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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 | ⊢ 𝑀 = {〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉} |
Ref | Expression |
---|---|
grp1inv | ⊢ (𝐼 ∈ 𝑉 → (invg‘𝑀) = ( I ↾ {𝐼})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grp1.m | . . . 4 ⊢ 𝑀 = {〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉} | |
2 | 1 | grp1 17522 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Grp) |
3 | snex 4908 | . . . . 5 ⊢ {𝐼} ∈ V | |
4 | 1 | grpbase 15991 | . . . . 5 ⊢ ({𝐼} ∈ V → {𝐼} = (Base‘𝑀)) |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ {𝐼} = (Base‘𝑀) |
6 | eqid 2622 | . . . 4 ⊢ (invg‘𝑀) = (invg‘𝑀) | |
7 | 5, 6 | grpinvf 17466 | . . 3 ⊢ (𝑀 ∈ Grp → (invg‘𝑀):{𝐼}⟶{𝐼}) |
8 | 2, 7 | syl 17 | . 2 ⊢ (𝐼 ∈ 𝑉 → (invg‘𝑀):{𝐼}⟶{𝐼}) |
9 | fsng 6404 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ∈ 𝑉) → ((invg‘𝑀):{𝐼}⟶{𝐼} ↔ (invg‘𝑀) = {〈𝐼, 𝐼〉})) | |
10 | 9 | anidms 677 | . . 3 ⊢ (𝐼 ∈ 𝑉 → ((invg‘𝑀):{𝐼}⟶{𝐼} ↔ (invg‘𝑀) = {〈𝐼, 𝐼〉})) |
11 | simpr 477 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (invg‘𝑀) = {〈𝐼, 𝐼〉}) → (invg‘𝑀) = {〈𝐼, 𝐼〉}) | |
12 | restidsing 5458 | . . . . . . 7 ⊢ ( I ↾ {𝐼}) = ({𝐼} × {𝐼}) | |
13 | xpsng 6406 | . . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ∈ 𝑉) → ({𝐼} × {𝐼}) = {〈𝐼, 𝐼〉}) | |
14 | 13 | anidms 677 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → ({𝐼} × {𝐼}) = {〈𝐼, 𝐼〉}) |
15 | 12, 14 | syl5req 2669 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → {〈𝐼, 𝐼〉} = ( I ↾ {𝐼})) |
16 | 15 | adantr 481 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (invg‘𝑀) = {〈𝐼, 𝐼〉}) → {〈𝐼, 𝐼〉} = ( I ↾ {𝐼})) |
17 | 11, 16 | eqtrd 2656 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (invg‘𝑀) = {〈𝐼, 𝐼〉}) → (invg‘𝑀) = ( I ↾ {𝐼})) |
18 | 17 | ex 450 | . . 3 ⊢ (𝐼 ∈ 𝑉 → ((invg‘𝑀) = {〈𝐼, 𝐼〉} → (invg‘𝑀) = ( I ↾ {𝐼}))) |
19 | 10, 18 | sylbid 230 | . 2 ⊢ (𝐼 ∈ 𝑉 → ((invg‘𝑀):{𝐼}⟶{𝐼} → (invg‘𝑀) = ( I ↾ {𝐼}))) |
20 | 8, 19 | mpd 15 | 1 ⊢ (𝐼 ∈ 𝑉 → (invg‘𝑀) = ( I ↾ {𝐼})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 {csn 4177 {cpr 4179 〈cop 4183 I cid 5023 × cxp 5112 ↾ cres 5116 ⟶wf 5884 ‘cfv 5888 ndxcnx 15854 Basecbs 15857 +gcplusg 15941 Grpcgrp 17422 invgcminusg 17423 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 |
This theorem is referenced by: (None) |
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