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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 15650 | . . . . 5 ⊢ ℤ ≈ ℕ | |
2 | nnenom 13432 | . . . . 5 ⊢ ℕ ≈ ω | |
3 | 1, 2 | entr2i 8603 | . . . 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 18800 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝑂‘𝐴) = 0 ↔ 𝐹:ℤ–1-1→𝑋)) |
9 | 8 | biimp3a 1470 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → 𝐹:ℤ–1-1→𝑋) |
10 | f1f 6568 | . . . . . 6 ⊢ (𝐹:ℤ–1-1→𝑋 → 𝐹:ℤ⟶𝑋) | |
11 | zex 12064 | . . . . . . 7 ⊢ ℤ ∈ V | |
12 | 4 | fvexi 6682 | . . . . . . 7 ⊢ 𝑋 ∈ V |
13 | fex2 7657 | . . . . . . 7 ⊢ ((𝐹:ℤ⟶𝑋 ∧ ℤ ∈ V ∧ 𝑋 ∈ V) → 𝐹 ∈ V) | |
14 | 11, 12, 13 | mp3an23 1454 | . . . . . 6 ⊢ (𝐹:ℤ⟶𝑋 → 𝐹 ∈ V) |
15 | 9, 10, 14 | 3syl 18 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → 𝐹 ∈ V) |
16 | f1f1orn 6623 | . . . . . 6 ⊢ (𝐹:ℤ–1-1→𝑋 → 𝐹:ℤ–1-1-onto→ran 𝐹) | |
17 | 9, 16 | syl 17 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → 𝐹:ℤ–1-1-onto→ran 𝐹) |
18 | f1oen3g 8564 | . . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝐹:ℤ–1-1-onto→ran 𝐹) → ℤ ≈ ran 𝐹) | |
19 | 15, 17, 18 | syl2anc 587 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → ℤ ≈ ran 𝐹) |
20 | entr 8600 | . . . 4 ⊢ ((ω ≈ ℤ ∧ ℤ ≈ ran 𝐹) → ω ≈ ran 𝐹) | |
21 | 3, 19, 20 | sylancr 590 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → ω ≈ ran 𝐹) |
22 | endom 8575 | . . 3 ⊢ (ω ≈ ran 𝐹 → ω ≼ ran 𝐹) | |
23 | domnsym 8686 | . . 3 ⊢ (ω ≼ ran 𝐹 → ¬ ran 𝐹 ≺ ω) | |
24 | 21, 22, 23 | 3syl 18 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → ¬ ran 𝐹 ≺ ω) |
25 | isfinite 9181 | . 2 ⊢ (ran 𝐹 ∈ Fin ↔ ran 𝐹 ≺ ω) | |
26 | 24, 25 | sylnibr 332 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → ¬ ran 𝐹 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 Vcvv 3397 class class class wbr 5027 ↦ cmpt 5107 ran crn 5520 ⟶wf 6329 –1-1→wf1 6330 –1-1-onto→wf1o 6332 ‘cfv 6333 (class class class)co 7164 ωcom 7593 ≈ cen 8545 ≼ cdom 8546 ≺ csdm 8547 Fincfn 8548 0cc0 10608 ℕcn 11709 ℤcz 12055 Basecbs 16579 Grpcgrp 18212 .gcmg 18335 odcod 18763 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-inf2 9170 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-pre-sup 10686 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-isom 6342 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-oadd 8128 df-omul 8129 df-er 8313 df-map 8432 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-sup 8972 df-inf 8973 df-oi 9040 df-card 9434 df-acn 9437 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 df-nn 11710 df-2 11772 df-3 11773 df-n0 11970 df-z 12056 df-uz 12318 df-rp 12466 df-fz 12975 df-fl 13246 df-mod 13322 df-seq 13454 df-exp 13515 df-cj 14541 df-re 14542 df-im 14543 df-sqrt 14677 df-abs 14678 df-dvds 15693 df-0g 16811 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-grp 18215 df-minusg 18216 df-sbg 18217 df-mulg 18336 df-od 18767 |
This theorem is referenced by: dfod2 18802 odcl2 18803 |
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