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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 17970 and not on injsubmefmnd 18062 and sursubmefmnd 18061. (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 18054 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝑀 ∈ Mnd) |
3 | symgsubmefmndALT.g | . . . 4 ⊢ 𝐺 = (SymGrp‘𝐴) | |
4 | symgsubmefmndALT.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
5 | 3, 4, 1 | symgressbas 18510 | . . 3 ⊢ 𝐺 = (𝑀 ↾s 𝐵) |
6 | 3 | symggrp 18528 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ Grp) |
7 | grpmnd 18110 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ Mnd) |
9 | 5, 8 | eqeltrrid 2918 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑀 ↾s 𝐵) ∈ Mnd) |
10 | 3 | idresperm 18514 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ (Base‘𝐺)) |
11 | 1 | efmndid 18053 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) = (0g‘𝑀)) |
12 | 4 | eqcomi 2830 | . . . . 5 ⊢ (Base‘𝐺) = 𝐵 |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (Base‘𝐺) = 𝐵) |
14 | 10, 11, 13 | 3eltr3d 2927 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (0g‘𝑀) ∈ 𝐵) |
15 | 3, 4 | symgbasmap 18505 | . . . . 5 ⊢ (𝑓 ∈ 𝐵 → 𝑓 ∈ (𝐴 ↑m 𝐴)) |
16 | 15 | ssriv 3971 | . . . 4 ⊢ 𝐵 ⊆ (𝐴 ↑m 𝐴) |
17 | eqid 2821 | . . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
18 | 1, 17 | efmndbas 18036 | . . . 4 ⊢ (Base‘𝑀) = (𝐴 ↑m 𝐴) |
19 | 16, 18 | sseqtrri 4004 | . . 3 ⊢ 𝐵 ⊆ (Base‘𝑀) |
20 | 14, 19 | jctil 522 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵)) |
21 | eqid 2821 | . . 3 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
22 | 17, 21 | issubmndb 17970 | . 2 ⊢ (𝐵 ∈ (SubMnd‘𝑀) ↔ ((𝑀 ∈ Mnd ∧ (𝑀 ↾s 𝐵) ∈ Mnd) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵))) |
23 | 2, 9, 20, 22 | syl21anbrc 1340 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐵 ∈ (SubMnd‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 I cid 5459 ↾ cres 5557 ‘cfv 6355 (class class class)co 7156 ↑m cmap 8406 Basecbs 16483 ↾s cress 16484 0gc0g 16713 Mndcmnd 17911 SubMndcsubmnd 17955 EndoFMndcefmnd 18033 Grpcgrp 18103 SymGrpcsymg 18495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-tset 16584 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-efmnd 18034 df-grp 18106 df-symg 18496 |
This theorem is referenced by: (None) |
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