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Mirrors > Home > ILE Home > Th. List > grp1inv | 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 | . . . . 5 ⊢ 𝑀 = {〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉} | |
2 | 1 | grp1 12852 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Grp) |
3 | eqid 2177 | . . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
4 | eqid 2177 | . . . . 5 ⊢ (invg‘𝑀) = (invg‘𝑀) | |
5 | 3, 4 | grpinvf 12797 | . . . 4 ⊢ (𝑀 ∈ Grp → (invg‘𝑀):(Base‘𝑀)⟶(Base‘𝑀)) |
6 | 2, 5 | syl 14 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (invg‘𝑀):(Base‘𝑀)⟶(Base‘𝑀)) |
7 | snexg 4181 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → {𝐼} ∈ V) | |
8 | opexg 4224 | . . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ∈ 𝑉) → 〈𝐼, 𝐼〉 ∈ V) | |
9 | 8 | anidms 397 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → 〈𝐼, 𝐼〉 ∈ V) |
10 | opexg 4224 | . . . . . . 7 ⊢ ((〈𝐼, 𝐼〉 ∈ V ∧ 𝐼 ∈ 𝑉) → 〈〈𝐼, 𝐼〉, 𝐼〉 ∈ V) | |
11 | 9, 10 | mpancom 422 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → 〈〈𝐼, 𝐼〉, 𝐼〉 ∈ V) |
12 | snexg 4181 | . . . . . 6 ⊢ (〈〈𝐼, 𝐼〉, 𝐼〉 ∈ V → {〈〈𝐼, 𝐼〉, 𝐼〉} ∈ V) | |
13 | 11, 12 | syl 14 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → {〈〈𝐼, 𝐼〉, 𝐼〉} ∈ V) |
14 | 1 | grpbaseg 12551 | . . . . 5 ⊢ (({𝐼} ∈ V ∧ {〈〈𝐼, 𝐼〉, 𝐼〉} ∈ V) → {𝐼} = (Base‘𝑀)) |
15 | 7, 13, 14 | syl2anc 411 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → {𝐼} = (Base‘𝑀)) |
16 | 15, 15 | feq23d 5356 | . . 3 ⊢ (𝐼 ∈ 𝑉 → ((invg‘𝑀):{𝐼}⟶{𝐼} ↔ (invg‘𝑀):(Base‘𝑀)⟶(Base‘𝑀))) |
17 | 6, 16 | mpbird 167 | . 2 ⊢ (𝐼 ∈ 𝑉 → (invg‘𝑀):{𝐼}⟶{𝐼}) |
18 | fsng 5684 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ∈ 𝑉) → ((invg‘𝑀):{𝐼}⟶{𝐼} ↔ (invg‘𝑀) = {〈𝐼, 𝐼〉})) | |
19 | 18 | anidms 397 | . . 3 ⊢ (𝐼 ∈ 𝑉 → ((invg‘𝑀):{𝐼}⟶{𝐼} ↔ (invg‘𝑀) = {〈𝐼, 𝐼〉})) |
20 | simpr 110 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (invg‘𝑀) = {〈𝐼, 𝐼〉}) → (invg‘𝑀) = {〈𝐼, 𝐼〉}) | |
21 | restidsing 4958 | . . . . . . 7 ⊢ ( I ↾ {𝐼}) = ({𝐼} × {𝐼}) | |
22 | xpsng 5686 | . . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ∈ 𝑉) → ({𝐼} × {𝐼}) = {〈𝐼, 𝐼〉}) | |
23 | 22 | anidms 397 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → ({𝐼} × {𝐼}) = {〈𝐼, 𝐼〉}) |
24 | 21, 23 | eqtr2id 2223 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → {〈𝐼, 𝐼〉} = ( I ↾ {𝐼})) |
25 | 24 | adantr 276 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (invg‘𝑀) = {〈𝐼, 𝐼〉}) → {〈𝐼, 𝐼〉} = ( I ↾ {𝐼})) |
26 | 20, 25 | eqtrd 2210 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (invg‘𝑀) = {〈𝐼, 𝐼〉}) → (invg‘𝑀) = ( I ↾ {𝐼})) |
27 | 26 | ex 115 | . . 3 ⊢ (𝐼 ∈ 𝑉 → ((invg‘𝑀) = {〈𝐼, 𝐼〉} → (invg‘𝑀) = ( I ↾ {𝐼}))) |
28 | 19, 27 | sylbid 150 | . 2 ⊢ (𝐼 ∈ 𝑉 → ((invg‘𝑀):{𝐼}⟶{𝐼} → (invg‘𝑀) = ( I ↾ {𝐼}))) |
29 | 17, 28 | mpd 13 | 1 ⊢ (𝐼 ∈ 𝑉 → (invg‘𝑀) = ( I ↾ {𝐼})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 Vcvv 2737 {csn 3591 {cpr 3592 〈cop 3594 I cid 4284 × cxp 4620 ↾ cres 4624 ⟶wf 5207 ‘cfv 5211 ndxcnx 12429 Basecbs 12432 +gcplusg 12505 Grpcgrp 12754 invgcminusg 12755 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-addcom 7889 ax-addass 7891 ax-i2m1 7894 ax-0lt1 7895 ax-0id 7897 ax-rnegex 7898 ax-pre-ltirr 7901 ax-pre-ltadd 7905 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-riota 5824 df-ov 5871 df-pnf 7971 df-mnf 7972 df-ltxr 7974 df-inn 8896 df-2 8954 df-ndx 12435 df-slot 12436 df-base 12438 df-plusg 12518 df-0g 12642 df-mgm 12654 df-sgrp 12687 df-mnd 12697 df-grp 12757 df-minusg 12758 |
This theorem is referenced by: (None) |
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