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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 25705 | . . . . 5 β’ πΈ = βͺ ran π» |
3 | omelon 9589 | . . . . . . . . 9 β’ Ο β On | |
4 | nn0ennn 13891 | . . . . . . . . . . 11 β’ β0 β β | |
5 | nnenom 13892 | . . . . . . . . . . 11 β’ β β Ο | |
6 | 4, 5 | entri 8955 | . . . . . . . . . 10 β’ β0 β Ο |
7 | 6 | ensymi 8951 | . . . . . . . . 9 β’ Ο β β0 |
8 | isnumi 9889 | . . . . . . . . 9 β’ ((Ο β On β§ Ο β β0) β β0 β dom card) | |
9 | 3, 7, 8 | mp2an 691 | . . . . . . . 8 β’ β0 β dom card |
10 | cnex 11139 | . . . . . . . . . . 11 β’ β β V | |
11 | 10 | rabex 5294 | . . . . . . . . . 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 6649 | . . . . . . . . 9 β’ π» Fn β0 |
13 | dffn4 6767 | . . . . . . . . 9 β’ (π» Fn β0 β π»:β0βontoβran π») | |
14 | 12, 13 | mpbi 229 | . . . . . . . 8 β’ π»:β0βontoβran π» |
15 | fodomnum 10000 | . . . . . . . 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 8960 | . . . . . . 7 β’ ((ran π» βΌ β0 β§ β0 β Ο) β ran π» βΌ Ο) | |
18 | 16, 6, 17 | mp2an 691 | . . . . . 6 β’ ran π» βΌ Ο |
19 | fvelrnb 6908 | . . . . . . . . 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 25704 | . . . . . . . . . 10 β’ (π β β0 β (π»βπ) β Fin) |
22 | eleq1 2826 | . . . . . . . . . 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 3146 | . . . . . . . 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 3953 | . . . . . 6 β’ ran π» β Fin |
27 | aasscn 25694 | . . . . . . . 8 β’ πΈ β β | |
28 | 2, 27 | eqsstrri 3984 | . . . . . . 7 β’ βͺ ran π» β β |
29 | soss 5570 | . . . . . . 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 10057 | . . . . . 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 1466 | . . . . 5 β’ (π Or β β βͺ ran π» βΌ Ο) |
33 | 2, 32 | eqbrtrid 5145 | . . . 4 β’ (π Or β β πΈ βΌ Ο) |
34 | cnso 16136 | . . . 4 β’ βπ π Or β | |
35 | 33, 34 | exlimiiv 1935 | . . 3 β’ πΈ βΌ Ο |
36 | 5 | ensymi 8951 | . . 3 β’ Ο β β |
37 | domentr 8960 | . . 3 β’ ((πΈ βΌ Ο β§ Ο β β) β πΈ βΌ β) | |
38 | 35, 36, 37 | mp2an 691 | . 2 β’ πΈ βΌ β |
39 | 10, 27 | ssexi 5284 | . . 3 β’ πΈ β V |
40 | nnssq 12890 | . . . 4 β’ β β β | |
41 | qssaa 25700 | . . . 4 β’ β β πΈ | |
42 | 40, 41 | sstri 3958 | . . 3 β’ β β πΈ |
43 | ssdomg 8947 | . . 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 9044 | . 2 β’ ((πΈ βΌ β β§ β βΌ πΈ) β πΈ β β) | |
46 | 38, 44, 45 | mp2an 691 | 1 β’ πΈ β β |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2944 βwral 3065 βwrex 3074 {crab 3410 Vcvv 3448 β wss 3915 βͺ cuni 4870 class class class wbr 5110 β¦ cmpt 5193 Or wor 5549 dom cdm 5638 ran crn 5639 Oncon0 6322 Fn wfn 6496 βontoβwfo 6499 βcfv 6501 Οcom 7807 β cen 8887 βΌ cdom 8888 Fincfn 8890 cardccrd 9878 βcc 11056 0cc0 11058 β€ cle 11197 βcn 12160 β0cn0 12420 β€cz 12506 βcq 12880 abscabs 15126 0πc0p 25049 Polycply 25561 coeffccoe 25563 degcdgr 25564 πΈcaa 25690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-oadd 8421 df-omul 8422 df-er 8655 df-map 8774 df-pm 8775 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-inf 9386 df-oi 9453 df-dju 9844 df-card 9882 df-acn 9885 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-xnn0 12493 df-z 12507 df-uz 12771 df-q 12881 df-rp 12923 df-ico 13277 df-icc 13278 df-fz 13432 df-fzo 13575 df-fl 13704 df-mod 13782 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-limsup 15360 df-clim 15377 df-rlim 15378 df-sum 15578 df-0p 25050 df-ply 25565 df-idp 25566 df-coe 25567 df-dgr 25568 df-quot 25667 df-aa 25691 |
This theorem is referenced by: aannen 25707 |
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