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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 15567 | . . . . 5 ⊢ ℤ ≈ ℕ | |
2 | nnenom 13351 | . . . . 5 ⊢ ℕ ≈ ω | |
3 | 1, 2 | entr2i 8566 | . . . 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 18691 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝑂‘𝐴) = 0 ↔ 𝐹:ℤ–1-1→𝑋)) |
9 | 8 | biimp3a 1465 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → 𝐹:ℤ–1-1→𝑋) |
10 | f1f 6577 | . . . . . 6 ⊢ (𝐹:ℤ–1-1→𝑋 → 𝐹:ℤ⟶𝑋) | |
11 | zex 11993 | . . . . . . 7 ⊢ ℤ ∈ V | |
12 | 4 | fvexi 6686 | . . . . . . 7 ⊢ 𝑋 ∈ V |
13 | fex2 7640 | . . . . . . 7 ⊢ ((𝐹:ℤ⟶𝑋 ∧ ℤ ∈ V ∧ 𝑋 ∈ V) → 𝐹 ∈ V) | |
14 | 11, 12, 13 | mp3an23 1449 | . . . . . 6 ⊢ (𝐹:ℤ⟶𝑋 → 𝐹 ∈ V) |
15 | 9, 10, 14 | 3syl 18 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → 𝐹 ∈ V) |
16 | f1f1orn 6628 | . . . . . 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 8527 | . . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝐹:ℤ–1-1-onto→ran 𝐹) → ℤ ≈ ran 𝐹) | |
19 | 15, 17, 18 | syl2anc 586 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → ℤ ≈ ran 𝐹) |
20 | entr 8563 | . . . 4 ⊢ ((ω ≈ ℤ ∧ ℤ ≈ ran 𝐹) → ω ≈ ran 𝐹) | |
21 | 3, 19, 20 | sylancr 589 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → ω ≈ ran 𝐹) |
22 | endom 8538 | . . 3 ⊢ (ω ≈ ran 𝐹 → ω ≼ ran 𝐹) | |
23 | domnsym 8645 | . . 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 9117 | . 2 ⊢ (ran 𝐹 ∈ Fin ↔ ran 𝐹 ≺ ω) | |
26 | 24, 25 | sylnibr 331 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → ¬ ran 𝐹 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 Vcvv 3496 class class class wbr 5068 ↦ cmpt 5148 ran crn 5558 ⟶wf 6353 –1-1→wf1 6354 –1-1-onto→wf1o 6356 ‘cfv 6357 (class class class)co 7158 ωcom 7582 ≈ cen 8508 ≼ cdom 8509 ≺ csdm 8510 Fincfn 8511 0cc0 10539 ℕcn 11640 ℤcz 11984 Basecbs 16485 Grpcgrp 18105 .gcmg 18226 odcod 18654 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-omul 8109 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-acn 9373 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-dvds 15610 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-od 18658 |
This theorem is referenced by: dfod2 18693 odcl2 18694 |
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