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| Mirrors > Home > MPE Home > Th. List > finodsubmsubg | Structured version Visualization version GIF version | ||
| Description: A submonoid whose elements have finite order is a subgroup. (Contributed by SN, 31-Jan-2025.) |
| Ref | Expression |
|---|---|
| finodsubmsubg.o | ⊢ 𝑂 = (od‘𝐺) |
| finodsubmsubg.g | ⊢ (𝜑 → 𝐺 ∈ Grp) |
| finodsubmsubg.s | ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺)) |
| finodsubmsubg.1 | ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 (𝑂‘𝑎) ∈ ℕ) |
| Ref | Expression |
|---|---|
| finodsubmsubg | ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | finodsubmsubg.s | . 2 ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺)) | |
| 2 | finodsubmsubg.1 | . . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 (𝑂‘𝑎) ∈ ℕ) | |
| 3 | eqid 2737 | . . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
| 4 | finodsubmsubg.o | . . . . . . . 8 ⊢ 𝑂 = (od‘𝐺) | |
| 5 | eqid 2737 | . . . . . . . 8 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
| 6 | eqid 2737 | . . . . . . . 8 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
| 7 | finodsubmsubg.g | . . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
| 8 | 7 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝐺 ∈ Grp) |
| 9 | 3 | submss 18822 | . . . . . . . . . 10 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 ⊆ (Base‘𝐺)) |
| 10 | 1, 9 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐺)) |
| 11 | 10 | sselda 3983 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ (Base‘𝐺)) |
| 12 | 3, 4, 5, 6, 8, 11 | odm1inv 19571 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (((𝑂‘𝑎) − 1)(.g‘𝐺)𝑎) = ((invg‘𝐺)‘𝑎)) |
| 13 | 12 | adantr 480 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑂‘𝑎) ∈ ℕ) → (((𝑂‘𝑎) − 1)(.g‘𝐺)𝑎) = ((invg‘𝐺)‘𝑎)) |
| 14 | eqid 2737 | . . . . . . . 8 ⊢ (Base‘(𝐺 ↾s 𝑆)) = (Base‘(𝐺 ↾s 𝑆)) | |
| 15 | eqid 2737 | . . . . . . . 8 ⊢ (.g‘(𝐺 ↾s 𝑆)) = (.g‘(𝐺 ↾s 𝑆)) | |
| 16 | eqid 2737 | . . . . . . . . . . 11 ⊢ (𝐺 ↾s 𝑆) = (𝐺 ↾s 𝑆) | |
| 17 | 16 | submmnd 18826 | . . . . . . . . . 10 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → (𝐺 ↾s 𝑆) ∈ Mnd) |
| 18 | 1, 17 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (𝐺 ↾s 𝑆) ∈ Mnd) |
| 19 | 18 | ad2antrr 726 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑂‘𝑎) ∈ ℕ) → (𝐺 ↾s 𝑆) ∈ Mnd) |
| 20 | nnm1nn0 12567 | . . . . . . . . 9 ⊢ ((𝑂‘𝑎) ∈ ℕ → ((𝑂‘𝑎) − 1) ∈ ℕ0) | |
| 21 | 20 | adantl 481 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑂‘𝑎) ∈ ℕ) → ((𝑂‘𝑎) − 1) ∈ ℕ0) |
| 22 | simplr 769 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑂‘𝑎) ∈ ℕ) → 𝑎 ∈ 𝑆) | |
| 23 | 16, 3 | ressbas2 17283 | . . . . . . . . . . 11 ⊢ (𝑆 ⊆ (Base‘𝐺) → 𝑆 = (Base‘(𝐺 ↾s 𝑆))) |
| 24 | 10, 23 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑆 = (Base‘(𝐺 ↾s 𝑆))) |
| 25 | 24 | ad2antrr 726 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑂‘𝑎) ∈ ℕ) → 𝑆 = (Base‘(𝐺 ↾s 𝑆))) |
| 26 | 22, 25 | eleqtrd 2843 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑂‘𝑎) ∈ ℕ) → 𝑎 ∈ (Base‘(𝐺 ↾s 𝑆))) |
| 27 | 14, 15, 19, 21, 26 | mulgnn0cld 19113 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑂‘𝑎) ∈ ℕ) → (((𝑂‘𝑎) − 1)(.g‘(𝐺 ↾s 𝑆))𝑎) ∈ (Base‘(𝐺 ↾s 𝑆))) |
| 28 | 1 | ad2antrr 726 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑂‘𝑎) ∈ ℕ) → 𝑆 ∈ (SubMnd‘𝐺)) |
| 29 | 5, 16, 15 | submmulg 19136 | . . . . . . . 8 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ ((𝑂‘𝑎) − 1) ∈ ℕ0 ∧ 𝑎 ∈ 𝑆) → (((𝑂‘𝑎) − 1)(.g‘𝐺)𝑎) = (((𝑂‘𝑎) − 1)(.g‘(𝐺 ↾s 𝑆))𝑎)) |
| 30 | 28, 21, 22, 29 | syl3anc 1373 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑂‘𝑎) ∈ ℕ) → (((𝑂‘𝑎) − 1)(.g‘𝐺)𝑎) = (((𝑂‘𝑎) − 1)(.g‘(𝐺 ↾s 𝑆))𝑎)) |
| 31 | 27, 30, 25 | 3eltr4d 2856 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑂‘𝑎) ∈ ℕ) → (((𝑂‘𝑎) − 1)(.g‘𝐺)𝑎) ∈ 𝑆) |
| 32 | 13, 31 | eqeltrrd 2842 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑂‘𝑎) ∈ ℕ) → ((invg‘𝐺)‘𝑎) ∈ 𝑆) |
| 33 | 32 | ex 412 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ((𝑂‘𝑎) ∈ ℕ → ((invg‘𝐺)‘𝑎) ∈ 𝑆)) |
| 34 | 33 | ralimdva 3167 | . . 3 ⊢ (𝜑 → (∀𝑎 ∈ 𝑆 (𝑂‘𝑎) ∈ ℕ → ∀𝑎 ∈ 𝑆 ((invg‘𝐺)‘𝑎) ∈ 𝑆)) |
| 35 | 2, 34 | mpd 15 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 ((invg‘𝐺)‘𝑎) ∈ 𝑆) |
| 36 | 6 | issubg3 19162 | . . 3 ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆 ∈ (SubMnd‘𝐺) ∧ ∀𝑎 ∈ 𝑆 ((invg‘𝐺)‘𝑎) ∈ 𝑆))) |
| 37 | 7, 36 | syl 17 | . 2 ⊢ (𝜑 → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆 ∈ (SubMnd‘𝐺) ∧ ∀𝑎 ∈ 𝑆 ((invg‘𝐺)‘𝑎) ∈ 𝑆))) |
| 38 | 1, 35, 37 | mpbir2and 713 | 1 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 ‘cfv 6561 (class class class)co 7431 1c1 11156 − cmin 11492 ℕcn 12266 ℕ0cn0 12526 Basecbs 17247 ↾s cress 17274 Mndcmnd 18747 SubMndcsubmnd 18795 Grpcgrp 18951 invgcminusg 18952 .gcmg 19085 SubGrpcsubg 19138 odcod 19542 |
| 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-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 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 |
| 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-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-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-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-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-seq 14043 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-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-od 19546 |
| This theorem is referenced by: 0subgALT 19586 finsubmsubg 42520 |
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