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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 19231 | . 2 ⊢ (𝐺 ≃𝑔 𝐻 → 𝐵 ≈ 𝐶) |
4 | eqid 2728 | . . . . . 6 ⊢ if(𝐵 ∈ Fin, (♯‘𝐵), 0) = if(𝐵 ∈ Fin, (♯‘𝐵), 0) | |
5 | eqid 2728 | . . . . . 6 ⊢ (ℤ/nℤ‘if(𝐵 ∈ Fin, (♯‘𝐵), 0)) = (ℤ/nℤ‘if(𝐵 ∈ Fin, (♯‘𝐵), 0)) | |
6 | 1, 4, 5 | cygzn 21503 | . . . . 5 ⊢ (𝐺 ∈ CycGrp → 𝐺 ≃𝑔 (ℤ/nℤ‘if(𝐵 ∈ Fin, (♯‘𝐵), 0))) |
7 | 6 | ad2antrr 725 | . . . 4 ⊢ (((𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp) ∧ 𝐵 ≈ 𝐶) → 𝐺 ≃𝑔 (ℤ/nℤ‘if(𝐵 ∈ Fin, (♯‘𝐵), 0))) |
8 | enfi 9214 | . . . . . . . 8 ⊢ (𝐵 ≈ 𝐶 → (𝐵 ∈ Fin ↔ 𝐶 ∈ Fin)) | |
9 | 8 | adantl 481 | . . . . . . 7 ⊢ (((𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp) ∧ 𝐵 ≈ 𝐶) → (𝐵 ∈ Fin ↔ 𝐶 ∈ Fin)) |
10 | hasheni 14339 | . . . . . . . 8 ⊢ (𝐵 ≈ 𝐶 → (♯‘𝐵) = (♯‘𝐶)) | |
11 | 10 | adantl 481 | . . . . . . 7 ⊢ (((𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp) ∧ 𝐵 ≈ 𝐶) → (♯‘𝐵) = (♯‘𝐶)) |
12 | 9, 11 | ifbieq1d 4553 | . . . . . 6 ⊢ (((𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp) ∧ 𝐵 ≈ 𝐶) → if(𝐵 ∈ Fin, (♯‘𝐵), 0) = if(𝐶 ∈ Fin, (♯‘𝐶), 0)) |
13 | 12 | fveq2d 6901 | . . . . 5 ⊢ (((𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp) ∧ 𝐵 ≈ 𝐶) → (ℤ/nℤ‘if(𝐵 ∈ Fin, (♯‘𝐵), 0)) = (ℤ/nℤ‘if(𝐶 ∈ Fin, (♯‘𝐶), 0))) |
14 | eqid 2728 | . . . . . . . 8 ⊢ if(𝐶 ∈ Fin, (♯‘𝐶), 0) = if(𝐶 ∈ Fin, (♯‘𝐶), 0) | |
15 | eqid 2728 | . . . . . . . 8 ⊢ (ℤ/nℤ‘if(𝐶 ∈ Fin, (♯‘𝐶), 0)) = (ℤ/nℤ‘if(𝐶 ∈ Fin, (♯‘𝐶), 0)) | |
16 | 2, 14, 15 | cygzn 21503 | . . . . . . 7 ⊢ (𝐻 ∈ CycGrp → 𝐻 ≃𝑔 (ℤ/nℤ‘if(𝐶 ∈ Fin, (♯‘𝐶), 0))) |
17 | 16 | ad2antlr 726 | . . . . . 6 ⊢ (((𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp) ∧ 𝐵 ≈ 𝐶) → 𝐻 ≃𝑔 (ℤ/nℤ‘if(𝐶 ∈ Fin, (♯‘𝐶), 0))) |
18 | gicsym 19228 | . . . . . 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 5170 | . . . 4 ⊢ (((𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp) ∧ 𝐵 ≈ 𝐶) → (ℤ/nℤ‘if(𝐵 ∈ Fin, (♯‘𝐵), 0)) ≃𝑔 𝐻) |
21 | gictr 19229 | . . . 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 583 | . . 3 ⊢ (((𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp) ∧ 𝐵 ≈ 𝐶) → 𝐺 ≃𝑔 𝐻) |
23 | 22 | ex 412 | . 2 ⊢ ((𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp) → (𝐵 ≈ 𝐶 → 𝐺 ≃𝑔 𝐻)) |
24 | 3, 23 | impbid2 225 | 1 ⊢ ((𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp) → (𝐺 ≃𝑔 𝐻 ↔ 𝐵 ≈ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ifcif 4529 class class class wbr 5148 ‘cfv 6548 ≈ cen 8960 Fincfn 8963 0cc0 11138 ♯chash 14321 Basecbs 17179 ≃𝑔 cgic 19211 CycGrpccyg 19831 ℤ/nℤczn 21427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 ax-mulf 11218 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-tpos 8231 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-oadd 8490 df-omul 8491 df-er 8724 df-ec 8726 df-qs 8730 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-sup 9465 df-inf 9466 df-oi 9533 df-card 9962 df-acn 9965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-rp 13007 df-fz 13517 df-fl 13789 df-mod 13867 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-dvds 16231 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-0g 17422 df-imas 17489 df-qus 17490 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-grp 18892 df-minusg 18893 df-sbg 18894 df-mulg 19023 df-subg 19077 df-nsg 19078 df-eqg 19079 df-ghm 19167 df-gim 19212 df-gic 19213 df-od 19482 df-cmn 19736 df-abl 19737 df-cyg 19832 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-cring 20175 df-oppr 20272 df-dvdsr 20295 df-rhm 20410 df-subrng 20482 df-subrg 20507 df-lmod 20744 df-lss 20815 df-lsp 20855 df-sra 21057 df-rgmod 21058 df-lidl 21103 df-rsp 21104 df-2idl 21143 df-cnfld 21279 df-zring 21372 df-zrh 21428 df-zn 21431 |
This theorem is referenced by: (None) |
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