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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 16180 | . . . . 5 β’ β€ β β | |
2 | nnenom 13969 | . . . . 5 β’ β β Ο | |
3 | 1, 2 | entr2i 9021 | . . . 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 19508 | . . . . . . 7 β’ ((πΊ β Grp β§ π΄ β π) β ((πβπ΄) = 0 β πΉ:β€β1-1βπ)) |
9 | 8 | biimp3a 1466 | . . . . . 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 12589 | . . . . . . 7 β’ β€ β V | |
12 | 4 | fvexi 6905 | . . . . . . 7 β’ π β V |
13 | fex2 7935 | . . . . . . 7 β’ ((πΉ:β€βΆπ β§ β€ β V β§ π β V) β πΉ β V) | |
14 | 11, 12, 13 | mp3an23 1450 | . . . . . 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 8978 | . . . . 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 9018 | . . . 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 8991 | . . 3 β’ (Ο β ran πΉ β Ο βΌ ran πΉ) | |
23 | domnsym 9115 | . . 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 9667 | . 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 1085 = wceq 1534 β wcel 2099 Vcvv 3469 class class class wbr 5142 β¦ cmpt 5225 ran crn 5673 βΆwf 6538 β1-1βwf1 6539 β1-1-ontoβwf1o 6541 βcfv 6542 (class class class)co 7414 Οcom 7864 β cen 8952 βΌ cdom 8953 βΊ csdm 8954 Fincfn 8955 0cc0 11130 βcn 12234 β€cz 12580 Basecbs 17171 Grpcgrp 18881 .gcmg 19014 odcod 19470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9656 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-omul 8485 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9457 df-inf 9458 df-oi 9525 df-card 9954 df-acn 9957 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-n0 12495 df-z 12581 df-uz 12845 df-rp 12999 df-fz 13509 df-fl 13781 df-mod 13859 df-seq 13991 df-exp 14051 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-dvds 16223 df-0g 17414 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-grp 18884 df-minusg 18885 df-sbg 18886 df-mulg 19015 df-od 19474 |
This theorem is referenced by: dfod2 19510 odcl2 19511 |
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