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Mirrors > Home > MPE Home > Th. List > aannenlem3 | Structured version Visualization version GIF version |
Description: The algebraic numbers are countable. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
Ref | Expression |
---|---|
aannenlem.a | β’ π» = (π β β0 β¦ {π β β β£ βπ β {π β (Polyββ€) β£ (π β 0π β§ (degβπ) β€ π β§ βπ β β0 (absβ((coeffβπ)βπ)) β€ π)} (πβπ) = 0}) |
Ref | Expression |
---|---|
aannenlem3 | β’ πΈ β β |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aannenlem.a | . . . . . 6 β’ π» = (π β β0 β¦ {π β β β£ βπ β {π β (Polyββ€) β£ (π β 0π β§ (degβπ) β€ π β§ βπ β β0 (absβ((coeffβπ)βπ)) β€ π)} (πβπ) = 0}) | |
2 | 1 | aannenlem2 26280 | . . . . 5 β’ πΈ = βͺ ran π» |
3 | omelon 9667 | . . . . . . . . 9 β’ Ο β On | |
4 | nn0ennn 13974 | . . . . . . . . . . 11 β’ β0 β β | |
5 | nnenom 13975 | . . . . . . . . . . 11 β’ β β Ο | |
6 | 4, 5 | entri 9025 | . . . . . . . . . 10 β’ β0 β Ο |
7 | 6 | ensymi 9021 | . . . . . . . . 9 β’ Ο β β0 |
8 | isnumi 9967 | . . . . . . . . 9 β’ ((Ο β On β§ Ο β β0) β β0 β dom card) | |
9 | 3, 7, 8 | mp2an 690 | . . . . . . . 8 β’ β0 β dom card |
10 | cnex 11217 | . . . . . . . . . . 11 β’ β β V | |
11 | 10 | rabex 5329 | . . . . . . . . . 10 β’ {π β β β£ βπ β {π β (Polyββ€) β£ (π β 0π β§ (degβπ) β€ π β§ βπ β β0 (absβ((coeffβπ)βπ)) β€ π)} (πβπ) = 0} β V |
12 | 11, 1 | fnmpti 6692 | . . . . . . . . 9 β’ π» Fn β0 |
13 | dffn4 6811 | . . . . . . . . 9 β’ (π» Fn β0 β π»:β0βontoβran π») | |
14 | 12, 13 | mpbi 229 | . . . . . . . 8 β’ π»:β0βontoβran π» |
15 | fodomnum 10078 | . . . . . . . 8 β’ (β0 β dom card β (π»:β0βontoβran π» β ran π» βΌ β0)) | |
16 | 9, 14, 15 | mp2 9 | . . . . . . 7 β’ ran π» βΌ β0 |
17 | domentr 9030 | . . . . . . 7 β’ ((ran π» βΌ β0 β§ β0 β Ο) β ran π» βΌ Ο) | |
18 | 16, 6, 17 | mp2an 690 | . . . . . 6 β’ ran π» βΌ Ο |
19 | fvelrnb 6953 | . . . . . . . . 9 β’ (π» Fn β0 β (π β ran π» β βπ β β0 (π»βπ) = π)) | |
20 | 12, 19 | ax-mp 5 | . . . . . . . 8 β’ (π β ran π» β βπ β β0 (π»βπ) = π) |
21 | 1 | aannenlem1 26279 | . . . . . . . . . 10 β’ (π β β0 β (π»βπ) β Fin) |
22 | eleq1 2813 | . . . . . . . . . 10 β’ ((π»βπ) = π β ((π»βπ) β Fin β π β Fin)) | |
23 | 21, 22 | syl5ibcom 244 | . . . . . . . . 9 β’ (π β β0 β ((π»βπ) = π β π β Fin)) |
24 | 23 | rexlimiv 3138 | . . . . . . . 8 β’ (βπ β β0 (π»βπ) = π β π β Fin) |
25 | 20, 24 | sylbi 216 | . . . . . . 7 β’ (π β ran π» β π β Fin) |
26 | 25 | ssriv 3976 | . . . . . 6 β’ ran π» β Fin |
