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Mirrors > Home > MPE Home > Th. List > fincygsubgodexd | Structured version Visualization version GIF version |
Description: A finite cyclic group has subgroups of every possible order. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
Ref | Expression |
---|---|
fincygsubgodexd.1 | ⊢ 𝐵 = (Base‘𝐺) |
fincygsubgodexd.2 | ⊢ (𝜑 → 𝐺 ∈ CycGrp) |
fincygsubgodexd.3 | ⊢ (𝜑 → 𝐶 ∥ (♯‘𝐵)) |
fincygsubgodexd.4 | ⊢ (𝜑 → 𝐵 ∈ Fin) |
fincygsubgodexd.5 | ⊢ (𝜑 → 𝐶 ∈ ℕ) |
Ref | Expression |
---|---|
fincygsubgodexd | ⊢ (𝜑 → ∃𝑥 ∈ (SubGrp‘𝐺)(♯‘𝑥) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fincygsubgodexd.2 | . . 3 ⊢ (𝜑 → 𝐺 ∈ CycGrp) | |
2 | fincygsubgodexd.1 | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
3 | eqid 2734 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
4 | 2, 3 | iscyg 19235 | . . . 4 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑦 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵)) |
5 | 4 | simprbi 500 | . . 3 ⊢ (𝐺 ∈ CycGrp → ∃𝑦 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵) |
6 | 1, 5 | syl 17 | . 2 ⊢ (𝜑 → ∃𝑦 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵) |
7 | eqid 2734 | . . . 4 ⊢ (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)(((♯‘𝐵) / 𝐶)(.g‘𝐺)𝑦))) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)(((♯‘𝐵) / 𝐶)(.g‘𝐺)𝑦))) | |
8 | cyggrp 19246 | . . . . . 6 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Grp) | |
9 | 1, 8 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ Grp) |
10 | 9 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵)) → 𝐺 ∈ Grp) |
11 | simprl 771 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵)) → 𝑦 ∈ 𝐵) | |
12 | fincygsubgodexd.3 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∥ (♯‘𝐵)) | |
13 | fincygsubgodexd.4 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ Fin) | |
14 | 2, 9, 13 | hashfingrpnn 18372 | . . . . . . 7 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ) |
15 | fincygsubgodexd.5 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℕ) | |
16 | nndivdvds 15805 | . . . . . . 7 ⊢ (((♯‘𝐵) ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 ∥ (♯‘𝐵) ↔ ((♯‘𝐵) / 𝐶) ∈ ℕ)) | |
17 | 14, 15, 16 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → (𝐶 ∥ (♯‘𝐵) ↔ ((♯‘𝐵) / 𝐶) ∈ ℕ)) |
18 | 12, 17 | mpbid 235 | . . . . 5 ⊢ (𝜑 → ((♯‘𝐵) / 𝐶) ∈ ℕ) |
19 | 18 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵)) → ((♯‘𝐵) / 𝐶) ∈ ℕ) |
20 | 2, 3, 7, 10, 11, 19 | fincygsubgd 19470 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵)) → ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)(((♯‘𝐵) / 𝐶)(.g‘𝐺)𝑦))) ∈ (SubGrp‘𝐺)) |
21 | simpr 488 | . . . . 5 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵)) ∧ 𝑥 = ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)(((♯‘𝐵) / 𝐶)(.g‘𝐺)𝑦)))) → 𝑥 = ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)(((♯‘𝐵) / 𝐶)(.g‘𝐺)𝑦)))) | |
22 | 21 | fveq2d 6710 | . . . 4 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵)) ∧ 𝑥 = ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)(((♯‘𝐵) / 𝐶)(.g‘𝐺)𝑦)))) → (♯‘𝑥) = (♯‘ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)(((♯‘𝐵) / 𝐶)(.g‘𝐺)𝑦))))) |
