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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 18086 and not on injsubmefmnd 18178 and sursubmefmnd 18177. (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 18170 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝑀 ∈ Mnd) |
3 | symgsubmefmndALT.g | . . . 4 ⊢ 𝐺 = (SymGrp‘𝐴) | |
4 | symgsubmefmndALT.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
5 | 3, 4, 1 | symgressbas 18628 | . . 3 ⊢ 𝐺 = (𝑀 ↾s 𝐵) |
6 | 3 | symggrp 18646 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ Grp) |
7 | grpmnd 18226 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ Mnd) |
9 | 5, 8 | eqeltrrid 2838 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑀 ↾s 𝐵) ∈ Mnd) |
10 | 3 | idresperm 18632 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ (Base‘𝐺)) |
11 | 1 | efmndid 18169 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) = (0g‘𝑀)) |
12 | 4 | eqcomi 2747 | . . . . 5 ⊢ (Base‘𝐺) = 𝐵 |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (Base‘𝐺) = 𝐵) |
14 | 10, 11, 13 | 3eltr3d 2847 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (0g‘𝑀) ∈ 𝐵) |
15 | 3, 4 | symgbasmap 18623 | . . . . 5 ⊢ (𝑓 ∈ 𝐵 → 𝑓 ∈ (𝐴 ↑m 𝐴)) |
16 | 15 | ssriv 3881 | . . . 4 ⊢ 𝐵 ⊆ (𝐴 ↑m 𝐴) |
17 | eqid 2738 | . . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
18 | 1, 17 | efmndbas 18152 | . . . 4 ⊢ (Base‘𝑀) = (𝐴 ↑m 𝐴) |
19 | 16, 18 | sseqtrri 3914 | . . 3 ⊢ 𝐵 ⊆ (Base‘𝑀) |
20 | 14, 19 | jctil 523 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵)) |
21 | eqid 2738 | . . 3 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
22 | 17, 21 | issubmndb 18086 | . 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 399 = wceq 1542 ∈ wcel 2114 ⊆ wss 3843 I cid 5428 ↾ cres 5527 ‘cfv 6339 (class class class)co 7170 ↑m cmap 8437 Basecbs 16586 ↾s cress 16587 0gc0g 16816 Mndcmnd 18027 SubMndcsubmnd 18071 EndoFMndcefmnd 18149 Grpcgrp 18219 SymGrpcsymg 18613 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-uz 12325 df-fz 12982 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-tset 16687 df-0g 16818 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-submnd 18073 df-efmnd 18150 df-grp 18222 df-symg 18614 |
This theorem is referenced by: (None) |
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