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Mirrors > Home > MPE Home > Th. List > quartlem4 | Structured version Visualization version GIF version |
Description: Closure lemmas for quart 25442. (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) |
quart.m0 | ⊢ (𝜑 → 𝑀 ≠ 0) |
quart.i | ⊢ (𝜑 → 𝐼 = (√‘((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)))) |
quart.j | ⊢ (𝜑 → 𝐽 = (√‘((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)))) |
Ref | Expression |
---|---|
quartlem4 | ⊢ (𝜑 → (𝑆 ≠ 0 ∧ 𝐼 ∈ ℂ ∧ 𝐽 ∈ ℂ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | quart.s | . . 3 ⊢ (𝜑 → 𝑆 = ((√‘𝑀) / 2)) | |
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.e | . . . . . . 7 ⊢ (𝜑 → 𝐸 = -(𝐴 / 4)) | |
7 | quart.p | . . . . . . 7 ⊢ (𝜑 → 𝑃 = (𝐵 − ((3 / 8) · (𝐴↑2)))) | |
8 | quart.q | . . . . . . 7 ⊢ (𝜑 → 𝑄 = ((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))) | |
9 | quart.r | . . . . . . 7 ⊢ (𝜑 → 𝑅 = ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / ;16) − ((3 / ;;256) · (𝐴↑4))))) | |
10 | quart.u | . . . . . . 7 ⊢ (𝜑 → 𝑈 = ((𝑃↑2) + (;12 · 𝑅))) | |
11 | quart.v | . . . . . . 7 ⊢ (𝜑 → 𝑉 = ((-(2 · (𝑃↑3)) − (;27 · (𝑄↑2))) + (;72 · (𝑃 · 𝑅)))) | |
12 | quart.w | . . . . . . 7 ⊢ (𝜑 → 𝑊 = (√‘((𝑉↑2) − (4 · (𝑈↑3))))) | |
13 | quart.m | . . . . . . 7 ⊢ (𝜑 → 𝑀 = -((((2 · 𝑃) + 𝑇) + (𝑈 / 𝑇)) / 3)) | |
14 | quart.t | . . . . . . 7 ⊢ (𝜑 → 𝑇 = (((𝑉 + 𝑊) / 2)↑𝑐(1 / 3))) | |
15 | quart.t0 | . . . . . . 7 ⊢ (𝜑 → 𝑇 ≠ 0) | |
16 | 2, 3, 4, 5, 2, 6, 7, 8, 9, 10, 11, 12, 1, 13, 14, 15 | quartlem3 25440 | . . . . . 6 ⊢ (𝜑 → (𝑆 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑇 ∈ ℂ)) |
17 | 16 | simp2d 1139 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
18 | 17 | sqrtcld 14800 | . . . 4 ⊢ (𝜑 → (√‘𝑀) ∈ ℂ) |
19 | 2cnd 11718 | . . . 4 ⊢ (𝜑 → 2 ∈ ℂ) | |
20 | 17 | sqsqrtd 14802 | . . . . . 6 ⊢ (𝜑 → ((√‘𝑀)↑2) = 𝑀) |
21 | quart.m0 | . . . . . 6 ⊢ (𝜑 → 𝑀 ≠ 0) | |
22 | 20, 21 | eqnetrd 3086 | . . . . 5 ⊢ (𝜑 → ((√‘𝑀)↑2) ≠ 0) |
23 | sqne0 13492 | . . . . . 6 ⊢ ((√‘𝑀) ∈ ℂ → (((√‘𝑀)↑2) ≠ 0 ↔ (√‘𝑀) ≠ 0)) | |
24 | 18, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → (((√‘𝑀)↑2) ≠ 0 ↔ (√‘𝑀) ≠ 0)) |
25 | 22, 24 | mpbid 234 | . . . 4 ⊢ (𝜑 → (√‘𝑀) ≠ 0) |
26 | 2ne0 11744 | . . . . 5 ⊢ 2 ≠ 0 | |
27 | 26 | a1i 11 | . . . 4 ⊢ (𝜑 → 2 ≠ 0) |
28 | 18, 19, 25, 27 | divne0d 11435 | . . 3 ⊢ (𝜑 → ((√‘𝑀) / 2) ≠ 0) |
29 | 1, 28 | eqnetrd 3086 | . 2 ⊢ (𝜑 → 𝑆 ≠ 0) |
30 | quart.i | . . 3 ⊢ (𝜑 → 𝐼 = (√‘((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)))) | |
31 | 16 | simp1d 1138 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℂ) |
32 | 31 | sqcld 13511 | . . . . . . 7 ⊢ (𝜑 → (𝑆↑2) ∈ ℂ) |
33 | 32 | negcld 10987 | . . . . . 6 ⊢ (𝜑 → -(𝑆↑2) ∈ ℂ) |
34 | 2, 3, 4, 5, 7, 8, 9 | quart1cl 25435 | . . . . . . . 8 ⊢ (𝜑 → (𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ)) |
