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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 16162 | . . . . 5 ⊢ ℤ ≈ ℕ | |
2 | nnenom 13952 | . . . . 5 ⊢ ℕ ≈ ω | |
3 | 1, 2 | entr2i 9011 | . . . 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 19475 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝑂‘𝐴) = 0 ↔ 𝐹:ℤ–1-1→𝑋)) |
9 | 8 | biimp3a 1468 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → 𝐹:ℤ–1-1→𝑋) |
10 | f1f 6787 | . . . . . 6 ⊢ (𝐹:ℤ–1-1→𝑋 → 𝐹:ℤ⟶𝑋) | |
11 | zex 12574 | . . . . . . 7 ⊢ ℤ ∈ V | |
12 | 4 | fvexi 6905 | . . . . . . 7 ⊢ 𝑋 ∈ V |
13 | fex2 7928 | . . . . . . 7 ⊢ ((𝐹:ℤ⟶𝑋 ∧ ℤ ∈ V ∧ 𝑋 ∈ V) → 𝐹 ∈ V) | |
14 | 11, 12, 13 | mp3an23 1452 | . . . . . 6 ⊢ (𝐹:ℤ⟶𝑋 → 𝐹 ∈ V) |
15 | 9, 10, 14 | 3syl 18 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → 𝐹 ∈ V) |
16 | f1f1orn 6844 | . . . . . 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 8968 | . . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝐹:ℤ–1-1-onto→ran 𝐹) → ℤ ≈ ran 𝐹) | |
19 | 15, 17, 18 | syl2anc 583 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → ℤ ≈ ran 𝐹) |
20 | entr 9008 | . . . 4 ⊢ ((ω ≈ ℤ ∧ ℤ ≈ ran 𝐹) → ω ≈ ran 𝐹) | |
21 | 3, 19, 20 | sylancr 586 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → ω ≈ ran 𝐹) |
22 | endom 8981 | . . 3 ⊢ (ω ≈ ran 𝐹 → ω ≼ ran 𝐹) | |
23 | domnsym 9105 | . . 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 9653 | . 2 ⊢ (ran 𝐹 ∈ Fin ↔ ran 𝐹 ≺ ω) | |
26 | 24, 25 | sylnibr 329 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → ¬ ran 𝐹 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 Vcvv 3473 class class class wbr 5148 ↦ cmpt 5231 ran crn 5677 ⟶wf 6539 –1-1→wf1 6540 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7412 ωcom 7859 ≈ cen 8942 ≼ cdom 8943 ≺ csdm 8944 Fincfn 8945 0cc0 11116 ℕcn 12219 ℤcz 12565 Basecbs 17151 Grpcgrp 18858 .gcmg 18990 odcod 19437 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-oadd 8476 df-omul 8477 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-inf 9444 df-oi 9511 df-card 9940 df-acn 9943 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-z 12566 df-uz 12830 df-rp 12982 df-fz 13492 df-fl 13764 df-mod 13842 df-seq 13974 df-exp 14035 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-dvds 16205 df-0g 17394 df-mgm 18568 df-sgrp 18647 df-mnd 18663 df-grp 18861 df-minusg 18862 df-sbg 18863 df-mulg 18991 df-od 19441 |
This theorem is referenced by: dfod2 19477 odcl2 19478 |
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