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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 2735 | . . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | finodsubmsubg.o | . . . . . . . 8 ⊢ 𝑂 = (od‘𝐺) | |
5 | eqid 2735 | . . . . . . . 8 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
6 | eqid 2735 | . . . . . . . 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 18835 | . . . . . . . . . 10 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 ⊆ (Base‘𝐺)) |
10 | 1, 9 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐺)) |
11 | 10 | sselda 3995 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ (Base‘𝐺)) |
12 | 3, 4, 5, 6, 8, 11 | odm1inv 19586 | . . . . . . 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 2735 | . . . . . . . 8 ⊢ (Base‘(𝐺 ↾s 𝑆)) = (Base‘(𝐺 ↾s 𝑆)) | |
15 | eqid 2735 | . . . . . . . 8 ⊢ (.g‘(𝐺 ↾s 𝑆)) = (.g‘(𝐺 ↾s 𝑆)) | |
16 | eqid 2735 | . . . . . . . . . . 11 ⊢ (𝐺 ↾s 𝑆) = (𝐺 ↾s 𝑆) | |
17 | 16 | submmnd 18839 | . . . . . . . . . 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 12565 | . . . . . . . . 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 2841 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑂‘𝑎) ∈ ℕ) → 𝑎 ∈ (Base‘(𝐺 ↾s 𝑆))) |
27 | 14, 15, 19, 21, 26 | mulgnn0cld 19126 | . . . . . . 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 19149 | . . . . . . . 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 1370 | . . . . . . 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 2854 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ (𝑂‘𝑎) ∈ ℕ) → (((𝑂‘𝑎) − 1)(.g‘𝐺)𝑎) ∈ 𝑆) |
32 | 13, 31 | eqeltrrd 2840 | . . . . 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 3165 | . . 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 19175 | . . 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 1537 ∈ wcel 2106 ∀wral 3059 ⊆ wss 3963 ‘cfv 6563 (class class class)co 7431 1c1 11154 − cmin 11490 ℕcn 12264 ℕ0cn0 12524 Basecbs 17245 ↾s cress 17274 Mndcmnd 18760 SubMndcsubmnd 18808 Grpcgrp 18964 invgcminusg 18965 .gcmg 19098 SubGrpcsubg 19151 odcod 19557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-seq 14040 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-od 19561 |
This theorem is referenced by: 0subgALT 19601 finsubmsubg 42497 |
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