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| Mirrors > Home > MPE Home > Th. List > quartlem4 | Structured version Visualization version GIF version | ||
| Description: Closure lemmas for quart 26905. (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 26903 | . . . . . 6 ⊢ (𝜑 → (𝑆 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑇 ∈ ℂ)) | 
| 17 | 16 | simp2d 1143 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℂ) | 
| 18 | 17 | sqrtcld 15477 | . . . 4 ⊢ (𝜑 → (√‘𝑀) ∈ ℂ) | 
| 19 | 2cnd 12345 | . . . 4 ⊢ (𝜑 → 2 ∈ ℂ) | |
| 20 | 17 | sqsqrtd 15479 | . . . . . 6 ⊢ (𝜑 → ((√‘𝑀)↑2) = 𝑀) | 
| 21 | quart.m0 | . . . . . 6 ⊢ (𝜑 → 𝑀 ≠ 0) | |
| 22 | 20, 21 | eqnetrd 3007 | . . . . 5 ⊢ (𝜑 → ((√‘𝑀)↑2) ≠ 0) | 
| 23 | sqne0 14164 | . . . . . 6 ⊢ ((√‘𝑀) ∈ ℂ → (((√‘𝑀)↑2) ≠ 0 ↔ (√‘𝑀) ≠ 0)) | |
| 24 | 18, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → (((√‘𝑀)↑2) ≠ 0 ↔ (√‘𝑀) ≠ 0)) | 
| 25 | 22, 24 | mpbid 232 | . . . 4 ⊢ (𝜑 → (√‘𝑀) ≠ 0) | 
| 26 | 2ne0 12371 | . . . . 5 ⊢ 2 ≠ 0 | |
| 27 | 26 | a1i 11 | . . . 4 ⊢ (𝜑 → 2 ≠ 0) | 
| 28 | 18, 19, 25, 27 | divne0d 12060 | . . 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 1142 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℂ) | 
| 32 | 31 | sqcld 14185 | . . . . . . 7 ⊢ (𝜑 → (𝑆↑2) ∈ ℂ) | 
| 33 | 32 | negcld 11608 | . . . . . 6 ⊢ (𝜑 → -(𝑆↑2) ∈ ℂ) | 
| 34 | 2, 3, 4, 5, 7, 8, 9 | quart1cl 26898 | . . . . . . . 8 ⊢ (𝜑 → (𝑃 ∈ ℂ ∧ 𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ)) | 
| 35 | 34 | simp1d 1142 | . . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℂ) | 
| 36 | 35 | halfcld 12513 | . . . . . 6 ⊢ (𝜑 → (𝑃 / 2) ∈ ℂ) | 
| 37 | 33, 36 | subcld 11621 | . . . . 5 ⊢ (𝜑 → (-(𝑆↑2) − (𝑃 / 2)) ∈ ℂ) | 
| 38 | 34 | simp2d 1143 | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℂ) | 
| 39 | 4cn 12352 | . . . . . . . 8 ⊢ 4 ∈ ℂ | |
| 40 | 39 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 4 ∈ ℂ) | 
| 41 | 4ne0 12375 | . . . . . . . 8 ⊢ 4 ≠ 0 | |
| 42 | 41 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 4 ≠ 0) | 
| 43 | 38, 40, 42 | divcld 12044 | . . . . . 6 ⊢ (𝜑 → (𝑄 / 4) ∈ ℂ) | 
| 44 | 43, 31, 29 | divcld 12044 | . . . . 5 ⊢ (𝜑 → ((𝑄 / 4) / 𝑆) ∈ ℂ) | 
| 45 | 37, 44 | addcld 11281 | . . . 4 ⊢ (𝜑 → ((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆)) ∈ ℂ) | 
| 46 | 45 | sqrtcld 15477 | . . 3 ⊢ (𝜑 → (√‘((-(𝑆↑2) − (𝑃 / 2)) + ((𝑄 / 4) / 𝑆))) ∈ ℂ) | 
| 47 | 30, 46 | eqeltrd 2840 | . 2 ⊢ (𝜑 → 𝐼 ∈ ℂ) | 
| 48 | quart.j | . . 3 ⊢ (𝜑 → 𝐽 = (√‘((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)))) | |
| 49 | 37, 44 | subcld 11621 | . . . 4 ⊢ (𝜑 → ((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆)) ∈ ℂ) | 
| 50 | 49 | sqrtcld 15477 | . . 3 ⊢ (𝜑 → (√‘((-(𝑆↑2) − (𝑃 / 2)) − ((𝑄 / 4) / 𝑆))) ∈ ℂ) | 
| 51 | 48, 50 | eqeltrd 2840 | . 2 ⊢ (𝜑 → 𝐽 ∈ ℂ) | 
| 52 | 29, 47, 51 | 3jca 1128 | 1 ⊢ (𝜑 → (𝑆 ≠ 0 ∧ 𝐼 ∈ ℂ ∧ 𝐽 ∈ ℂ)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ‘cfv 6560 (class class class)co 7432 ℂcc 11154 0cc0 11156 1c1 11157 + caddc 11159 · cmul 11161 − cmin 11493 -cneg 11494 / cdiv 11921 2c2 12322 3c3 12323 4c4 12324 5c5 12325 6c6 12326 7c7 12327 8c8 12328 ;cdc 12735 ↑cexp 14103 √csqrt 15273 ↑𝑐ccxp 26598 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ioc 13393 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-seq 14044 df-exp 14104 df-fac 14314 df-bc 14343 df-hash 14371 df-shft 15107 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-limsup 15508 df-clim 15525 df-rlim 15526 df-sum 15724 df-ef 16104 df-sin 16106 df-cos 16107 df-pi 16109 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-mulg 19087 df-cntz 19336 df-cmn 19801 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-fbas 21362 df-fg 21363 df-cnfld 21366 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-lp 23145 df-perf 23146 df-cn 23236 df-cnp 23237 df-haus 23324 df-tx 23571 df-hmeo 23764 df-fil 23855 df-fm 23947 df-flim 23948 df-flf 23949 df-xms 24331 df-ms 24332 df-tms 24333 df-cncf 24905 df-limc 25902 df-dv 25903 df-log 26599 df-cxp 26600 | 
| This theorem is referenced by: quart 26905 | 
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