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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 7445 | . . . . 5 β’ (π΄ βm π΄) β V | |
3 | eqid 2731 | . . . . . 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 17312 | . . . . 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 2731 | . . . . . 6 β’ (π΄ βm π΄) = (π΄ βm π΄) | |
8 | eqid 2731 | . . . . . 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 2731 | . . . . . 6 β’ (βtβ(π΄ Γ {π« π΄})) = (βtβ(π΄ Γ {π« π΄})) | |
10 | 6, 7, 8, 9 | efmnd 18788 | . . . . 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 6895 | . . . 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 2774 | . . 3 β’ (π΄ β V β (π΄ βm π΄) = (BaseβπΊ)) |
13 | base0 17154 | . . . 4 β’ β = (Baseββ ) | |
14 | reldmmap 8833 | . . . . 5 β’ Rel dom βm | |
15 | 14 | ovprc1 7451 | . . . 4 β’ (Β¬ π΄ β V β (π΄ βm π΄) = β ) |
16 | fvprc 6883 | . . . . . 6 β’ (Β¬ π΄ β V β (EndoFMndβπ΄) = β ) | |
17 | 6, 16 | eqtrid 2783 | . . . . 5 β’ (Β¬ π΄ β V β πΊ = β ) |
18 | 17 | fveq2d 6895 | . . . 4 β’ (Β¬ π΄ β V β (BaseβπΊ) = (Baseββ )) |
19 | 13, 15, 18 | 3eqtr4a 2797 | . . 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 2762 | 1 β’ π΅ = (π΄ βm π΄) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 = wceq 1540 β wcel 2105 Vcvv 3473 β c0 4322 π« cpw 4602 {csn 4628 {ctp 4632 β¨cop 4634 Γ cxp 5674 β ccom 5680 βcfv 6543 (class class class)co 7412 β cmpo 7414 βm cmap 8824 ndxcnx 17131 Basecbs 17149 +gcplusg 17202 TopSetcts 17208 βtcpt 17389 EndoFMndcefmnd 18786 |
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 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-uz 12828 df-fz 13490 df-struct 17085 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-tset 17221 df-efmnd 18787 |
This theorem is referenced by: efmndbasabf 18790 elefmndbas 18791 efmndhash 18794 efmndbasfi 18795 efmndplusg 18798 efmndbas0 18809 efmnd1bas 18811 smndex1ibas 18818 smndex1gbas 18820 symgplusg 19292 symgpssefmnd 19305 symgvalstruct 19306 symgvalstructOLD 19307 symgsubmefmndALT 19313 efmndtmd 23826 1aryenefmnd 47420 |
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