Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > efmndcl | Structured version Visualization version GIF version |
Description: The group operation of the monoid of endofunctions on 𝐴 is closed. (Contributed by AV, 27-Jan-2024.) |
Ref | Expression |
---|---|
efmndtset.g | ⊢ 𝐺 = (EndoFMnd‘𝐴) |
efmndplusg.b | ⊢ 𝐵 = (Base‘𝐺) |
efmndplusg.p | ⊢ + = (+g‘𝐺) |
Ref | Expression |
---|---|
efmndcl | ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | efmndtset.g | . . 3 ⊢ 𝐺 = (EndoFMnd‘𝐴) | |
2 | efmndplusg.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | efmndplusg.p | . . 3 ⊢ + = (+g‘𝐺) | |
4 | 1, 2, 3 | efmndov 18162 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑋 ∘ 𝑌)) |
5 | 1, 2 | efmndbasf 18156 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → 𝑋:𝐴⟶𝐴) |
6 | 1, 2 | efmndbasf 18156 | . . . 4 ⊢ (𝑌 ∈ 𝐵 → 𝑌:𝐴⟶𝐴) |
7 | fco 6528 | . . . 4 ⊢ ((𝑋:𝐴⟶𝐴 ∧ 𝑌:𝐴⟶𝐴) → (𝑋 ∘ 𝑌):𝐴⟶𝐴) | |
8 | 5, 6, 7 | syl2an 599 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∘ 𝑌):𝐴⟶𝐴) |
9 | coexg 7660 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∘ 𝑌) ∈ V) | |
10 | 1, 2 | elefmndbas2 18155 | . . . 4 ⊢ ((𝑋 ∘ 𝑌) ∈ V → ((𝑋 ∘ 𝑌) ∈ 𝐵 ↔ (𝑋 ∘ 𝑌):𝐴⟶𝐴)) |
11 | 9, 10 | syl 17 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∘ 𝑌) ∈ 𝐵 ↔ (𝑋 ∘ 𝑌):𝐴⟶𝐴)) |
12 | 8, 11 | mpbird 260 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∘ 𝑌) ∈ 𝐵) |
13 | 4, 12 | eqeltrd 2833 | 1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 Vcvv 3398 ∘ ccom 5529 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 Basecbs 16586 +gcplusg 16668 EndoFMndcefmnd 18149 |
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-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-plusg 16681 df-tset 16687 df-efmnd 18150 |
This theorem is referenced by: efmndmgm 18166 efmndsgrp 18167 |
Copyright terms: Public domain | W3C validator |