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Mirrors > Home > MPE Home > Th. List > quartlem4 | Structured version Visualization version GIF version |
Description: Closure lemmas for quart 26603. (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 26601 | . . . . . 6 ⊢ (𝜑 → (𝑆 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑇 ∈ ℂ)) |
17 | 16 | simp2d 1142 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
18 | 17 | sqrtcld 15389 | . . . 4 ⊢ (𝜑 → (√‘𝑀) ∈ ℂ) |
19 | 2cnd 12295 | . . . 4 ⊢ (𝜑 → 2 ∈ ℂ) | |
20 | 17 | sqsqrtd 15391 | . . . . . 6 ⊢ (𝜑 → ((√‘𝑀)↑2) = 𝑀) |
21 | quart.m0 | . . . . . 6 ⊢ (𝜑 → 𝑀 ≠ 0) | |
22 | 20, 21 | eqnetrd 3007 | . . . . 5 ⊢ (𝜑 → ((√‘𝑀)↑2) ≠ 0) |
23 | sqne0 14093 | . . . . . 6 ⊢ ((√‘𝑀) ∈ ℂ → (((√‘𝑀)↑2) ≠ 0 ↔ (√‘𝑀) ≠ 0)) | |
24 | 18, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → (((√‘𝑀)↑2) ≠ 0 ↔ (√‘𝑀) ≠ 0)) |
25 | 22, 24 | mpbid 231 | . . . 4 ⊢ (𝜑 → (√‘𝑀) ≠ 0) |
26 | 2ne0 12321 | . . . . 5 ⊢ 2 ≠ 0 | |
27 | 26 | a1i 11 | . . . 4 ⊢ (𝜑 → 2 ≠ 0) |
28 | 18, 19, 25, 27 | divne0d 12011 | . . 3 ⊢ (𝜑 → ((√‘𝑀) / 2) ≠ 0) |
29 | 1, 28 | eqnetrd 3007 | . 2 ⊢ (𝜑 → 𝑆 ≠ 0) |
30 | quart.i | . . 3 ⊢ (𝜑 → 𝐼 = (√‘((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)))) | |
31 | 16 | simp1d 1141 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℂ) |
32 | 31 | sqcld 14114 | . . . . . . 7 ⊢ (𝜑 → (𝑆↑2) ∈ ℂ) |
33 | 32 | negcld 11563 | . . . . . 6 ⊢ (𝜑 → -(𝑆↑2) ∈ ℂ) |
34 | 2, 3, 4, 5, 7, 8, 9 | quart1cl 26596 | . . . . . . . 8 ⊢ (𝜑 → (𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ)) |
35 | 34 | simp1d 1141 | . . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
36 | 35 | halfcld 12462 | . . . . . 6 ⊢ (𝜑 → (𝑃 / 2) ∈ ℂ) |
37 | 33, 36 | subcld 11576 | . . . . 5 ⊢ (𝜑 → (-(𝑆↑2) − (𝑃 / 2)) ∈ ℂ) |
38 | 34 | simp2d 1142 | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℂ) |
39 | 4cn 12302 | . . . . . . . 8 ⊢ 4 ∈ ℂ | |
40 | 39 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 4 ∈ ℂ) |
41 | 4ne0 12325 | . . . . . . . 8 ⊢ 4 ≠ 0 | |
42 | 41 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 4 ≠ 0) |
43 | 38, 40, 42 | divcld 11995 | . . . . . 6 ⊢ (𝜑 → (𝑄 / 4) ∈ ℂ) |
44 | 43, 31, 29 | divcld 11995 | . . . . 5 ⊢ (𝜑 → ((𝑄 / 4) / 𝑆) ∈ ℂ) |
45 | 37, 44 | addcld 11238 | . . . 4 ⊢ (𝜑 → ((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)) ∈ ℂ) |
46 | 45 | sqrtcld 15389 | . . 3 ⊢ (𝜑 → (√‘((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆))) ∈ ℂ) |
47 | 30, 46 | eqeltrd 2832 | . 2 ⊢ (𝜑 → 𝐼 ∈ ℂ) |
48 | quart.j | . . 3 ⊢ (𝜑 → 𝐽 = (√‘((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)))) | |
49 | 37, 44 | subcld 11576 | . . . 4 ⊢ (𝜑 → ((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)) ∈ ℂ) |
50 | 49 | sqrtcld 15389 | . . 3 ⊢ (𝜑 → (√‘((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆))) ∈ ℂ) |
51 | 48, 50 | eqeltrd 2832 | . 2 ⊢ (𝜑 → 𝐽 ∈ ℂ) |
52 | 29, 47, 51 | 3jca 1127 | 1 ⊢ (𝜑 → (𝑆 ≠ 0 ∧ 𝐼 ∈ ℂ ∧ 𝐽 ∈ ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ‘cfv 6543 (class class class)co 7412 ℂcc 11112 0cc0 11114 1c1 11115 + caddc 11117 · cmul 11119 − cmin 11449 -cneg 11450 / cdiv 11876 2c2 12272 3c3 12273 4c4 12274 5c5 12275 6c6 12276 7c7 12277 8c8 12278 ;cdc 12682 ↑cexp 14032 √csqrt 15185 ↑𝑐ccxp 26301 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 ax-addf 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-er 8707 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9366 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9509 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ioo 13333 df-ioc 13334 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-fl 13762 df-mod 13840 df-seq 13972 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15019 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-limsup 15420 df-clim 15437 df-rlim 15438 df-sum 15638 df-ef 16016 df-sin 16018 df-cos 16019 df-pi 16021 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-mulg 18988 df-cntz 19223 df-cmn 19692 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-fbas 21142 df-fg 21143 df-cnfld 21146 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-cld 22744 df-ntr 22745 df-cls 22746 df-nei 22823 df-lp 22861 df-perf 22862 df-cn 22952 df-cnp 22953 df-haus 23040 df-tx 23287 df-hmeo 23480 df-fil 23571 df-fm 23663 df-flim 23664 df-flf 23665 df-xms 24047 df-ms 24048 df-tms 24049 df-cncf 24619 df-limc 25616 df-dv 25617 df-log 26302 df-cxp 26303 |
This theorem is referenced by: quart 26603 |
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