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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 26219 | . . . . 5 β’ πΈ = βͺ ran π» |
3 | omelon 9643 | . . . . . . . . 9 β’ Ο β On | |
4 | nn0ennn 13950 | . . . . . . . . . . 11 β’ β0 β β | |
5 | nnenom 13951 | . . . . . . . . . . 11 β’ β β Ο | |
6 | 4, 5 | entri 9006 | . . . . . . . . . 10 β’ β0 β Ο |
7 | 6 | ensymi 9002 | . . . . . . . . 9 β’ Ο β β0 |
8 | isnumi 9943 | . . . . . . . . 9 β’ ((Ο β On β§ Ο β β0) β β0 β dom card) | |
9 | 3, 7, 8 | mp2an 689 | . . . . . . . 8 β’ β0 β dom card |
10 | cnex 11193 | . . . . . . . . . . 11 β’ β β V | |
11 | 10 | rabex 5325 | . . . . . . . . . 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 6687 | . . . . . . . . 9 β’ π» Fn β0 |
13 | dffn4 6805 | . . . . . . . . 9 β’ (π» Fn β0 β π»:β0βontoβran π») | |
14 | 12, 13 | mpbi 229 | . . . . . . . 8 β’ π»:β0βontoβran π» |
15 | fodomnum 10054 | . . . . . . . 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 9011 | . . . . . . 7 β’ ((ran π» βΌ β0 β§ β0 β Ο) β ran π» βΌ Ο) | |
18 | 16, 6, 17 | mp2an 689 | . . . . . 6 β’ ran π» βΌ Ο |
19 | fvelrnb 6946 | . . . . . . . . 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 26218 | . . . . . . . . . 10 β’ (π β β0 β (π»βπ) β Fin) |
22 | eleq1 2815 | . . . . . . . . . 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 3142 | . . . . . . . 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 3981 | . . . . . 6 β’ ran π» β Fin |
27 | aasscn 26208 | . . . . . . . 8 β’ πΈ β β | |
28 | 2, 27 | eqsstrri 4012 | . . . . . . 7 β’ βͺ ran π» β β |
29 | soss 5601 | . . . . . . 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 10111 | . . . . . 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 5176 | . . . 4 β’ (π Or β β πΈ βΌ Ο) |
34 | cnso 16197 | . . . 4 β’ βπ π Or β | |
35 | 33, 34 | exlimiiv 1926 | . . 3 β’ πΈ βΌ Ο |
36 | 5 | ensymi 9002 | . . 3 β’ Ο β β |
37 | domentr 9011 | . . 3 β’ ((πΈ βΌ Ο β§ Ο β β) β πΈ βΌ β) | |
38 | 35, 36, 37 | mp2an 689 | . 2 β’ πΈ βΌ β |
39 | 10, 27 | ssexi 5315 | . . 3 β’ πΈ β V |
40 | nnssq 12946 | . . . 4 β’ β β β | |
41 | qssaa 26214 | . . . 4 β’ β β πΈ | |
42 | 40, 41 | sstri 3986 | . . 3 β’ β β πΈ |
43 | ssdomg 8998 | . . 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 9095 | . 2 β’ ((πΈ βΌ β β§ β βΌ πΈ) β πΈ β β) | |
46 | 38, 44, 45 | mp2an 689 | 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 2934 βwral 3055 βwrex 3064 {crab 3426 Vcvv 3468 β wss 3943 βͺ cuni 4902 class class class wbr 5141 β¦ cmpt 5224 Or wor 5580 dom cdm 5669 ran crn 5670 Oncon0 6358 Fn wfn 6532 βontoβwfo 6535 βcfv 6537 Οcom 7852 β cen 8938 βΌ cdom 8939 Fincfn 8941 cardccrd 9932 βcc 11110 0cc0 11112 β€ cle 11253 βcn 12216 β0cn0 12476 β€cz 12562 βcq 12936 abscabs 15187 0πc0p 25553 Polycply 26073 coeffccoe 26075 degcdgr 26076 πΈcaa 26204 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-oadd 8471 df-omul 8472 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-acn 9939 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-q 12937 df-rp 12981 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15421 df-clim 15438 df-rlim 15439 df-sum 15639 df-0p 25554 df-ply 26077 df-idp 26078 df-coe 26079 df-dgr 26080 df-quot 26181 df-aa 26205 |
This theorem is referenced by: aannen 26221 |
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