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Mirrors > Home > MPE Home > Th. List > efmndplusg | Structured version Visualization version GIF version |
Description: The group operation of a monoid of endofunctions is the function composition. (Contributed by AV, 27-Jan-2024.) |
Ref | Expression |
---|---|
efmndtset.g | β’ πΊ = (EndoFMndβπ΄) |
efmndplusg.b | β’ π΅ = (BaseβπΊ) |
efmndplusg.p | β’ + = (+gβπΊ) |
Ref | Expression |
---|---|
efmndplusg | β’ + = (π β π΅, π β π΅ β¦ (π β π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | efmndtset.g | . . . . 5 β’ πΊ = (EndoFMndβπ΄) | |
2 | efmndplusg.b | . . . . . 6 β’ π΅ = (BaseβπΊ) | |
3 | 1, 2 | efmndbas 18682 | . . . . 5 β’ π΅ = (π΄ βm π΄) |
4 | eqid 2737 | . . . . 5 β’ (π β π΅, π β π΅ β¦ (π β π)) = (π β π΅, π β π΅ β¦ (π β π)) | |
5 | eqid 2737 | . . . . 5 β’ (βtβ(π΄ Γ {π« π΄})) = (βtβ(π΄ Γ {π« π΄})) | |
6 | 1, 3, 4, 5 | efmnd 18681 | . . . 4 β’ (π΄ β V β πΊ = {β¨(Baseβndx), π΅β©, β¨(+gβndx), (π β π΅, π β π΅ β¦ (π β π))β©, β¨(TopSetβndx), (βtβ(π΄ Γ {π« π΄}))β©}) |
7 | 6 | fveq2d 6847 | . . 3 β’ (π΄ β V β (+gβπΊ) = (+gβ{β¨(Baseβndx), π΅β©, β¨(+gβndx), (π β π΅, π β π΅ β¦ (π β π))β©, β¨(TopSetβndx), (βtβ(π΄ Γ {π« π΄}))β©})) |
8 | efmndplusg.p | . . 3 β’ + = (+gβπΊ) | |
9 | 2 | fvexi 6857 | . . . . 5 β’ π΅ β V |
10 | 9, 9 | mpoex 8013 | . . . 4 β’ (π β π΅, π β π΅ β¦ (π β π)) β V |
11 | eqid 2737 | . . . . 5 β’ {β¨(Baseβndx), π΅β©, β¨(+gβndx), (π β π΅, π β π΅ β¦ (π β π))β©, β¨(TopSetβndx), (βtβ(π΄ Γ {π« π΄}))β©} = {β¨(Baseβndx), π΅β©, β¨(+gβndx), (π β π΅, π β π΅ β¦ (π β π))β©, β¨(TopSetβndx), (βtβ(π΄ Γ {π« π΄}))β©} | |
12 | 11 | topgrpplusg 17245 | . . . 4 β’ ((π β π΅, π β π΅ β¦ (π β π)) β V β (π β π΅, π β π΅ β¦ (π β π)) = (+gβ{β¨(Baseβndx), π΅β©, β¨(+gβndx), (π β π΅, π β π΅ β¦ (π β π))β©, β¨(TopSetβndx), (βtβ(π΄ Γ {π« π΄}))β©})) |
13 | 10, 12 | ax-mp 5 | . . 3 β’ (π β π΅, π β π΅ β¦ (π β π)) = (+gβ{β¨(Baseβndx), π΅β©, β¨(+gβndx), (π β π΅, π β π΅ β¦ (π β π))β©, β¨(TopSetβndx), (βtβ(π΄ Γ {π« π΄}))β©}) |
14 | 7, 8, 13 | 3eqtr4g 2802 | . 2 β’ (π΄ β V β + = (π β π΅, π β π΅ β¦ (π β π))) |
15 | fvprc 6835 | . . . . . 6 β’ (Β¬ π΄ β V β (EndoFMndβπ΄) = β ) | |
16 | 1, 15 | eqtrid 2789 | . . . . 5 β’ (Β¬ π΄ β V β πΊ = β ) |
17 | 16 | fveq2d 6847 | . . . 4 β’ (Β¬ π΄ β V β (+gβπΊ) = (+gββ )) |
18 | plusgid 17161 | . . . . 5 β’ +g = Slot (+gβndx) | |
19 | 18 | str0 17062 | . . . 4 β’ β = (+gββ ) |
20 | 17, 8, 19 | 3eqtr4g 2802 | . . 3 β’ (Β¬ π΄ β V β + = β ) |
21 | 16 | fveq2d 6847 | . . . . . 6 β’ (Β¬ π΄ β V β (BaseβπΊ) = (Baseββ )) |
22 | base0 17089 | . . . . . 6 β’ β = (Baseββ ) | |
23 | 21, 2, 22 | 3eqtr4g 2802 | . . . . 5 β’ (Β¬ π΄ β V β π΅ = β ) |
24 | 23 | olcd 873 | . . . 4 β’ (Β¬ π΄ β V β (π΅ = β β¨ π΅ = β )) |
25 | 0mpo0 7441 | . . . 4 β’ ((π΅ = β β¨ π΅ = β ) β (π β π΅, π β π΅ β¦ (π β π)) = β ) | |
26 | 24, 25 | syl 17 | . . 3 β’ (Β¬ π΄ β V β (π β π΅, π β π΅ β¦ (π β π)) = β ) |
27 | 20, 26 | eqtr4d 2780 | . 2 β’ (Β¬ π΄ β V β + = (π β π΅, π β π΅ β¦ (π β π))) |
28 | 14, 27 | pm2.61i 182 | 1 β’ + = (π β π΅, π β π΅ β¦ (π β π)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β¨ wo 846 = wceq 1542 β wcel 2107 Vcvv 3446 β c0 4283 π« cpw 4561 {csn 4587 {ctp 4591 β¨cop 4593 Γ cxp 5632 β ccom 5638 βcfv 6497 β cmpo 7360 ndxcnx 17066 Basecbs 17084 +gcplusg 17134 TopSetcts 17140 βtcpt 17321 EndoFMndcefmnd 18679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-7 12222 df-8 12223 df-9 12224 df-n0 12415 df-z 12501 df-uz 12765 df-fz 13426 df-struct 17020 df-slot 17055 df-ndx 17067 df-base 17085 df-plusg 17147 df-tset 17153 df-efmnd 18680 |
This theorem is referenced by: efmndov 18692 submefmnd 18706 symgplusg 19165 efmndtmd 23455 |
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