27 | aasscn 26269 | . . . . . . . 8 β’ πΈ β β | |
28 | 2, 27 | eqsstrri 4008 | . . . . . . 7 β’ βͺ ran π» β β |
29 | soss 5604 | . . . . . . 7 β’ (βͺ ran π» β β β (π Or β β π Or βͺ ran π»)) | |
30 | 28, 29 | ax-mp 5 | . . . . . 6 β’ (π Or β β π Or βͺ ran π») |
31 | iunfictbso 10135 | . . . . . 6 β’ ((ran π» βΌ Ο β§ ran π» β Fin β§ π Or βͺ ran π») β βͺ ran π» βΌ Ο) | |
32 | 18, 26, 30, 31 | mp3an12i 1461 | . . . . 5 β’ (π Or β β βͺ ran π» βΌ Ο) |
33 | 2, 32 | eqbrtrid 5178 | . . . 4 β’ (π Or β β πΈ βΌ Ο) |
34 | cnso 16221 | . . . 4 β’ βπ π Or β | |
35 | 33, 34 | exlimiiv 1926 | . . 3 β’ πΈ βΌ Ο |
36 | 5 | ensymi 9021 | . . 3 β’ Ο β β |
37 | domentr 9030 | . . 3 β’ ((πΈ βΌ Ο β§ Ο β β) β πΈ βΌ β) | |
38 | 35, 36, 37 | mp2an 690 | . 2 β’ πΈ βΌ β |
39 | 10, 27 | ssexi 5317 | . . 3 β’ πΈ β V |
40 | nnssq 12970 | . . . 4 β’ β β β | |
41 | qssaa 26275 | . . . 4 β’ β β πΈ | |
42 | 40, 41 | sstri 3982 | . . 3 β’ β β πΈ |
43 | ssdomg 9017 | . . 3 β’ (πΈ β V β (β β πΈ β β βΌ πΈ)) | |
44 | 39, 42, 43 | mp2 9 | . 2 β’ β βΌ πΈ |
45 | sbth 9114 | . 2 β’ ((πΈ βΌ β β§ β βΌ πΈ) β πΈ β β) | |
46 | 38, 44, 45 | mp2an 690 | 1 β’ πΈ β β |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2930 βwral 3051 βwrex 3060 {crab 3419 Vcvv 3463 β wss 3940 βͺ cuni 4903 class class class wbr 5143 β¦ cmpt 5226 Or wor 5583 dom cdm 5672 ran crn 5673 Oncon0 6364 Fn wfn 6537 βontoβwfo 6540 βcfv 6542 Οcom 7867 β cen 8957 βΌ cdom 8958 Fincfn 8960 cardccrd 9956 βcc 11134 0cc0 11136 β€ cle 11277 βcn 12240 β0cn0 12500 β€cz 12586 βcq 12960 abscabs 15211 0πc0p 25614 Polycply 26134 coeffccoe 26136 degcdgr 26137 πΈcaa 26265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-inf2 9662 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-2o 8484 df-oadd 8487 df-omul 8488 df-er 8721 df-map 8843 df-pm 8844 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-sup 9463 df-inf 9464 df-oi 9531 df-dju 9922 df-card 9960 df-acn 9963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-n0 12501 df-xnn0 12573 df-z 12587 df-uz 12851 df-q 12961 df-rp 13005 df-ico 13360 df-icc 13361 df-fz 13515 df-fzo 13658 df-fl 13787 df-mod 13865 df-seq 13997 df-exp 14057 df-hash 14320 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-limsup 15445 df-clim 15462 df-rlim 15463 df-sum 15663 df-0p 25615 df-ply 26138 df-idp 26139 df-coe 26140 df-dgr 26141 df-quot 26242 df-aa 26266 |
This theorem is referenced by: aannen 26282 |
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