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Mirrors > Home > MPE Home > Th. List > quartlem4 | Structured version Visualization version GIF version |
Description: Closure lemmas for quart 25433. (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 25431 | . . . . . 6 ⊢ (𝜑 → (𝑆 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑇 ∈ ℂ)) |
17 | 16 | simp2d 1139 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
18 | 17 | sqrtcld 14791 | . . . 4 ⊢ (𝜑 → (√‘𝑀) ∈ ℂ) |
19 | 2cnd 11709 | . . . 4 ⊢ (𝜑 → 2 ∈ ℂ) | |
20 | 17 | sqsqrtd 14793 | . . . . . 6 ⊢ (𝜑 → ((√‘𝑀)↑2) = 𝑀) |
21 | quart.m0 | . . . . . 6 ⊢ (𝜑 → 𝑀 ≠ 0) | |
22 | 20, 21 | eqnetrd 3083 | . . . . 5 ⊢ (𝜑 → ((√‘𝑀)↑2) ≠ 0) |
23 | sqne0 13483 | . . . . . 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 11735 | . . . . 5 ⊢ 2 ≠ 0 | |
27 | 26 | a1i 11 | . . . 4 ⊢ (𝜑 → 2 ≠ 0) |
28 | 18, 19, 25, 27 | divne0d 11426 | . . 3 ⊢ (𝜑 → ((√‘𝑀) / 2) ≠ 0) |
29 | 1, 28 | eqnetrd 3083 | . 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 13502 | . . . . . . 7 ⊢ (𝜑 → (𝑆↑2) ∈ ℂ) |
33 | 32 | negcld 10978 | . . . . . 6 ⊢ (𝜑 → -(𝑆↑2) ∈ ℂ) |
34 | 2, 3, 4, 5, 7, 8, 9 | quart1cl 25426 | . . . . . . . 8 ⊢ (𝜑 → (𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ)) |
35 | 34 | simp1d 1138 | . . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
36 | 35 | halfcld 11876 | . . . . . 6 ⊢ (𝜑 → (𝑃 / 2) ∈ ℂ) |
37 | 33, 36 | subcld 10991 | . . . . 5 ⊢ (𝜑 → (-(𝑆↑2) − (𝑃 / 2)) ∈ ℂ) |
38 | 34 | simp2d 1139 | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℂ) |
39 | 4cn 11716 | . . . . . . . 8 ⊢ 4 ∈ ℂ | |
40 | 39 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 4 ∈ ℂ) |
41 | 4ne0 11739 | . . . . . . . 8 ⊢ 4 ≠ 0 | |
42 | 41 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 4 ≠ 0) |
43 | 38, 40, 42 | divcld 11410 | . . . . . 6 ⊢ (𝜑 → (𝑄 / 4) ∈ ℂ) |
44 | 43, 31, 29 | divcld 11410 | . . . . 5 ⊢ (𝜑 → ((𝑄 / 4) / 𝑆) ∈ ℂ) |
45 | 37, 44 | addcld 10654 | . . . 4 ⊢ (𝜑 → ((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)) ∈ ℂ) |
46 | 45 | sqrtcld 14791 | . . 3 ⊢ (𝜑 → (√‘((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆))) ∈ ℂ) |
47 | 30, 46 | eqeltrd 2913 | . 2 ⊢ (𝜑 → 𝐼 ∈ ℂ) |
48 | quart.j | . . 3 ⊢ (𝜑 → 𝐽 = (√‘((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)))) | |
49 | 37, 44 | subcld 10991 | . . . 4 ⊢ (𝜑 → ((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)) ∈ ℂ) |
50 | 49 | sqrtcld 14791 | . . 3 ⊢ (𝜑 → (√‘((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆))) ∈ ℂ) |
51 | 48, 50 | eqeltrd 2913 | . 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 1533 ∈ wcel 2110 ≠ wne 3016 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 0cc0 10531 1c1 10532 + caddc 10534 · cmul 10536 − cmin 10864 -cneg 10865 / cdiv 11291 2c2 11686 3c3 11687 4c4 11688 5c5 11689 6c6 11690 7c7 11691 8c8 11692 ;cdc 12092 ↑cexp 13423 √csqrt 14586 ↑𝑐ccxp 25133 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-fac 13628 df-bc 13657 df-hash 13685 df-shft 14420 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-limsup 14822 df-clim 14839 df-rlim 14840 df-sum 15037 df-ef 15415 df-sin 15417 df-cos 15418 df-pi 15420 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-mulg 18219 df-cntz 18441 df-cmn 18902 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-fbas 20536 df-fg 20537 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-lp 21738 df-perf 21739 df-cn 21829 df-cnp 21830 df-haus 21917 df-tx 22164 df-hmeo 22357 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-xms 22924 df-ms 22925 df-tms 22926 df-cncf 23480 df-limc 24458 df-dv 24459 df-log 25134 df-cxp 25135 |
This theorem is referenced by: quart 25433 |
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