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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 2737 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
| 4 | 2, 3 | iscyg 19897 | . . . 4 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑦 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵)) |
| 5 | 4 | simprbi 496 | . . 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 2737 | . . . 4 ⊢ (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)(((♯‘𝐵) / 𝐶)(.g‘𝐺)𝑦))) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)(((♯‘𝐵) / 𝐶)(.g‘𝐺)𝑦))) | |
| 8 | cyggrp 19908 | . . . . . 6 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Grp) | |
| 9 | 1, 8 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ Grp) |
| 10 | 9 | adantr 480 | . . . 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 18990 | . . . . . . 7 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ) |
| 15 | fincygsubgodexd.5 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℕ) | |
| 16 | nndivdvds 16299 | . . . . . . 7 ⊢ (((♯‘𝐵) ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 ∥ (♯‘𝐵) ↔ ((♯‘𝐵) / 𝐶) ∈ ℕ)) | |
| 17 | 14, 15, 16 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝐶 ∥ (♯‘𝐵) ↔ ((♯‘𝐵) / 𝐶) ∈ ℕ)) |
| 18 | 12, 17 | mpbid 232 | . . . . 5 ⊢ (𝜑 → ((♯‘𝐵) / 𝐶) ∈ ℕ) |
| 19 | 18 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵)) → ((♯‘𝐵) / 𝐶) ∈ ℕ) |
| 20 | 2, 3, 7, 10, 11, 19 | fincygsubgd 20131 | . . 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 484 | . . . . 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 6910 | . . . 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 2737 | . . . . . 6 ⊢ ((♯‘𝐵) / ((♯‘𝐵) / 𝐶)) = ((♯‘𝐵) / ((♯‘𝐵) / 𝐶)) | |
| 24 | eqid 2737 | . . . . . 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 12316 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ≠ 0) |
| 27 | divconjdvds 16352 | . . . . . . . 8 ⊢ ((𝐶 ∥ (♯‘𝐵) ∧ 𝐶 ≠ 0) → ((♯‘𝐵) / 𝐶) ∥ (♯‘𝐵)) | |
| 28 | 12, 26, 27 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → ((♯‘𝐵) / 𝐶) ∥ (♯‘𝐵)) |
| 29 | 28 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵)) → ((♯‘𝐵) / 𝐶) ∥ (♯‘𝐵)) |
| 30 | 13 | adantr 480 | . . . . . 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 20132 | . . . . 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 480 | . . . 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 12282 | . . . . . 6 ⊢ (𝜑 → (♯‘𝐵) ∈ ℂ) |
| 34 | 15 | nncnd 12282 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
| 35 | 14 | nnne0d 12316 | . . . . . 6 ⊢ (𝜑 → (♯‘𝐵) ≠ 0) |
| 36 | 33, 34, 35, 26 | ddcand 12063 | . . . . 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 2781 | . . 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 3629 | . 2 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑦)) = 𝐵)) → ∃𝑥 ∈ (SubGrp‘𝐺)(♯‘𝑥) = 𝐶) |
| 40 | 6, 39 | rexlimddv 3161 | 1 ⊢ (𝜑 → ∃𝑥 ∈ (SubGrp‘𝐺)(♯‘𝑥) = 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 class class class wbr 5143 ↦ cmpt 5225 ran crn 5686 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 0cc0 11155 / cdiv 11920 ℕcn 12266 ℤcz 12613 ♯chash 14369 ∥ cdvds 16290 Basecbs 17247 Grpcgrp 18951 .gcmg 19085 SubGrpcsubg 19138 CycGrpccyg 19895 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-omul 8511 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-acn 9982 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-gcd 16532 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-od 19546 df-cyg 19896 |
| This theorem is referenced by: ablsimpgprmd 20135 |
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