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Mirrors > Home > MPE Home > Th. List > efmndbas | Structured version Visualization version GIF version |
Description: The base set of the monoid of endofunctions on class π΄. (Contributed by AV, 25-Jan-2024.) |
Ref | Expression |
---|---|
efmndbas.g | β’ πΊ = (EndoFMndβπ΄) |
efmndbas.b | β’ π΅ = (BaseβπΊ) |
Ref | Expression |
---|---|
efmndbas | β’ π΅ = (π΄ βm π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | efmndbas.b | . 2 β’ π΅ = (BaseβπΊ) | |
2 | ovex 7340 | . . . . 5 β’ (π΄ βm π΄) β V | |
3 | eqid 2736 | . . . . . 6 β’ {β¨(Baseβndx), (π΄ βm π΄)β©, β¨(+gβndx), (π β (π΄ βm π΄), π β (π΄ βm π΄) β¦ (π β π))β©, β¨(TopSetβndx), (βtβ(π΄ Γ {π« π΄}))β©} = {β¨(Baseβndx), (π΄ βm π΄)β©, β¨(+gβndx), (π β (π΄ βm π΄), π β (π΄ βm π΄) β¦ (π β π))β©, β¨(TopSetβndx), (βtβ(π΄ Γ {π« π΄}))β©} | |
4 | 3 | topgrpbas 17117 | . . . . 5 β’ ((π΄ βm π΄) β V β (π΄ βm π΄) = (Baseβ{β¨(Baseβndx), (π΄ βm π΄)β©, β¨(+gβndx), (π β (π΄ βm π΄), π β (π΄ βm π΄) β¦ (π β π))β©, β¨(TopSetβndx), (βtβ(π΄ Γ {π« π΄}))β©})) |
5 | 2, 4 | mp1i 13 | . . . 4 β’ (π΄ β V β (π΄ βm π΄) = (Baseβ{β¨(Baseβndx), (π΄ βm π΄)β©, β¨(+gβndx), (π β (π΄ βm π΄), π β (π΄ βm π΄) β¦ (π β π))β©, β¨(TopSetβndx), (βtβ(π΄ Γ {π« π΄}))β©})) |
6 | efmndbas.g | . . . . . 6 β’ πΊ = (EndoFMndβπ΄) | |
7 | eqid 2736 | . . . . . 6 β’ (π΄ βm π΄) = (π΄ βm π΄) | |
8 | eqid 2736 | . . . . . 6 β’ (π β (π΄ βm π΄), π β (π΄ βm π΄) β¦ (π β π)) = (π β (π΄ βm π΄), π β (π΄ βm π΄) β¦ (π β π)) | |
9 | eqid 2736 | . . . . . 6 β’ (βtβ(π΄ Γ {π« π΄})) = (βtβ(π΄ Γ {π« π΄})) | |
10 | 6, 7, 8, 9 | efmnd 18554 | . . . . 5 β’ (π΄ β V β πΊ = {β¨(Baseβndx), (π΄ βm π΄)β©, β¨(+gβndx), (π β (π΄ βm π΄), π β (π΄ βm π΄) β¦ (π β π))β©, β¨(TopSetβndx), (βtβ(π΄ Γ {π« π΄}))β©}) |
11 | 10 | fveq2d 6808 | . . . 4 β’ (π΄ β V β (BaseβπΊ) = (Baseβ{β¨(Baseβndx), (π΄ βm π΄)β©, β¨(+gβndx), (π β (π΄ βm π΄), π β (π΄ βm π΄) β¦ (π β π))β©, β¨(TopSetβndx), (βtβ(π΄ Γ {π« π΄}))β©})) |
12 | 5, 11 | eqtr4d 2779 | . . 3 β’ (π΄ β V β (π΄ βm π΄) = (BaseβπΊ)) |
13 | base0 16962 | . . . 4 β’ β = (Baseββ ) | |
14 | reldmmap 8655 | . . . . 5 β’ Rel dom βm | |
15 | 14 | ovprc1 7346 | . . . 4 β’ (Β¬ π΄ β V β (π΄ βm π΄) = β ) |
16 | fvprc 6796 | . . . . . 6 β’ (Β¬ π΄ β V β (EndoFMndβπ΄) = β ) | |
17 | 6, 16 | eqtrid 2788 | . . . . 5 β’ (Β¬ π΄ β V β πΊ = β ) |
18 | 17 | fveq2d 6808 | . . . 4 β’ (Β¬ π΄ β V β (BaseβπΊ) = (Baseββ )) |
19 | 13, 15, 18 | 3eqtr4a 2802 | . . 3 β’ (Β¬ π΄ β V β (π΄ βm π΄) = (BaseβπΊ)) |
20 | 12, 19 | pm2.61i 182 | . 2 β’ (π΄ βm π΄) = (BaseβπΊ) |
21 | 1, 20 | eqtr4i 2767 | 1 β’ π΅ = (π΄ βm π΄) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 = wceq 1539 β wcel 2104 Vcvv 3437 β c0 4262 π« cpw 4539 {csn 4565 {ctp 4569 β¨cop 4571 Γ cxp 5598 β ccom 5604 βcfv 6458 (class class class)co 7307 β cmpo 7309 βm cmap 8646 ndxcnx 16939 Basecbs 16957 +gcplusg 17007 TopSetcts 17013 βtcpt 17194 EndoFMndcefmnd 18552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 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 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-1st 7863 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-er 8529 df-map 8648 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-nn 12020 df-2 12082 df-3 12083 df-4 12084 df-5 12085 df-6 12086 df-7 12087 df-8 12088 df-9 12089 df-n0 12280 df-z 12366 df-uz 12629 df-fz 13286 df-struct 16893 df-slot 16928 df-ndx 16940 df-base 16958 df-plusg 17020 df-tset 17026 df-efmnd 18553 |
This theorem is referenced by: efmndbasabf 18556 elefmndbas 18557 efmndhash 18560 efmndbasfi 18561 efmndplusg 18564 efmndbas0 18575 efmnd1bas 18577 smndex1ibas 18584 smndex1gbas 18586 symgplusg 19035 symgpssefmnd 19048 symgvalstruct 19049 symgvalstructOLD 19050 symgsubmefmndALT 19056 efmndtmd 23297 1aryenefmnd 46050 |
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