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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 16154 | . . . . 5 β’ β€ β β | |
2 | nnenom 13944 | . . . . 5 β’ β β Ο | |
3 | 1, 2 | entr2i 9004 | . . . 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 19429 | . . . . . . 7 β’ ((πΊ β Grp β§ π΄ β π) β ((πβπ΄) = 0 β πΉ:β€β1-1βπ)) |
9 | 8 | biimp3a 1469 | . . . . . 6 β’ ((πΊ β Grp β§ π΄ β π β§ (πβπ΄) = 0) β πΉ:β€β1-1βπ) |
10 | f1f 6787 | . . . . . 6 β’ (πΉ:β€β1-1βπ β πΉ:β€βΆπ) | |
11 | zex 12566 | . . . . . . 7 β’ β€ β V | |
12 | 4 | fvexi 6905 | . . . . . . 7 β’ π β V |
13 | fex2 7923 | . . . . . . 7 β’ ((πΉ:β€βΆπ β§ β€ β V β§ π β V) β πΉ β V) | |
14 | 11, 12, 13 | mp3an23 1453 | . . . . . 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 6844 | . . . . . 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 8961 | . . . . 5 β’ ((πΉ β V β§ πΉ:β€β1-1-ontoβran πΉ) β β€ β ran πΉ) | |
19 | 15, 17, 18 | syl2anc 584 | . . . 4 β’ ((πΊ β Grp β§ π΄ β π β§ (πβπ΄) = 0) β β€ β ran πΉ) |
20 | entr 9001 | . . . 4 β’ ((Ο β β€ β§ β€ β ran πΉ) β Ο β ran πΉ) | |
21 | 3, 19, 20 | sylancr 587 | . . 3 β’ ((πΊ β Grp β§ π΄ β π β§ (πβπ΄) = 0) β Ο β ran πΉ) |
22 | endom 8974 | . . 3 β’ (Ο β ran πΉ β Ο βΌ ran πΉ) | |
23 | domnsym 9098 | . . 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 9646 | . 2 β’ (ran πΉ β Fin β ran πΉ βΊ Ο) | |
26 | 24, 25 | sylnibr 328 | 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 1087 = wceq 1541 β wcel 2106 Vcvv 3474 class class class wbr 5148 β¦ cmpt 5231 ran crn 5677 βΆwf 6539 β1-1βwf1 6540 β1-1-ontoβwf1o 6542 βcfv 6543 (class class class)co 7408 Οcom 7854 β cen 8935 βΌ cdom 8936 βΊ csdm 8937 Fincfn 8938 0cc0 11109 βcn 12211 β€cz 12557 Basecbs 17143 Grpcgrp 18818 .gcmg 18949 odcod 19391 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-oadd 8469 df-omul 8470 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-acn 9936 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-z 12558 df-uz 12822 df-rp 12974 df-fz 13484 df-fl 13756 df-mod 13834 df-seq 13966 df-exp 14027 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-dvds 16197 df-0g 17386 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-grp 18821 df-minusg 18822 df-sbg 18823 df-mulg 18950 df-od 19395 |
This theorem is referenced by: dfod2 19431 odcl2 19432 |
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