Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > efmndov | Structured version Visualization version GIF version |
Description: The value of the group operation of the monoid of endofunctions on 𝐴. (Contributed by AV, 27-Jan-2024.) |
Ref | Expression |
---|---|
efmndtset.g | ⊢ 𝐺 = (EndoFMnd‘𝐴) |
efmndplusg.b | ⊢ 𝐵 = (Base‘𝐺) |
efmndplusg.p | ⊢ + = (+g‘𝐺) |
Ref | Expression |
---|---|
efmndov | ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑋 ∘ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coexg 7771 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∘ 𝑌) ∈ V) | |
2 | coeq1 5765 | . . 3 ⊢ (𝑓 = 𝑋 → (𝑓 ∘ 𝑔) = (𝑋 ∘ 𝑔)) | |
3 | coeq2 5766 | . . 3 ⊢ (𝑔 = 𝑌 → (𝑋 ∘ 𝑔) = (𝑋 ∘ 𝑌)) | |
4 | efmndtset.g | . . . 4 ⊢ 𝐺 = (EndoFMnd‘𝐴) | |
5 | efmndplusg.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
6 | efmndplusg.p | . . . 4 ⊢ + = (+g‘𝐺) | |
7 | 4, 5, 6 | efmndplusg 18530 | . . 3 ⊢ + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) |
8 | 2, 3, 7 | ovmpog 7427 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 ∘ 𝑌) ∈ V) → (𝑋 + 𝑌) = (𝑋 ∘ 𝑌)) |
9 | 1, 8 | mpd3an3 1461 | 1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑋 ∘ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 Vcvv 3431 ∘ ccom 5594 ‘cfv 6432 (class class class)co 7272 Basecbs 16923 +gcplusg 16973 EndoFMndcefmnd 18518 |
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 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-om 7708 df-1st 7825 df-2nd 7826 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-1o 8289 df-er 8490 df-map 8609 df-en 8726 df-dom 8727 df-sdom 8728 df-fin 8729 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-nn 11985 df-2 12047 df-3 12048 df-4 12049 df-5 12050 df-6 12051 df-7 12052 df-8 12053 df-9 12054 df-n0 12245 df-z 12331 df-uz 12594 df-fz 13251 df-struct 16859 df-slot 16894 df-ndx 16906 df-base 16924 df-plusg 16986 df-tset 16992 df-efmnd 18519 |
This theorem is referenced by: efmndcl 18532 efmndsgrp 18536 efmndid 18538 efmndmnd 18539 sursubmefmnd 18546 injsubmefmnd 18547 smndex1mgm 18557 smndex1sgrp 18558 smndex1mnd 18560 smndex1id 18561 |
Copyright terms: Public domain | W3C validator |