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| Mirrors > Home > MPE Home > Th. List > efmndcl | Structured version Visualization version GIF version | ||
| Description: The group operation of the monoid of endofunctions on π΄ is closed. (Contributed by AV, 27-Jan-2024.) |
| Ref | Expression |
|---|---|
| efmndtset.g | β’ πΊ = (EndoFMndβπ΄) |
| efmndplusg.b | β’ π΅ = (BaseβπΊ) |
| efmndplusg.p | β’ + = (+gβπΊ) |
| Ref | Expression |
|---|---|
| efmndcl | β’ ((π β π΅ β§ π β π΅) β (π + π) β π΅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | efmndtset.g | . . 3 β’ πΊ = (EndoFMndβπ΄) | |
| 2 | efmndplusg.b | . . 3 β’ π΅ = (BaseβπΊ) | |
| 3 | efmndplusg.p | . . 3 β’ + = (+gβπΊ) | |
| 4 | 1, 2, 3 | efmndov 18840 | . 2 β’ ((π β π΅ β§ π β π΅) β (π + π) = (π β π)) |
| 5 | 1, 2 | efmndbasf 18834 | . . . 4 β’ (π β π΅ β π:π΄βΆπ΄) |
| 6 | 1, 2 | efmndbasf 18834 | . . . 4 β’ (π β π΅ β π:π΄βΆπ΄) |
| 7 | fco 6752 | . . . 4 β’ ((π:π΄βΆπ΄ β§ π:π΄βΆπ΄) β (π β π):π΄βΆπ΄) | |
| 8 | 5, 6, 7 | syl2an 594 | . . 3 β’ ((π β π΅ β§ π β π΅) β (π β π):π΄βΆπ΄) |
| 9 | coexg 7943 | . . . 4 β’ ((π β π΅ β§ π β π΅) β (π β π) β V) | |
| 10 | 1, 2 | elefmndbas2 18833 | . . . 4 β’ ((π β π) β V β ((π β π) β π΅ β (π β π):π΄βΆπ΄)) |
| 11 | 9, 10 | syl 17 | . . 3 β’ ((π β π΅ β§ π β π΅) β ((π β π) β π΅ β (π β π):π΄βΆπ΄)) |
| 12 | 8, 11 | mpbird 256 | . 2 β’ ((π β π΅ β§ π β π΅) β (π β π) β π΅) |
| 13 | 4, 12 | eqeltrd 2829 | 1 β’ ((π β π΅ β§ π β π΅) β (π + π) β π΅) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 Vcvv 3473 β ccom 5686 βΆwf 6549 βcfv 6553 (class class class)co 7426 Basecbs 17187 +gcplusg 17240 EndoFMndcefmnd 18827 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-tset 17259 df-efmnd 18828 |
| This theorem is referenced by: efmndmgm 18844 efmndsgrp 18845 |
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