35 | 34 | simp1d 1138 | . . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
36 | 35 | halfcld 11885 | . . . . . 6 ⊢ (𝜑 → (𝑃 / 2) ∈ ℂ) |
37 | 33, 36 | subcld 11000 | . . . . 5 ⊢ (𝜑 → (-(𝑆↑2) − (𝑃 / 2)) ∈ ℂ) |
38 | 34 | simp2d 1139 | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℂ) |
39 | 4cn 11725 | . . . . . . . 8 ⊢ 4 ∈ ℂ | |
40 | 39 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 4 ∈ ℂ) |
41 | 4ne0 11748 | . . . . . . . 8 ⊢ 4 ≠ 0 | |
42 | 41 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 4 ≠ 0) |
43 | 38, 40, 42 | divcld 11419 | . . . . . 6 ⊢ (𝜑 → (𝑄 / 4) ∈ ℂ) |
44 | 43, 31, 29 | divcld 11419 | . . . . 5 ⊢ (𝜑 → ((𝑄 / 4) / 𝑆) ∈ ℂ) |
45 | 37, 44 | addcld 10663 | . . . 4 ⊢ (𝜑 → ((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)) ∈ ℂ) |
46 | 45 | sqrtcld 14800 | . . 3 ⊢ (𝜑 → (√‘((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆))) ∈ ℂ) |
47 | 30, 46 | eqeltrd 2916 | . 2 ⊢ (𝜑 → 𝐼 ∈ ℂ) |
48 | quart.j | . . 3 ⊢ (𝜑 → 𝐽 = (√‘((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)))) | |
49 | 37, 44 | subcld 11000 | . . . 4 ⊢ (𝜑 → ((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)) ∈ ℂ) |
50 | 49 | sqrtcld 14800 | . . 3 ⊢ (𝜑 → (√‘((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆))) ∈ ℂ) |
51 | 48, 50 | eqeltrd 2916 | . 2 ⊢ (𝜑 → 𝐽 ∈ ℂ) |
52 | 29, 47, 51 | 3jca 1124 | 1 ⊢ (𝜑 → (𝑆 ≠ 0 ∧ 𝐼 ∈ ℂ ∧ 𝐽 ∈ ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 0cc0 10540 1c1 10541 + caddc 10543 · cmul 10545 − cmin 10873 -cneg 10874 / cdiv 11300 2c2 11695 3c3 11696 4c4 11697 5c5 11698 6c6 11699 7c7 11700 8c8 11701 ;cdc 12101 ↑cexp 13432 √csqrt 14595 ↑𝑐ccxp 25142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-fi 8878 df-sup 8909 df-inf 8910 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14429 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-limsup 14831 df-clim 14848 df-rlim 14849 df-sum 15046 df-ef 15424 df-sin 15426 df-cos 15427 df-pi 15429 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-hom 16592 df-cco 16593 df-rest 16699 df-topn 16700 df-0g 16718 df-gsum 16719 df-topgen 16720 df-pt 16721 df-prds 16724 df-xrs 16778 df-qtop 16783 df-imas 16784 df-xps 16786 df-mre 16860 df-mrc 16861 df-acs 16863 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-mulg 18228 df-cntz 18450 df-cmn 18911 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-fbas 20545 df-fg 20546 df-cnfld 20549 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 df-cld 21630 df-ntr 21631 df-cls 21632 df-nei 21709 df-lp 21747 df-perf 21748 df-cn 21838 df-cnp 21839 df-haus 21926 df-tx 22173 df-hmeo 22366 df-fil 22457 df-fm 22549 df-flim 22550 df-flf 22551 df-xms 22933 df-ms 22934 df-tms 22935 df-cncf 23489 df-limc 24467 df-dv 24468 df-log 25143 df-cxp 25144 |
This theorem is referenced by: quart 25442 |
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