured version Visualization version GIF version |
Description: Closure lemmas for quart 25439. (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 25432 | . . . . . 6 ⊢ (𝜑 → (𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ)) |
10 | 9 | simp1d 1138 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
11 | 10 | sqcld 13509 | . . . 4 ⊢ (𝜑 → (𝑃↑2) ∈ ℂ) |
12 | 1nn0 11914 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
13 | 2nn 11711 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
14 | 12, 13 | decnncl 12119 | . . . . . 6 ⊢ ;12 ∈ ℕ |
15 | 14 | nncni 11648 | . . . . 5 ⊢ ;12 ∈ ℂ |
16 | 9 | simp3d 1140 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℂ) |
17 | mulcl 10621 | . . . . 5 ⊢ ((;12 ∈ ℂ ∧ 𝑅 ∈ ℂ) → (;12 · 𝑅) ∈ ℂ) | |
18 | 15, 16, 17 | sylancr 589 | . . . 4 ⊢ (𝜑 → (;12 · 𝑅) ∈ ℂ) |
19 | 11, 18 | addcld 10660 | . . 3 ⊢ (𝜑 → ((𝑃↑2) + (;12 · 𝑅)) ∈ ℂ) |
20 | 1, 19 | eqeltrd 2913 | . 2 ⊢ (𝜑 → 𝑈 ∈ ℂ) |
21 | quart.v | . . 3 ⊢ (𝜑 → 𝑉 = ((-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅)))) | |
22 | 2cn 11713 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
23 | 3nn0 11916 | . . . . . . . 8 ⊢ 3 ∈ ℕ0 | |
24 | expcl 13448 | . . . . . . . 8 ⊢ ((𝑃 ∈ ℂ ∧ 3 ∈ ℕ0) → (𝑃↑3) ∈ ℂ) | |
25 | 10, 23, 24 | sylancl 588 | . . . . . . 7 ⊢ (𝜑 → (𝑃↑3) ∈ ℂ) |
26 | mulcl 10621 | . . . . . . 7 ⊢ ((2 ∈ ℂ ∧ (𝑃↑3) ∈ ℂ) → (2 · (𝑃↑3)) ∈ ℂ) | |
27 | 22, 25, 26 | sylancr 589 | . . . . . 6 ⊢ (𝜑 → (2 · (𝑃↑3)) ∈ ℂ) |
28 | 27 | negcld 10984 | . . . . 5 ⊢ (𝜑 → -(2 · (𝑃↑3)) ∈ ℂ) |
29 | 2nn0 11915 | . . . . . . . 8 ⊢ 2 ∈ ℕ0 | |
30 | 7nn 11730 | . . . . . . . 8 ⊢ 7 ∈ ℕ | |
31 | 29, 30 | decnncl 12119 | . . . . . . 7 ⊢ ;27 ∈ ℕ |
32 | 31 | nncni 11648 | . . . . . 6 ⊢ ;27 ∈ ℂ |
33 | 9 | simp2d 1139 | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℂ) |
34 | 33 | sqcld 13509 | . . . . . 6 ⊢ (𝜑 → (𝑄↑2) ∈ ℂ) |
35 | mulcl 10621 | . . . . . 6 ⊢ ((;27 ∈ ℂ ∧ (𝑄↑2) ∈ ℂ) → (;27 · (𝑄↑2)) ∈ ℂ) | |
36 | 32, 34, 35 | sylancr 589 | . . . . 5 ⊢ (𝜑 → (;27 · (𝑄↑2)) ∈ ℂ) |
37 | 28, 36 | subcld 10997 | . . . 4 ⊢ (𝜑 → (-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) ∈ ℂ) |
38 | 7nn0 11920 | . . . . . . 7 ⊢ 7 ∈ ℕ0 | |
39 | 38, 13 | decnncl 12119 | . . . . . 6 ⊢ ;72 ∈ ℕ |
40 | 39 | nncni 11648 | . . . . 5 ⊢ ;72 ∈ ℂ |
41 | 10, 16 | mulcld 10661 | . . . . 5 ⊢ (𝜑 → (𝑃 · 𝑅) ∈ ℂ) |
42 | mulcl 10621 | . . . . 5 ⊢ ((;72 ∈ ℂ ∧ (𝑃 · 𝑅) ∈ ℂ) → (;72 · (𝑃 · 𝑅)) ∈ ℂ) | |
43 | 40, 41, 42 | sylancr 589 | . . . 4 ⊢ (𝜑 → (;72 · (𝑃 · 𝑅)) ∈ ℂ) |
44 | 37, 43 | addcld 10660 | . . 3 ⊢ (𝜑 → ((-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅))) ∈ ℂ) |
45 | 21, 44 | eqeltrd 2913 | . 2 ⊢ (𝜑 → 𝑉 ∈ ℂ) |
46 | quart.w | . . 3 ⊢ (𝜑 → 𝑊 = (√‘((𝑉↑2) − (4 · (𝑈↑3))))) | |
47 | 45 | sqcld 13509 | . . . . 5 ⊢ (𝜑 → (𝑉↑2) ∈ ℂ) |
48 | 4cn 11723 | . . . . . 6 ⊢ 4 ∈ ℂ | |
49 | expcl 13448 | . . . . . . 7 ⊢ ((𝑈 ∈ ℂ ∧ 3 ∈ ℕ0) → (𝑈↑3) ∈ ℂ) | |
50 | 20, 23, 49 | sylancl 588 | . . . . . 6 ⊢ (𝜑 → (𝑈↑3) ∈ ℂ) |
51 | mulcl 10621 | . . . . . 6 ⊢ ((4 ∈ ℂ ∧ (𝑈↑3) ∈ ℂ) → (4 · (𝑈↑3)) ∈ ℂ) | |
52 | 48, 50, 51 | sylancr 589 | . . . . 5 ⊢ (𝜑 → (4 · (𝑈↑3)) ∈ ℂ) |
53 | 47, 52 | subcld 10997 | . . . 4 ⊢ (𝜑 → ((𝑉↑2) − (4 · (𝑈↑3))) ∈ ℂ) |
54 | 53 | sqrtcld 14797 | . . 3 ⊢ (𝜑 → (√‘((𝑉↑2) − (4 · (𝑈↑3)))) ∈ ℂ) |
55 | 46, 54 | eqeltrd 2913 | . 2 ⊢ (𝜑 → 𝑊 ∈ ℂ) |
56 | 20, 45, 55 | 3jca 1124 | 1 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑉 ∈ ℂ ∧ 𝑊 ∈ ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 1c1 10538 + caddc 10540 · cmul 10542 − cmin 10870 -cneg 10871 / cdiv 11297 2c2 11693 3c3 11694 4c4 11695 5c5 11696 6c6 11697 7c7 11698 8c8 11699 ℕ0cn0 11898 ;cdc 12099 ↑cexp 13430 √csqrt 14592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-rp 12391 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 |
This theorem is referenced by: quartlem3 25437 quart 25439 |
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