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Mirrors > Home > MPE Home > Th. List > efmndid | Structured version Visualization version GIF version |
Description: The identity function restricted to a set π΄ is the identity element of the monoid of endofunctions on π΄. (Contributed by AV, 25-Jan-2024.) |
Ref | Expression |
---|---|
ielefmnd.g | β’ πΊ = (EndoFMndβπ΄) |
Ref | Expression |
---|---|
efmndid | β’ (π΄ β π β ( I βΎ π΄) = (0gβπΊ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . 2 β’ (BaseβπΊ) = (BaseβπΊ) | |
2 | eqid 2737 | . 2 β’ (0gβπΊ) = (0gβπΊ) | |
3 | eqid 2737 | . 2 β’ (+gβπΊ) = (+gβπΊ) | |
4 | ielefmnd.g | . . 3 β’ πΊ = (EndoFMndβπ΄) | |
5 | 4 | ielefmnd 18698 | . 2 β’ (π΄ β π β ( I βΎ π΄) β (BaseβπΊ)) |
6 | 4, 1, 3 | efmndov 18692 | . . . 4 β’ ((( I βΎ π΄) β (BaseβπΊ) β§ π β (BaseβπΊ)) β (( I βΎ π΄)(+gβπΊ)π) = (( I βΎ π΄) β π)) |
7 | 5, 6 | sylan 581 | . . 3 β’ ((π΄ β π β§ π β (BaseβπΊ)) β (( I βΎ π΄)(+gβπΊ)π) = (( I βΎ π΄) β π)) |
8 | 4, 1 | efmndbasf 18686 | . . . . 5 β’ (π β (BaseβπΊ) β π:π΄βΆπ΄) |
9 | 8 | adantl 483 | . . . 4 β’ ((π΄ β π β§ π β (BaseβπΊ)) β π:π΄βΆπ΄) |
10 | fcoi2 6718 | . . . 4 β’ (π:π΄βΆπ΄ β (( I βΎ π΄) β π) = π) | |
11 | 9, 10 | syl 17 | . . 3 β’ ((π΄ β π β§ π β (BaseβπΊ)) β (( I βΎ π΄) β π) = π) |
12 | 7, 11 | eqtrd 2777 | . 2 β’ ((π΄ β π β§ π β (BaseβπΊ)) β (( I βΎ π΄)(+gβπΊ)π) = π) |
13 | 5 | anim1ci 617 | . . . 4 β’ ((π΄ β π β§ π β (BaseβπΊ)) β (π β (BaseβπΊ) β§ ( I βΎ π΄) β (BaseβπΊ))) |
14 | 4, 1, 3 | efmndov 18692 | . . . 4 β’ ((π β (BaseβπΊ) β§ ( I βΎ π΄) β (BaseβπΊ)) β (π(+gβπΊ)( I βΎ π΄)) = (π β ( I βΎ π΄))) |
15 | 13, 14 | syl 17 | . . 3 β’ ((π΄ β π β§ π β (BaseβπΊ)) β (π(+gβπΊ)( I βΎ π΄)) = (π β ( I βΎ π΄))) |
16 | fcoi1 6717 | . . . 4 β’ (π:π΄βΆπ΄ β (π β ( I βΎ π΄)) = π) | |
17 | 9, 16 | syl 17 | . . 3 β’ ((π΄ β π β§ π β (BaseβπΊ)) β (π β ( I βΎ π΄)) = π) |
18 | 15, 17 | eqtrd 2777 | . 2 β’ ((π΄ β π β§ π β (BaseβπΊ)) β (π(+gβπΊ)( I βΎ π΄)) = π) |
19 | 1, 2, 3, 5, 12, 18 | ismgmid2 18524 | 1 β’ (π΄ β π β ( I βΎ π΄) = (0gβπΊ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 I cid 5531 βΎ cres 5636 β ccom 5638 βΆwf 6493 βcfv 6497 (class class class)co 7358 Basecbs 17084 +gcplusg 17134 0gc0g 17322 EndoFMndcefmnd 18679 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-7 12222 df-8 12223 df-9 12224 df-n0 12415 df-z 12501 df-uz 12765 df-fz 13426 df-struct 17020 df-slot 17055 df-ndx 17067 df-base 17085 df-plusg 17147 df-tset 17153 df-0g 17324 df-efmnd 18680 |
This theorem is referenced by: sursubmefmnd 18707 injsubmefmnd 18708 idressubmefmnd 18709 smndex1n0mnd 18723 smndex2dnrinv 18726 smndex2dlinvh 18728 symgid 19184 symgsubmefmndALT 19186 |
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