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Mirrors > Home > MPE Home > Th. List > efmndmnd | Structured version Visualization version GIF version |
Description: The monoid of endofunctions on a set 𝐴 is actually a monoid. (Contributed by AV, 31-Jan-2024.) |
Ref | Expression |
---|---|
ielefmnd.g | ⊢ 𝐺 = (EndoFMnd‘𝐴) |
Ref | Expression |
---|---|
efmndmnd | ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ Mnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ielefmnd.g | . . . 4 ⊢ 𝐺 = (EndoFMnd‘𝐴) | |
2 | 1 | efmndsgrp 18592 | . . 3 ⊢ 𝐺 ∈ Smgrp |
3 | 2 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ Smgrp) |
4 | 1 | ielefmnd 18593 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ (Base‘𝐺)) |
5 | oveq1 7320 | . . . . . . 7 ⊢ (𝑖 = ( I ↾ 𝐴) → (𝑖(+g‘𝐺)𝑓) = (( I ↾ 𝐴)(+g‘𝐺)𝑓)) | |
6 | 5 | eqeq1d 2739 | . . . . . 6 ⊢ (𝑖 = ( I ↾ 𝐴) → ((𝑖(+g‘𝐺)𝑓) = 𝑓 ↔ (( I ↾ 𝐴)(+g‘𝐺)𝑓) = 𝑓)) |
7 | oveq2 7321 | . . . . . . 7 ⊢ (𝑖 = ( I ↾ 𝐴) → (𝑓(+g‘𝐺)𝑖) = (𝑓(+g‘𝐺)( I ↾ 𝐴))) | |
8 | 7 | eqeq1d 2739 | . . . . . 6 ⊢ (𝑖 = ( I ↾ 𝐴) → ((𝑓(+g‘𝐺)𝑖) = 𝑓 ↔ (𝑓(+g‘𝐺)( I ↾ 𝐴)) = 𝑓)) |
9 | 6, 8 | anbi12d 631 | . . . . 5 ⊢ (𝑖 = ( I ↾ 𝐴) → (((𝑖(+g‘𝐺)𝑓) = 𝑓 ∧ (𝑓(+g‘𝐺)𝑖) = 𝑓) ↔ ((( I ↾ 𝐴)(+g‘𝐺)𝑓) = 𝑓 ∧ (𝑓(+g‘𝐺)( I ↾ 𝐴)) = 𝑓))) |
10 | 9 | ralbidv 3171 | . . . 4 ⊢ (𝑖 = ( I ↾ 𝐴) → (∀𝑓 ∈ (Base‘𝐺)((𝑖(+g‘𝐺)𝑓) = 𝑓 ∧ (𝑓(+g‘𝐺)𝑖) = 𝑓) ↔ ∀𝑓 ∈ (Base‘𝐺)((( I ↾ 𝐴)(+g‘𝐺)𝑓) = 𝑓 ∧ (𝑓(+g‘𝐺)( I ↾ 𝐴)) = 𝑓))) |
11 | 10 | adantl 482 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑖 = ( I ↾ 𝐴)) → (∀𝑓 ∈ (Base‘𝐺)((𝑖(+g‘𝐺)𝑓) = 𝑓 ∧ (𝑓(+g‘𝐺)𝑖) = 𝑓) ↔ ∀𝑓 ∈ (Base‘𝐺)((( I ↾ 𝐴)(+g‘𝐺)𝑓) = 𝑓 ∧ (𝑓(+g‘𝐺)( I ↾ 𝐴)) = 𝑓))) |
12 | eqid 2737 | . . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
13 | 1, 12 | efmndbasf 18581 | . . . . . . 7 ⊢ (𝑓 ∈ (Base‘𝐺) → 𝑓:𝐴⟶𝐴) |
14 | 13 | adantl 482 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑓:𝐴⟶𝐴) |
15 | fcoi2 6684 | . . . . . . 7 ⊢ (𝑓:𝐴⟶𝐴 → (( I ↾ 𝐴) ∘ 𝑓) = 𝑓) | |
16 | fcoi1 6683 | . . . . . . 7 ⊢ (𝑓:𝐴⟶𝐴 → (𝑓 ∘ ( I ↾ 𝐴)) = 𝑓) | |
17 | 15, 16 | jca 512 | . . . . . 6 ⊢ (𝑓:𝐴⟶𝐴 → ((( I ↾ 𝐴) ∘ 𝑓) = 𝑓 ∧ (𝑓 ∘ ( I ↾ 𝐴)) = 𝑓)) |
18 | 14, 17 | syl 17 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘𝐺)) → ((( I ↾ 𝐴) ∘ 𝑓) = 𝑓 ∧ (𝑓 ∘ ( I ↾ 𝐴)) = 𝑓)) |
19 | eqid 2737 | . . . . . . . . 9 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
20 | 1, 12, 19 | efmndov 18587 | . . . . . . . 8 ⊢ ((( I ↾ 𝐴) ∈ (Base‘𝐺) ∧ 𝑓 ∈ (Base‘𝐺)) → (( I ↾ 𝐴)(+g‘𝐺)𝑓) = (( I ↾ 𝐴) ∘ 𝑓)) |
21 | 4, 20 | sylan 580 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘𝐺)) → (( I ↾ 𝐴)(+g‘𝐺)𝑓) = (( I ↾ 𝐴) ∘ 𝑓)) |
22 | 21 | eqeq1d 2739 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘𝐺)) → ((( I ↾ 𝐴)(+g‘𝐺)𝑓) = 𝑓 ↔ (( I ↾ 𝐴) ∘ 𝑓) = 𝑓)) |
23 | 4 | anim1ci 616 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑓 ∈ (Base‘𝐺) ∧ ( I ↾ 𝐴) ∈ (Base‘𝐺))) |
24 | 1, 12, 19 | efmndov 18587 | . . . . . . . 8 ⊢ ((𝑓 ∈ (Base‘𝐺) ∧ ( I ↾ 𝐴) ∈ (Base‘𝐺)) → (𝑓(+g‘𝐺)( I ↾ 𝐴)) = (𝑓 ∘ ( I ↾ 𝐴))) |
25 | 23, 24 | syl 17 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑓(+g‘𝐺)( I ↾ 𝐴)) = (𝑓 ∘ ( I ↾ 𝐴))) |
