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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 18198 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Grp) |
3 | snex 5297 | . . . . 5 ⊢ {𝐼} ∈ V | |
4 | 1 | grpbase 16602 | . . . . 5 ⊢ ({𝐼} ∈ V → {𝐼} = (Base‘𝑀)) |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ {𝐼} = (Base‘𝑀) |
6 | eqid 2798 | . . . 4 ⊢ (invg‘𝑀) = (invg‘𝑀) | |
7 | 5, 6 | grpinvf 18142 | . . 3 ⊢ (𝑀 ∈ Grp → (invg‘𝑀):{𝐼}⟶{𝐼}) |
8 | 2, 7 | syl 17 | . 2 ⊢ (𝐼 ∈ 𝑉 → (invg‘𝑀):{𝐼}⟶{𝐼}) |
9 | fsng 6876 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ∈ 𝑉) → ((invg‘𝑀):{𝐼}⟶{𝐼} ↔ (invg‘𝑀) = {〈𝐼, 𝐼〉})) | |
10 | 9 | anidms 570 | . . 3 ⊢ (𝐼 ∈ 𝑉 → ((invg‘𝑀):{𝐼}⟶{𝐼} ↔ (invg‘𝑀) = {〈𝐼, 𝐼〉})) |
11 | simpr 488 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (invg‘𝑀) = {〈𝐼, 𝐼〉}) → (invg‘𝑀) = {〈𝐼, 𝐼〉}) | |
12 | restidsing 5889 | . . . . . . 7 ⊢ ( I ↾ {𝐼}) = ({𝐼} × {𝐼}) | |
13 | xpsng 6878 | . . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ∈ 𝑉) → ({𝐼} × {𝐼}) = {〈𝐼, 𝐼〉}) | |
14 | 13 | anidms 570 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → ({𝐼} × {𝐼}) = {〈𝐼, 𝐼〉}) |
15 | 12, 14 | syl5req 2846 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → {〈𝐼, 𝐼〉} = ( I ↾ {𝐼})) |
16 | 15 | adantr 484 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (invg‘𝑀) = {〈𝐼, 𝐼〉}) → {〈𝐼, 𝐼〉} = ( I ↾ {𝐼})) |
17 | 11, 16 | eqtrd 2833 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (invg‘𝑀) = {〈𝐼, 𝐼〉}) → (invg‘𝑀) = ( I ↾ {𝐼})) |
18 | 17 | ex 416 | . . 3 ⊢ (𝐼 ∈ 𝑉 → ((invg‘𝑀) = {〈𝐼, 𝐼〉} → (invg‘𝑀) = ( I ↾ {𝐼}))) |
19 | 10, 18 | sylbid 243 | . 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 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 {csn 4525 {cpr 4527 〈cop 4531 I cid 5424 × cxp 5517 ↾ cres 5521 ⟶wf 6320 ‘cfv 6324 ndxcnx 16472 Basecbs 16475 +gcplusg 16557 Grpcgrp 18095 invgcminusg 18096 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 |
This theorem is referenced by: (None) |
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