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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 16152 | . . . . 5 β’ β€ β β | |
2 | nnenom 13942 | . . . . 5 β’ β β Ο | |
3 | 1, 2 | entr2i 9001 | . . . 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 19472 | . . . . . . 7 β’ ((πΊ β Grp β§ π΄ β π) β ((πβπ΄) = 0 β πΉ:β€β1-1βπ)) |
9 | 8 | biimp3a 1465 | . . . . . 6 β’ ((πΊ β Grp β§ π΄ β π β§ (πβπ΄) = 0) β πΉ:β€β1-1βπ) |
10 | f1f 6777 | . . . . . 6 β’ (πΉ:β€β1-1βπ β πΉ:β€βΆπ) | |
11 | zex 12564 | . . . . . . 7 β’ β€ β V | |
12 | 4 | fvexi 6895 | . . . . . . 7 β’ π β V |
13 | fex2 7917 | . . . . . . 7 β’ ((πΉ:β€βΆπ β§ β€ β V β§ π β V) β πΉ β V) | |
14 | 11, 12, 13 | mp3an23 1449 | . . . . . 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 6834 | . . . . . 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 8958 | . . . . 5 β’ ((πΉ β V β§ πΉ:β€β1-1-ontoβran πΉ) β β€ β ran πΉ) | |
19 | 15, 17, 18 | syl2anc 583 | . . . 4 β’ ((πΊ β Grp β§ π΄ β π β§ (πβπ΄) = 0) β β€ β ran πΉ) |
20 | entr 8998 | . . . 4 β’ ((Ο β β€ β§ β€ β ran πΉ) β Ο β ran πΉ) | |
21 | 3, 19, 20 | sylancr 586 | . . 3 β’ ((πΊ β Grp β§ π΄ β π β§ (πβπ΄) = 0) β Ο β ran πΉ) |
22 | endom 8971 | . . 3 β’ (Ο β ran πΉ β Ο βΌ ran πΉ) | |
23 | domnsym 9095 | . . 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 9643 | . 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 1084 = wceq 1533 β wcel 2098 Vcvv 3466 class class class wbr 5138 β¦ cmpt 5221 ran crn 5667 βΆwf 6529 β1-1βwf1 6530 β1-1-ontoβwf1o 6532 βcfv 6533 (class class class)co 7401 Οcom 7848 β cen 8932 βΌ cdom 8933 βΊ csdm 8934 Fincfn 8935 0cc0 11106 βcn 12209 β€cz 12555 Basecbs 17143 Grpcgrp 18853 .gcmg 18985 odcod 19434 |
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 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-oadd 8465 df-omul 8466 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-acn 9933 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-rp 12972 df-fz 13482 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-dvds 16195 df-0g 17386 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-grp 18856 df-minusg 18857 df-sbg 18858 df-mulg 18986 df-od 19438 |
This theorem is referenced by: dfod2 19474 odcl2 19475 |
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