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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 17920 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Grp) |
3 | snex 5142 | . . . . 5 ⊢ {𝐼} ∈ V | |
4 | 1 | grpbase 16394 | . . . . 5 ⊢ ({𝐼} ∈ V → {𝐼} = (Base‘𝑀)) |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ {𝐼} = (Base‘𝑀) |
6 | eqid 2778 | . . . 4 ⊢ (invg‘𝑀) = (invg‘𝑀) | |
7 | 5, 6 | grpinvf 17864 | . . 3 ⊢ (𝑀 ∈ Grp → (invg‘𝑀):{𝐼}⟶{𝐼}) |
8 | 2, 7 | syl 17 | . 2 ⊢ (𝐼 ∈ 𝑉 → (invg‘𝑀):{𝐼}⟶{𝐼}) |
9 | fsng 6671 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ∈ 𝑉) → ((invg‘𝑀):{𝐼}⟶{𝐼} ↔ (invg‘𝑀) = {〈𝐼, 𝐼〉})) | |
10 | 9 | anidms 562 | . . 3 ⊢ (𝐼 ∈ 𝑉 → ((invg‘𝑀):{𝐼}⟶{𝐼} ↔ (invg‘𝑀) = {〈𝐼, 𝐼〉})) |
11 | simpr 479 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (invg‘𝑀) = {〈𝐼, 𝐼〉}) → (invg‘𝑀) = {〈𝐼, 𝐼〉}) | |
12 | restidsing 5716 | . . . . . . 7 ⊢ ( I ↾ {𝐼}) = ({𝐼} × {𝐼}) | |
13 | xpsng 6673 | . . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ∈ 𝑉) → ({𝐼} × {𝐼}) = {〈𝐼, 𝐼〉}) | |
14 | 13 | anidms 562 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → ({𝐼} × {𝐼}) = {〈𝐼, 𝐼〉}) |
15 | 12, 14 | syl5req 2827 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → {〈𝐼, 𝐼〉} = ( I ↾ {𝐼})) |
16 | 15 | adantr 474 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (invg‘𝑀) = {〈𝐼, 𝐼〉}) → {〈𝐼, 𝐼〉} = ( I ↾ {𝐼})) |
17 | 11, 16 | eqtrd 2814 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (invg‘𝑀) = {〈𝐼, 𝐼〉}) → (invg‘𝑀) = ( I ↾ {𝐼})) |
18 | 17 | ex 403 | . . 3 ⊢ (𝐼 ∈ 𝑉 → ((invg‘𝑀) = {〈𝐼, 𝐼〉} → (invg‘𝑀) = ( I ↾ {𝐼}))) |
19 | 10, 18 | sylbid 232 | . 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 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 Vcvv 3398 {csn 4398 {cpr 4400 〈cop 4404 I cid 5262 × cxp 5355 ↾ cres 5359 ⟶wf 6133 ‘cfv 6137 ndxcnx 16263 Basecbs 16266 +gcplusg 16349 Grpcgrp 17820 invgcminusg 17821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-2 11443 df-n0 11648 df-z 11734 df-uz 11998 df-fz 12649 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-plusg 16362 df-0g 16499 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-grp 17823 df-minusg 17824 |
This theorem is referenced by: (None) |
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