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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 𝐺. (Contributed by Mario Carneiro, 14-Jan-2015.) |
Ref | Expression |
---|---|
odf1.1 | ⊢ 𝑋 = (Base‘𝐺) |
odf1.2 | ⊢ 𝑂 = (od‘𝐺) |
odf1.3 | ⊢ · = (.g‘𝐺) |
odf1.4 | ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴)) |
Ref | Expression |
---|---|
odinf | ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → ¬ ran 𝐹 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | znnen 14941 | . . . . 5 ⊢ ℤ ≈ ℕ | |
2 | nnenom 12779 | . . . . 5 ⊢ ℕ ≈ ω | |
3 | 1, 2 | entr2i 8011 | . . . 4 ⊢ ω ≈ ℤ |
4 | odf1.1 | . . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺) | |
5 | odf1.2 | . . . . . . . 8 ⊢ 𝑂 = (od‘𝐺) | |
6 | odf1.3 | . . . . . . . 8 ⊢ · = (.g‘𝐺) | |
7 | odf1.4 | . . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴)) | |
8 | 4, 5, 6, 7 | odf1 17979 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝑂‘𝐴) = 0 ↔ 𝐹:ℤ–1-1→𝑋)) |
9 | 8 | biimp3a 1432 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → 𝐹:ℤ–1-1→𝑋) |
10 | f1f 6101 | . . . . . 6 ⊢ (𝐹:ℤ–1-1→𝑋 → 𝐹:ℤ⟶𝑋) | |
11 | zex 11386 | . . . . . . 7 ⊢ ℤ ∈ V | |
12 | fvex 6201 | . . . . . . . 8 ⊢ (Base‘𝐺) ∈ V | |
13 | 4, 12 | eqeltri 2697 | . . . . . . 7 ⊢ 𝑋 ∈ V |
14 | fex2 7121 | . . . . . . 7 ⊢ ((𝐹:ℤ⟶𝑋 ∧ ℤ ∈ V ∧ 𝑋 ∈ V) → 𝐹 ∈ V) | |
15 | 11, 13, 14 | mp3an23 1416 | . . . . . 6 ⊢ (𝐹:ℤ⟶𝑋 → 𝐹 ∈ V) |
16 | 9, 10, 15 | 3syl 18 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → 𝐹 ∈ V) |
17 | f1f1orn 6148 | . . . . . 6 ⊢ (𝐹:ℤ–1-1→𝑋 → 𝐹:ℤ–1-1-onto→ran 𝐹) | |
18 | 9, 17 | syl 17 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → 𝐹:ℤ–1-1-onto→ran 𝐹) |
19 | f1oen3g 7971 | . . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝐹:ℤ–1-1-onto→ran 𝐹) → ℤ ≈ ran 𝐹) | |
20 | 16, 18, 19 | syl2anc 693 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → ℤ ≈ ran 𝐹) |
21 | entr 8008 | . . . 4 ⊢ ((ω ≈ ℤ ∧ ℤ ≈ ran 𝐹) → ω ≈ ran 𝐹) | |
22 | 3, 20, 21 | sylancr 695 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → ω ≈ ran 𝐹) |
23 | endom 7982 | . . 3 ⊢ (ω ≈ ran 𝐹 → ω ≼ ran 𝐹) | |
24 | domnsym 8086 | . . 3 ⊢ (ω ≼ ran 𝐹 → ¬ ran 𝐹 ≺ ω) | |
25 | 22, 23, 24 | 3syl 18 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → ¬ ran 𝐹 ≺ ω) |
26 | isfinite 8549 | . 2 ⊢ (ran 𝐹 ∈ Fin ↔ ran 𝐹 ≺ ω) | |
27 | 25, 26 | sylnibr 319 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → ¬ ran 𝐹 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 Vcvv 3200 class class class wbr 4653 ↦ cmpt 4729 ran crn 5115 ⟶wf 5884 –1-1→wf1 5885 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 ωcom 7065 ≈ cen 7952 ≼ cdom 7953 ≺ csdm 7954 Fincfn 7955 0cc0 9936 ℕcn 11020 ℤcz 11377 Basecbs 15857 Grpcgrp 17422 .gcmg 17540 odcod 17944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-od 17948 |
This theorem is referenced by: dfod2 17981 odcl2 17982 |
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