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Mirrors > Home > MPE Home > Th. List > grp1inv | Structured version Visualization version GIF version |
Description: The inverse function of the trivial group. (Contributed by FL, 21-Jun-2010.) (Revised by AV, 26-Aug-2021.) |
Ref | Expression |
---|---|
grp1.m | β’ π = {β¨(Baseβndx), {πΌ}β©, β¨(+gβndx), {β¨β¨πΌ, πΌβ©, πΌβ©}β©} |
Ref | Expression |
---|---|
grp1inv | β’ (πΌ β π β (invgβπ) = ( I βΎ {πΌ})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grp1.m | . . . 4 β’ π = {β¨(Baseβndx), {πΌ}β©, β¨(+gβndx), {β¨β¨πΌ, πΌβ©, πΌβ©}β©} | |
2 | 1 | grp1 19010 | . . 3 β’ (πΌ β π β π β Grp) |
3 | snex 5437 | . . . . 5 β’ {πΌ} β V | |
4 | 1 | grpbase 17274 | . . . . 5 β’ ({πΌ} β V β {πΌ} = (Baseβπ)) |
5 | 3, 4 | ax-mp 5 | . . . 4 β’ {πΌ} = (Baseβπ) |
6 | eqid 2728 | . . . 4 β’ (invgβπ) = (invgβπ) | |
7 | 5, 6 | grpinvf 18950 | . . 3 β’ (π β Grp β (invgβπ):{πΌ}βΆ{πΌ}) |
8 | 2, 7 | syl 17 | . 2 β’ (πΌ β π β (invgβπ):{πΌ}βΆ{πΌ}) |
9 | fsng 7152 | . . . 4 β’ ((πΌ β π β§ πΌ β π) β ((invgβπ):{πΌ}βΆ{πΌ} β (invgβπ) = {β¨πΌ, πΌβ©})) | |
10 | 9 | anidms 565 | . . 3 β’ (πΌ β π β ((invgβπ):{πΌ}βΆ{πΌ} β (invgβπ) = {β¨πΌ, πΌβ©})) |
11 | simpr 483 | . . . . 5 β’ ((πΌ β π β§ (invgβπ) = {β¨πΌ, πΌβ©}) β (invgβπ) = {β¨πΌ, πΌβ©}) | |
12 | restidsing 6061 | . . . . . . 7 β’ ( I βΎ {πΌ}) = ({πΌ} Γ {πΌ}) | |
13 | xpsng 7154 | . . . . . . . 8 β’ ((πΌ β π β§ πΌ β π) β ({πΌ} Γ {πΌ}) = {β¨πΌ, πΌβ©}) | |
14 | 13 | anidms 565 | . . . . . . 7 β’ (πΌ β π β ({πΌ} Γ {πΌ}) = {β¨πΌ, πΌβ©}) |
15 | 12, 14 | eqtr2id 2781 | . . . . . 6 β’ (πΌ β π β {β¨πΌ, πΌβ©} = ( I βΎ {πΌ})) |
16 | 15 | adantr 479 | . . . . 5 β’ ((πΌ β π β§ (invgβπ) = {β¨πΌ, πΌβ©}) β {β¨πΌ, πΌβ©} = ( I βΎ {πΌ})) |
17 | 11, 16 | eqtrd 2768 | . . . 4 β’ ((πΌ β π β§ (invgβπ) = {β¨πΌ, πΌβ©}) β (invgβπ) = ( I βΎ {πΌ})) |
18 | 17 | ex 411 | . . 3 β’ (πΌ β π β ((invgβπ) = {β¨πΌ, πΌβ©} β (invgβπ) = ( I βΎ {πΌ}))) |
19 | 10, 18 | sylbid 239 | . 2 β’ (πΌ β π β ((invgβπ):{πΌ}βΆ{πΌ} β (invgβπ) = ( I βΎ {πΌ}))) |
20 | 8, 19 | mpd 15 | 1 β’ (πΌ β π β (invgβπ) = ( I βΎ {πΌ})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 Vcvv 3473 {csn 4632 {cpr 4634 β¨cop 4638 I cid 5579 Γ cxp 5680 βΎ cres 5684 βΆwf 6549 βcfv 6553 ndxcnx 17169 Basecbs 17187 +gcplusg 17240 Grpcgrp 18897 invgcminusg 18898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 |
This theorem is referenced by: (None) |
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