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Mirrors > Home > MPE Home > Th. List > symgsubmefmndALT | Structured version Visualization version GIF version |
Description: The symmetric group on a set 𝐴 is a submonoid of the monoid of endofunctions on 𝐴. Alternate proof based on issubmndb 18681 and not on injsubmefmnd 18773 and sursubmefmnd 18772. (Contributed by AV, 18-Feb-2024.) (Revised by AV, 30-Mar-2024.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
symgsubmefmndALT.m | ⊢ 𝑀 = (EndoFMnd‘𝐴) |
symgsubmefmndALT.g | ⊢ 𝐺 = (SymGrp‘𝐴) |
symgsubmefmndALT.b | ⊢ 𝐵 = (Base‘𝐺) |
Ref | Expression |
---|---|
symgsubmefmndALT | ⊢ (𝐴 ∈ 𝑉 → 𝐵 ∈ (SubMnd‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | symgsubmefmndALT.m | . . 3 ⊢ 𝑀 = (EndoFMnd‘𝐴) | |
2 | 1 | efmndmnd 18765 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝑀 ∈ Mnd) |
3 | symgsubmefmndALT.g | . . . 4 ⊢ 𝐺 = (SymGrp‘𝐴) | |
4 | symgsubmefmndALT.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
5 | 3, 4, 1 | symgressbas 19241 | . . 3 ⊢ 𝐺 = (𝑀 ↾s 𝐵) |
6 | 3 | symggrp 19260 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ Grp) |
7 | grpmnd 18821 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ Mnd) |
9 | 5, 8 | eqeltrrid 2839 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑀 ↾s 𝐵) ∈ Mnd) |
10 | 3 | idresperm 19245 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ (Base‘𝐺)) |
11 | 1 | efmndid 18764 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) = (0g‘𝑀)) |
12 | 4 | eqcomi 2742 | . . . . 5 ⊢ (Base‘𝐺) = 𝐵 |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (Base‘𝐺) = 𝐵) |
14 | 10, 11, 13 | 3eltr3d 2848 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (0g‘𝑀) ∈ 𝐵) |
15 | 3, 4 | symgbasmap 19236 | . . . . 5 ⊢ (𝑓 ∈ 𝐵 → 𝑓 ∈ (𝐴 ↑m 𝐴)) |
16 | 15 | ssriv 3984 | . . . 4 ⊢ 𝐵 ⊆ (𝐴 ↑m 𝐴) |
17 | eqid 2733 | . . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
18 | 1, 17 | efmndbas 18747 | . . . 4 ⊢ (Base‘𝑀) = (𝐴 ↑m 𝐴) |
19 | 16, 18 | sseqtrri 4017 | . . 3 ⊢ 𝐵 ⊆ (Base‘𝑀) |
20 | 14, 19 | jctil 521 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵)) |
21 | eqid 2733 | . . 3 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
22 | 17, 21 | issubmndb 18681 | . 2 ⊢ (𝐵 ∈ (SubMnd‘𝑀) ↔ ((𝑀 ∈ Mnd ∧ (𝑀 ↾s 𝐵) ∈ Mnd) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵))) |
23 | 2, 9, 20, 22 | syl21anbrc 1345 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐵 ∈ (SubMnd‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3946 I cid 5571 ↾ cres 5676 ‘cfv 6539 (class class class)co 7403 ↑m cmap 8815 Basecbs 17139 ↾s cress 17168 0gc0g 17380 Mndcmnd 18620 SubMndcsubmnd 18665 EndoFMndcefmnd 18744 Grpcgrp 18814 SymGrpcsymg 19226 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-1o 8460 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-nn 12208 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-7 12275 df-8 12276 df-9 12277 df-n0 12468 df-z 12554 df-uz 12818 df-fz 13480 df-struct 17075 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17140 df-ress 17169 df-plusg 17205 df-tset 17211 df-0g 17382 df-mgm 18556 df-sgrp 18605 df-mnd 18621 df-submnd 18667 df-efmnd 18745 df-grp 18817 df-symg 19227 |
This theorem is referenced by: (None) |
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