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Mirrors > Home > MPE Home > Th. List > odinf | Structured version Visualization version GIF version |
Description: The multiples of an element with infinite order form an infinite cyclic subgroup of πΊ. (Contributed by Mario Carneiro, 14-Jan-2015.) |
Ref | Expression |
---|---|
odf1.1 | β’ π = (BaseβπΊ) |
odf1.2 | β’ π = (odβπΊ) |
odf1.3 | β’ Β· = (.gβπΊ) |
odf1.4 | β’ πΉ = (π₯ β β€ β¦ (π₯ Β· π΄)) |
Ref | Expression |
---|---|
odinf | β’ ((πΊ β Grp β§ π΄ β π β§ (πβπ΄) = 0) β Β¬ ran πΉ β Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | znnen 16102 | . . . . 5 β’ β€ β β | |
2 | nnenom 13894 | . . . . 5 β’ β β Ο | |
3 | 1, 2 | entr2i 8955 | . . . 4 β’ Ο β β€ |
4 | odf1.1 | . . . . . . . 8 β’ π = (BaseβπΊ) | |
5 | odf1.2 | . . . . . . . 8 β’ π = (odβπΊ) | |
6 | odf1.3 | . . . . . . . 8 β’ Β· = (.gβπΊ) | |
7 | odf1.4 | . . . . . . . 8 β’ πΉ = (π₯ β β€ β¦ (π₯ Β· π΄)) | |
8 | 4, 5, 6, 7 | odf1 19352 | . . . . . . 7 β’ ((πΊ β Grp β§ π΄ β π) β ((πβπ΄) = 0 β πΉ:β€β1-1βπ)) |
9 | 8 | biimp3a 1470 | . . . . . 6 β’ ((πΊ β Grp β§ π΄ β π β§ (πβπ΄) = 0) β πΉ:β€β1-1βπ) |
10 | f1f 6742 | . . . . . 6 β’ (πΉ:β€β1-1βπ β πΉ:β€βΆπ) | |
11 | zex 12516 | . . . . . . 7 β’ β€ β V | |
12 | 4 | fvexi 6860 | . . . . . . 7 β’ π β V |
13 | fex2 7874 | . . . . . . 7 β’ ((πΉ:β€βΆπ β§ β€ β V β§ π β V) β πΉ β V) | |
14 | 11, 12, 13 | mp3an23 1454 | . . . . . 6 β’ (πΉ:β€βΆπ β πΉ β V) |
15 | 9, 10, 14 | 3syl 18 | . . . . 5 β’ ((πΊ β Grp β§ π΄ β π β§ (πβπ΄) = 0) β πΉ β V) |
16 | f1f1orn 6799 | . . . . . 6 β’ (πΉ:β€β1-1βπ β πΉ:β€β1-1-ontoβran πΉ) | |
17 | 9, 16 | syl 17 | . . . . 5 β’ ((πΊ β Grp β§ π΄ β π β§ (πβπ΄) = 0) β πΉ:β€β1-1-ontoβran πΉ) |
18 | f1oen3g 8912 | . . . . 5 β’ ((πΉ β V β§ πΉ:β€β1-1-ontoβran πΉ) β β€ β ran πΉ) | |
19 | 15, 17, 18 | syl2anc 585 | . . . 4 β’ ((πΊ β Grp β§ π΄ β π β§ (πβπ΄) = 0) β β€ β ran πΉ) |
20 | entr 8952 | . . . 4 β’ ((Ο β β€ β§ β€ β ran πΉ) β Ο β ran πΉ) | |
21 | 3, 19, 20 | sylancr 588 | . . 3 β’ ((πΊ β Grp β§ π΄ β π β§ (πβπ΄) = 0) β Ο β ran πΉ) |
22 | endom 8925 | . . 3 β’ (Ο β ran πΉ β Ο βΌ ran πΉ) | |
23 | domnsym 9049 | . . 3 β’ (Ο βΌ ran πΉ β Β¬ ran πΉ βΊ Ο) | |
24 | 21, 22, 23 | 3syl 18 | . 2 β’ ((πΊ β Grp β§ π΄ β π β§ (πβπ΄) = 0) β Β¬ ran πΉ βΊ Ο) |
25 | isfinite 9596 | . 2 β’ (ran πΉ β Fin β ran πΉ βΊ Ο) | |
26 | 24, 25 | sylnibr 329 | 1 β’ ((πΊ β Grp β§ π΄ β π β§ (πβπ΄) = 0) β Β¬ ran πΉ β Fin) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 Vcvv 3447 class class class wbr 5109 β¦ cmpt 5192 ran crn 5638 βΆwf 6496 β1-1βwf1 6497 β1-1-ontoβwf1o 6499 βcfv 6500 (class class class)co 7361 Οcom 7806 β cen 8886 βΌ cdom 8887 βΊ csdm 8888 Fincfn 8889 0cc0 11059 βcn 12161 β€cz 12507 Basecbs 17091 Grpcgrp 18756 .gcmg 18880 odcod 19314 |
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 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-inf2 9585 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-oadd 8420 df-omul 8421 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-sup 9386 df-inf 9387 df-oi 9454 df-card 9883 df-acn 9886 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-n0 12422 df-z 12508 df-uz 12772 df-rp 12924 df-fz 13434 df-fl 13706 df-mod 13784 df-seq 13916 df-exp 13977 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-dvds 16145 df-0g 17331 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-minusg 18760 df-sbg 18761 df-mulg 18881 df-od 19318 |
This theorem is referenced by: dfod2 19354 odcl2 19355 |
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