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Mirrors > Home > MPE Home > Th. List > grp1 | Structured version Visualization version GIF version |
Description: The (smallest) structure representing a trivial group. According to Wikipedia ("Trivial group", 28-Apr-2019, https://en.wikipedia.org/wiki/Trivial_group) "In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element". (Contributed by AV, 28-Apr-2019.) |
Ref | Expression |
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grp1.m | β’ π = {β¨(Baseβndx), {πΌ}β©, β¨(+gβndx), {β¨β¨πΌ, πΌβ©, πΌβ©}β©} |
Ref | Expression |
---|---|
grp1 | β’ (πΌ β π β π β Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grp1.m | . . 3 β’ π = {β¨(Baseβndx), {πΌ}β©, β¨(+gβndx), {β¨β¨πΌ, πΌβ©, πΌβ©}β©} | |
2 | 1 | mnd1 18602 | . 2 β’ (πΌ β π β π β Mnd) |
3 | df-ov 7361 | . . . . 5 β’ (πΌ{β¨β¨πΌ, πΌβ©, πΌβ©}πΌ) = ({β¨β¨πΌ, πΌβ©, πΌβ©}ββ¨πΌ, πΌβ©) | |
4 | opex 5422 | . . . . . 6 β’ β¨πΌ, πΌβ© β V | |
5 | fvsng 7127 | . . . . . 6 β’ ((β¨πΌ, πΌβ© β V β§ πΌ β π) β ({β¨β¨πΌ, πΌβ©, πΌβ©}ββ¨πΌ, πΌβ©) = πΌ) | |
6 | 4, 5 | mpan 689 | . . . . 5 β’ (πΌ β π β ({β¨β¨πΌ, πΌβ©, πΌβ©}ββ¨πΌ, πΌβ©) = πΌ) |
7 | 3, 6 | eqtrid 2785 | . . . 4 β’ (πΌ β π β (πΌ{β¨β¨πΌ, πΌβ©, πΌβ©}πΌ) = πΌ) |
8 | 1 | mnd1id 18603 | . . . 4 β’ (πΌ β π β (0gβπ) = πΌ) |
9 | 7, 8 | eqtr4d 2776 | . . 3 β’ (πΌ β π β (πΌ{β¨β¨πΌ, πΌβ©, πΌβ©}πΌ) = (0gβπ)) |
10 | oveq2 7366 | . . . . . . 7 β’ (π = πΌ β (π{β¨β¨πΌ, πΌβ©, πΌβ©}π) = (π{β¨β¨πΌ, πΌβ©, πΌβ©}πΌ)) | |
11 | 10 | eqeq1d 2735 | . . . . . 6 β’ (π = πΌ β ((π{β¨β¨πΌ, πΌβ©, πΌβ©}π) = (0gβπ) β (π{β¨β¨πΌ, πΌβ©, πΌβ©}πΌ) = (0gβπ))) |
12 | 11 | rexbidv 3172 | . . . . 5 β’ (π = πΌ β (βπ β {πΌ} (π{β¨β¨πΌ, πΌβ©, πΌβ©}π) = (0gβπ) β βπ β {πΌ} (π{β¨β¨πΌ, πΌβ©, πΌβ©}πΌ) = (0gβπ))) |
13 | 12 | ralsng 4635 | . . . 4 β’ (πΌ β π β (βπ β {πΌ}βπ β {πΌ} (π{β¨β¨πΌ, πΌβ©, πΌβ©}π) = (0gβπ) β βπ β {πΌ} (π{β¨β¨πΌ, πΌβ©, πΌβ©}πΌ) = (0gβπ))) |
14 | oveq1 7365 | . . . . . 6 β’ (π = πΌ β (π{β¨β¨πΌ, πΌβ©, πΌβ©}πΌ) = (πΌ{β¨β¨πΌ, πΌβ©, πΌβ©}πΌ)) | |
15 | 14 | eqeq1d 2735 | . . . . 5 β’ (π = πΌ β ((π{β¨β¨πΌ, πΌβ©, πΌβ©}πΌ) = (0gβπ) β (πΌ{β¨β¨πΌ, πΌβ©, πΌβ©}πΌ) = (0gβπ))) |
16 | 15 | rexsng 4636 | . . . 4 β’ (πΌ β π β (βπ β {πΌ} (π{β¨β¨πΌ, πΌβ©, πΌβ©}πΌ) = (0gβπ) β (πΌ{β¨β¨πΌ, πΌβ©, πΌβ©}πΌ) = (0gβπ))) |
17 | 13, 16 | bitrd 279 | . . 3 β’ (πΌ β π β (βπ β {πΌ}βπ β {πΌ} (π{β¨β¨πΌ, πΌβ©, πΌβ©}π) = (0gβπ) β (πΌ{β¨β¨πΌ, πΌβ©, πΌβ©}πΌ) = (0gβπ))) |
18 | 9, 17 | mpbird 257 | . 2 β’ (πΌ β π β βπ β {πΌ}βπ β {πΌ} (π{β¨β¨πΌ, πΌβ©, πΌβ©}π) = (0gβπ)) |
19 | snex 5389 | . . . 4 β’ {πΌ} β V | |
20 | 1 | grpbase 17172 | . . . 4 β’ ({πΌ} β V β {πΌ} = (Baseβπ)) |
21 | 19, 20 | ax-mp 5 | . . 3 β’ {πΌ} = (Baseβπ) |
22 | snex 5389 | . . . 4 β’ {β¨β¨πΌ, πΌβ©, πΌβ©} β V | |
23 | 1 | grpplusg 17174 | . . . 4 β’ ({β¨β¨πΌ, πΌβ©, πΌβ©} β V β {β¨β¨πΌ, πΌβ©, πΌβ©} = (+gβπ)) |
24 | 22, 23 | ax-mp 5 | . . 3 β’ {β¨β¨πΌ, πΌβ©, πΌβ©} = (+gβπ) |
25 | eqid 2733 | . . 3 β’ (0gβπ) = (0gβπ) | |
26 | 21, 24, 25 | isgrp 18759 | . 2 β’ (π β Grp β (π β Mnd β§ βπ β {πΌ}βπ β {πΌ} (π{β¨β¨πΌ, πΌβ©, πΌβ©}π) = (0gβπ))) |
27 | 2, 18, 26 | sylanbrc 584 | 1 β’ (πΌ β π β π β Grp) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βwral 3061 βwrex 3070 Vcvv 3444 {csn 4587 {cpr 4589 β¨cop 4593 βcfv 6497 (class class class)co 7358 ndxcnx 17070 Basecbs 17088 +gcplusg 17138 0gc0g 17326 Mndcmnd 18561 Grpcgrp 18753 |
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 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 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-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 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-struct 17024 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 |
This theorem is referenced by: grp1inv 18860 abl1 19649 ring1 20031 lmod1 46659 |
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