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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 2726 | . 2 β’ (BaseβπΊ) = (BaseβπΊ) | |
2 | eqid 2726 | . 2 β’ (0gβπΊ) = (0gβπΊ) | |
3 | eqid 2726 | . 2 β’ (+gβπΊ) = (+gβπΊ) | |
4 | ielefmnd.g | . . 3 β’ πΊ = (EndoFMndβπ΄) | |
5 | 4 | ielefmnd 18812 | . 2 β’ (π΄ β π β ( I βΎ π΄) β (BaseβπΊ)) |
6 | 4, 1, 3 | efmndov 18806 | . . . 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 579 | . . 3 β’ ((π΄ β π β§ π β (BaseβπΊ)) β (( I βΎ π΄)(+gβπΊ)π) = (( I βΎ π΄) β π)) |
8 | 4, 1 | efmndbasf 18800 | . . . . 5 β’ (π β (BaseβπΊ) β π:π΄βΆπ΄) |
9 | 8 | adantl 481 | . . . 4 β’ ((π΄ β π β§ π β (BaseβπΊ)) β π:π΄βΆπ΄) |
10 | fcoi2 6760 | . . . 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 2766 | . 2 β’ ((π΄ β π β§ π β (BaseβπΊ)) β (( I βΎ π΄)(+gβπΊ)π) = π) |
13 | 5 | anim1ci 615 | . . . 4 β’ ((π΄ β π β§ π β (BaseβπΊ)) β (π β (BaseβπΊ) β§ ( I βΎ π΄) β (BaseβπΊ))) |
14 | 4, 1, 3 | efmndov 18806 | . . . 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 6759 | . . . 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 2766 | . 2 β’ ((π΄ β π β§ π β (BaseβπΊ)) β (π(+gβπΊ)( I βΎ π΄)) = π) |
19 | 1, 2, 3, 5, 12, 18 | ismgmid2 18601 | 1 β’ (π΄ β π β ( I βΎ π΄) = (0gβπΊ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 I cid 5566 βΎ cres 5671 β ccom 5673 βΆwf 6533 βcfv 6537 (class class class)co 7405 Basecbs 17153 +gcplusg 17206 0gc0g 17394 EndoFMndcefmnd 18793 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-struct 17089 df-slot 17124 df-ndx 17136 df-base 17154 df-plusg 17219 df-tset 17225 df-0g 17396 df-efmnd 18794 |
This theorem is referenced by: sursubmefmnd 18821 injsubmefmnd 18822 idressubmefmnd 18823 smndex1n0mnd 18837 smndex2dnrinv 18840 smndex2dlinvh 18842 symgid 19321 symgsubmefmndALT 19323 |
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