26 | 25 | eqeq1d 2739 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘𝐺)) → ((𝑓(+g‘𝐺)( I ↾ 𝐴)) = 𝑓 ↔ (𝑓 ∘ ( I ↾ 𝐴)) = 𝑓)) |
27 | 22, 26 | anbi12d 631 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘𝐺)) → (((( I ↾ 𝐴)(+g‘𝐺)𝑓) = 𝑓 ∧ (𝑓(+g‘𝐺)( I ↾ 𝐴)) = 𝑓) ↔ ((( I ↾ 𝐴) ∘ 𝑓) = 𝑓 ∧ (𝑓 ∘ ( I ↾ 𝐴)) = 𝑓))) |
28 | 18, 27 | mpbird 256 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ (Base‘𝐺)) → ((( I ↾ 𝐴)(+g‘𝐺)𝑓) = 𝑓 ∧ (𝑓(+g‘𝐺)( I ↾ 𝐴)) = 𝑓)) |
29 | 28 | ralrimiva 3140 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∀𝑓 ∈ (Base‘𝐺)((( I ↾ 𝐴)(+g‘𝐺)𝑓) = 𝑓 ∧ (𝑓(+g‘𝐺)( I ↾ 𝐴)) = 𝑓)) |
30 | 4, 11, 29 | rspcedvd 3572 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑖 ∈ (Base‘𝐺)∀𝑓 ∈ (Base‘𝐺)((𝑖(+g‘𝐺)𝑓) = 𝑓 ∧ (𝑓(+g‘𝐺)𝑖) = 𝑓)) |
31 | 12, 19 | ismnddef 18454 | . 2 ⊢ (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑖 ∈ (Base‘𝐺)∀𝑓 ∈ (Base‘𝐺)((𝑖(+g‘𝐺)𝑓) = 𝑓 ∧ (𝑓(+g‘𝐺)𝑖) = 𝑓))) |
32 | 3, 30, 31 | sylanbrc 583 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ∀wral 3062 ∃wrex 3071 I cid 5504 ↾ cres 5607 ∘ ccom 5609 ⟶wf 6459 ‘cfv 6463 (class class class)co 7313 Basecbs 16979 +gcplusg 17029 Smgrpcsgrp 18441 Mndcmnd 18452 EndoFMndcefmnd 18574 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2708 ax-rep 5222 ax-sep 5236 ax-nul 5243 ax-pow 5301 ax-pr 5365 ax-un 7626 ax-cnex 10997 ax-resscn 10998 ax-1cn 10999 ax-icn 11000 ax-addcl 11001 ax-addrcl 11002 ax-mulcl 11003 ax-mulrcl 11004 ax-mulcom 11005 ax-addass 11006 ax-mulass 11007 ax-distr 11008 ax-i2m1 11009 ax-1ne0 11010 ax-1rid 11011 ax-rnegex 11012 ax-rrecex 11013 ax-cnre 11014 ax-pre-lttri 11015 ax-pre-lttrn 11016 ax-pre-ltadd 11017 ax-pre-mulgt0 11018 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3351 df-rab 3405 df-v 3443 df-sbc 3726 df-csb 3842 df-dif 3899 df-un 3901 df-in 3903 df-ss 3913 df-pss 3915 df-nul 4267 df-if 4470 df-pw 4545 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4849 df-iun 4937 df-br 5086 df-opab 5148 df-mpt 5169 df-tr 5203 df-id 5505 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5560 df-we 5562 df-xp 5611 df-rel 5612 df-cnv 5613 df-co 5614 df-dm 5615 df-rn 5616 df-res 5617 df-ima 5618 df-pred 6222 df-ord 6289 df-on 6290 df-lim 6291 df-suc 6292 df-iota 6415 df-fun 6465 df-fn 6466 df-f 6467 df-f1 6468 df-fo 6469 df-f1o 6470 df-fv 6471 df-riota 7270 df-ov 7316 df-oprab 7317 df-mpo 7318 df-om 7756 df-1st 7874 df-2nd 7875 df-frecs 8142 df-wrecs 8173 df-recs 8247 df-rdg 8286 df-1o 8342 df-er 8544 df-map 8663 df-en 8780 df-dom 8781 df-sdom 8782 df-fin 8783 df-pnf 11081 df-mnf 11082 df-xr 11083 df-ltxr 11084 df-le 11085 df-sub 11277 df-neg 11278 df-nn 12044 df-2 12106 df-3 12107 df-4 12108 df-5 12109 df-6 12110 df-7 12111 df-8 12112 df-9 12113 df-n0 12304 df-z 12390 df-uz 12653 df-fz 13310 df-struct 16915 df-slot 16950 df-ndx 16962 df-base 16980 df-plusg 17042 df-tset 17048 df-mgm 18393 df-sgrp 18442 df-mnd 18453 df-efmnd 18575 |
This theorem is referenced by: efmnd0nmnd 18596 submefmnd 18601 sursubmefmnd 18602 injsubmefmnd 18603 idressubmefmnd 18604 idresefmnd 18605 symgsubmefmndALT 19078 efmndtmd 23323 |
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