23 | eqid 2734 | . . . . . 6 ⊢ ((♯‘𝐵) / ((♯‘𝐵) / 𝐶)) = ((♯‘𝐵) / ((♯‘𝐵) / 𝐶)) | |
24 | eqid 2734 | . . . . . 6 ⊢ (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) | |
25 | simprr 773 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵)) → ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵) | |
26 | 15 | nnne0d 11863 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ≠ 0) |
27 | divconjdvds 15857 | . . . . . . . 8 ⊢ ((𝐶 ∥ (♯‘𝐵) ∧ 𝐶 ≠ 0) → ((♯‘𝐵) / 𝐶) ∥ (♯‘𝐵)) | |
28 | 12, 26, 27 | syl2anc 587 | . . . . . . 7 ⊢ (𝜑 → ((♯‘𝐵) / 𝐶) ∥ (♯‘𝐵)) |
29 | 28 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵)) → ((♯‘𝐵) / 𝐶) ∥ (♯‘𝐵)) |
30 | 13 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵)) → 𝐵 ∈ Fin) |
31 | 2, 3, 23, 24, 7, 10, 11, 25, 29, 30, 19 | fincygsubgodd 19471 | . . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵)) → (♯‘ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)(((♯‘𝐵) / 𝐶)(.g‘𝐺)𝑦)))) = ((♯‘𝐵) / ((♯‘𝐵) / 𝐶))) |
32 | 31 | adantr 484 | . . . 4 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵)) ∧ 𝑥 = ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)(((♯‘𝐵) / 𝐶)(.g‘𝐺)𝑦)))) → (♯‘ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)(((♯‘𝐵) / 𝐶)(.g‘𝐺)𝑦)))) = ((♯‘𝐵) / ((♯‘𝐵) / 𝐶))) |
33 | 14 | nncnd 11829 | . . . . . 6 ⊢ (𝜑 → (♯‘𝐵) ∈ ℂ) |
34 | 15 | nncnd 11829 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
35 | 14 | nnne0d 11863 | . . . . . 6 ⊢ (𝜑 → (♯‘𝐵) ≠ 0) |
36 | 33, 34, 35, 26 | ddcand 11611 | . . . . 5 ⊢ (𝜑 → ((♯‘𝐵) / ((♯‘𝐵) / 𝐶)) = 𝐶) |
37 | 36 | ad2antrr 726 | . . . 4 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵)) ∧ 𝑥 = ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)(((♯‘𝐵) / 𝐶)(.g‘𝐺)𝑦)))) → ((♯‘𝐵) / ((♯‘𝐵) / 𝐶)) = 𝐶) |
38 | 22, 32, 37 | 3eqtrd 2778 | . . 3 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵)) ∧ 𝑥 = ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)(((♯‘𝐵) / 𝐶)(.g‘𝐺)𝑦)))) → (♯‘𝑥) = 𝐶) |
39 | 20, 38 | rspcedeq1vd 3536 | . 2 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵)) → ∃𝑥 ∈ (SubGrp‘𝐺)(♯‘𝑥) = 𝐶) |
40 | 6, 39 | rexlimddv 3203 | 1 ⊢ (𝜑 → ∃𝑥 ∈ (SubGrp‘𝐺)(♯‘𝑥) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 ∃wrex 3055 class class class wbr 5043 ↦ cmpt 5124 ran crn 5541 ‘cfv 6369 (class class class)co 7202 Fincfn 8615 0cc0 10712 / cdiv 11472 ℕcn 11813 ℤcz 12159 ♯chash 13879 ∥ cdvds 15796 Basecbs 16684 Grpcgrp 18337 .gcmg 18460 SubGrpcsubg 18509 CycGrpccyg 19233 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-oadd 8195 df-omul 8196 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-sup 9047 df-inf 9048 df-oi 9115 df-card 9538 df-acn 9541 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-n0 12074 df-z 12160 df-uz 12422 df-rp 12570 df-fz 13079 df-fl 13350 df-mod 13426 df-seq 13558 df-exp 13619 df-hash 13880 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-dvds 15797 df-gcd 16035 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-0g 16918 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-grp 18340 df-minusg 18341 df-sbg 18342 df-mulg 18461 df-subg 18512 df-od 18892 df-cyg 19234 |
This theorem is referenced by: ablsimpgprmd 19474 |
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