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Mirrors > Home > MPE Home > Th. List > cyggic | Structured version Visualization version GIF version |
Description: Cyclic groups are isomorphic precisely when they have the same order. (Contributed by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
cygctb.b | ⊢ 𝐵 = (Base‘𝐺) |
cygctb.c | ⊢ 𝐶 = (Base‘𝐻) |
Ref | Expression |
---|---|
cyggic | ⊢ ((𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp) → (𝐺 ≃𝑔 𝐻 ↔ 𝐵 ≈ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cygctb.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | cygctb.c | . . 3 ⊢ 𝐶 = (Base‘𝐻) | |
3 | 1, 2 | gicen 17766 | . 2 ⊢ (𝐺 ≃𝑔 𝐻 → 𝐵 ≈ 𝐶) |
4 | eqid 2651 | . . . . . 6 ⊢ if(𝐵 ∈ Fin, (#‘𝐵), 0) = if(𝐵 ∈ Fin, (#‘𝐵), 0) | |
5 | eqid 2651 | . . . . . 6 ⊢ (ℤ/nℤ‘if(𝐵 ∈ Fin, (#‘𝐵), 0)) = (ℤ/nℤ‘if(𝐵 ∈ Fin, (#‘𝐵), 0)) | |
6 | 1, 4, 5 | cygzn 19967 | . . . . 5 ⊢ (𝐺 ∈ CycGrp → 𝐺 ≃𝑔 (ℤ/nℤ‘if(𝐵 ∈ Fin, (#‘𝐵), 0))) |
7 | 6 | ad2antrr 762 | . . . 4 ⊢ (((𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp) ∧ 𝐵 ≈ 𝐶) → 𝐺 ≃𝑔 (ℤ/nℤ‘if(𝐵 ∈ Fin, (#‘𝐵), 0))) |
8 | enfi 8217 | . . . . . . . 8 ⊢ (𝐵 ≈ 𝐶 → (𝐵 ∈ Fin ↔ 𝐶 ∈ Fin)) | |
9 | 8 | adantl 481 | . . . . . . 7 ⊢ (((𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp) ∧ 𝐵 ≈ 𝐶) → (𝐵 ∈ Fin ↔ 𝐶 ∈ Fin)) |
10 | hasheni 13176 | . . . . . . . 8 ⊢ (𝐵 ≈ 𝐶 → (#‘𝐵) = (#‘𝐶)) | |
11 | 10 | adantl 481 | . . . . . . 7 ⊢ (((𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp) ∧ 𝐵 ≈ 𝐶) → (#‘𝐵) = (#‘𝐶)) |
12 | 9, 11 | ifbieq1d 4142 | . . . . . 6 ⊢ (((𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp) ∧ 𝐵 ≈ 𝐶) → if(𝐵 ∈ Fin, (#‘𝐵), 0) = if(𝐶 ∈ Fin, (#‘𝐶), 0)) |
13 | 12 | fveq2d 6233 | . . . . 5 ⊢ (((𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp) ∧ 𝐵 ≈ 𝐶) → (ℤ/nℤ‘if(𝐵 ∈ Fin, (#‘𝐵), 0)) = (ℤ/nℤ‘if(𝐶 ∈ Fin, (#‘𝐶), 0))) |
14 | eqid 2651 | . . . . . . . 8 ⊢ if(𝐶 ∈ Fin, (#‘𝐶), 0) = if(𝐶 ∈ Fin, (#‘𝐶), 0) | |
15 | eqid 2651 | . . . . . . . 8 ⊢ (ℤ/nℤ‘if(𝐶 ∈ Fin, (#‘𝐶), 0)) = (ℤ/nℤ‘if(𝐶 ∈ Fin, (#‘𝐶), 0)) | |
16 | 2, 14, 15 | cygzn 19967 | . . . . . . 7 ⊢ (𝐻 ∈ CycGrp → 𝐻 ≃𝑔 (ℤ/nℤ‘if(𝐶 ∈ Fin, (#‘𝐶), 0))) |
17 | 16 | ad2antlr 763 | . . . . . 6 ⊢ (((𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp) ∧ 𝐵 ≈ 𝐶) → 𝐻 ≃𝑔 (ℤ/nℤ‘if(𝐶 ∈ Fin, (#‘𝐶), 0))) |
18 | gicsym 17763 | . . . . . 6 ⊢ (𝐻 ≃𝑔 (ℤ/nℤ‘if(𝐶 ∈ Fin, (#‘𝐶), 0)) → (ℤ/nℤ‘if(𝐶 ∈ Fin, (#‘𝐶), 0)) ≃𝑔 𝐻) | |
19 | 17, 18 | syl 17 | . . . . 5 ⊢ (((𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp) ∧ 𝐵 ≈ 𝐶) → (ℤ/nℤ‘if(𝐶 ∈ Fin, (#‘𝐶), 0)) ≃𝑔 𝐻) |
20 | 13, 19 | eqbrtrd 4707 | . . . 4 ⊢ (((𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp) ∧ 𝐵 ≈ 𝐶) → (ℤ/nℤ‘if(𝐵 ∈ Fin, (#‘𝐵), 0)) ≃𝑔 𝐻) |
21 | gictr 17764 | . . . 4 ⊢ ((𝐺 ≃𝑔 (ℤ/nℤ‘if(𝐵 ∈ Fin, (#‘𝐵), 0)) ∧ (ℤ/nℤ‘if(𝐵 ∈ Fin, (#‘𝐵), 0)) ≃𝑔 𝐻) → 𝐺 ≃𝑔 𝐻) | |
22 | 7, 20, 21 | syl2anc 694 | . . 3 ⊢ (((𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp) ∧ 𝐵 ≈ 𝐶) → 𝐺 ≃𝑔 𝐻) |
23 | 22 | ex 449 | . 2 ⊢ ((𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp) → (𝐵 ≈ 𝐶 → 𝐺 ≃𝑔 𝐻)) |
24 | 3, 23 | impbid2 216 | 1 ⊢ ((𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp) → (𝐺 ≃𝑔 𝐻 ↔ 𝐵 ≈ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ifcif 4119 class class class wbr 4685 ‘cfv 5926 ≈ cen 7994 Fincfn 7997 0cc0 9974 #chash 13157 Basecbs 15904 ≃𝑔 cgic 17747 CycGrpccyg 18325 ℤ/nℤczn 19899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 ax-mulf 10054 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-tpos 7397 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-omul 7610 df-er 7787 df-ec 7789 df-qs 7793 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-acn 8806 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-rp 11871 df-fz 12365 df-fl 12633 df-mod 12709 df-seq 12842 df-exp 12901 df-hash 13158 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-dvds 15028 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-starv 16003 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-0g 16149 df-imas 16215 df-qus 16216 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-mhm 17382 df-grp 17472 df-minusg 17473 df-sbg 17474 df-mulg 17588 df-subg 17638 df-nsg 17639 df-eqg 17640 df-ghm 17705 df-gim 17748 df-gic 17749 df-od 17994 df-cmn 18241 df-abl 18242 df-cyg 18326 df-mgp 18536 df-ur 18548 df-ring 18595 df-cring 18596 df-oppr 18669 df-dvdsr 18687 df-rnghom 18763 df-subrg 18826 df-lmod 18913 df-lss 18981 df-lsp 19020 df-sra 19220 df-rgmod 19221 df-lidl 19222 df-rsp 19223 df-2idl 19280 df-cnfld 19795 df-zring 19867 df-zrh 19900 df-zn 19903 |
This theorem is referenced by: (None) |
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