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Mirrors > Home > MPE Home > Th. List > odhash | Structured version Visualization version GIF version |
Description: An element of zero order generates an infinite subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
Ref | Expression |
---|---|
odhash.x | ⊢ 𝑋 = (Base‘𝐺) |
odhash.o | ⊢ 𝑂 = (od‘𝐺) |
odhash.k | ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) |
Ref | Expression |
---|---|
odhash | ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → (#‘(𝐾‘{𝐴})) = +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zex 11263 | . . 3 ⊢ ℤ ∈ V | |
2 | ominf 8057 | . . . 4 ⊢ ¬ ω ∈ Fin | |
3 | znnen 14780 | . . . . . 6 ⊢ ℤ ≈ ℕ | |
4 | nnenom 12641 | . . . . . 6 ⊢ ℕ ≈ ω | |
5 | 3, 4 | entri 7896 | . . . . 5 ⊢ ℤ ≈ ω |
6 | enfi 8061 | . . . . 5 ⊢ (ℤ ≈ ω → (ℤ ∈ Fin ↔ ω ∈ Fin)) | |
7 | 5, 6 | ax-mp 5 | . . . 4 ⊢ (ℤ ∈ Fin ↔ ω ∈ Fin) |
8 | 2, 7 | mtbir 312 | . . 3 ⊢ ¬ ℤ ∈ Fin |
9 | hashinf 12984 | . . 3 ⊢ ((ℤ ∈ V ∧ ¬ ℤ ∈ Fin) → (#‘ℤ) = +∞) | |
10 | 1, 8, 9 | mp2an 704 | . 2 ⊢ (#‘ℤ) = +∞ |
11 | odhash.x | . . . 4 ⊢ 𝑋 = (Base‘𝐺) | |
12 | eqid 2610 | . . . 4 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
13 | odhash.o | . . . 4 ⊢ 𝑂 = (od‘𝐺) | |
14 | odhash.k | . . . 4 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) | |
15 | 11, 12, 13, 14 | odf1o1 17810 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → (𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴)):ℤ–1-1-onto→(𝐾‘{𝐴})) |
16 | 1 | f1oen 7862 | . . 3 ⊢ ((𝑥 ∈ ℤ ↦ (𝑥(.g‘𝐺)𝐴)):ℤ–1-1-onto→(𝐾‘{𝐴}) → ℤ ≈ (𝐾‘{𝐴})) |
17 | hasheni 12998 | . . 3 ⊢ (ℤ ≈ (𝐾‘{𝐴}) → (#‘ℤ) = (#‘(𝐾‘{𝐴}))) | |
18 | 15, 16, 17 | 3syl 18 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → (#‘ℤ) = (#‘(𝐾‘{𝐴}))) |
19 | 10, 18 | syl5reqr 2659 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → (#‘(𝐾‘{𝐴})) = +∞) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 class class class wbr 4583 ↦ cmpt 4643 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ωcom 6957 ≈ cen 7838 Fincfn 7841 0cc0 9815 +∞cpnf 9950 ℕcn 10897 ℤcz 11254 #chash 12979 Basecbs 15695 mrClscmrc 16066 Grpcgrp 17245 .gcmg 17363 SubGrpcsubg 17411 odcod 17767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-omul 7452 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-od 17771 |
This theorem is referenced by: odhash3 17814 |
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