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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 18208 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Grp) |
3 | snex 5334 | . . . . 5 ⊢ {𝐼} ∈ V | |
4 | 1 | grpbase 16612 | . . . . 5 ⊢ ({𝐼} ∈ V → {𝐼} = (Base‘𝑀)) |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ {𝐼} = (Base‘𝑀) |
6 | eqid 2823 | . . . 4 ⊢ (invg‘𝑀) = (invg‘𝑀) | |
7 | 5, 6 | grpinvf 18152 | . . 3 ⊢ (𝑀 ∈ Grp → (invg‘𝑀):{𝐼}⟶{𝐼}) |
8 | 2, 7 | syl 17 | . 2 ⊢ (𝐼 ∈ 𝑉 → (invg‘𝑀):{𝐼}⟶{𝐼}) |
9 | fsng 6901 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ∈ 𝑉) → ((invg‘𝑀):{𝐼}⟶{𝐼} ↔ (invg‘𝑀) = {〈𝐼, 𝐼〉})) | |
10 | 9 | anidms 569 | . . 3 ⊢ (𝐼 ∈ 𝑉 → ((invg‘𝑀):{𝐼}⟶{𝐼} ↔ (invg‘𝑀) = {〈𝐼, 𝐼〉})) |
11 | simpr 487 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (invg‘𝑀) = {〈𝐼, 𝐼〉}) → (invg‘𝑀) = {〈𝐼, 𝐼〉}) | |
12 | restidsing 5924 | . . . . . . 7 ⊢ ( I ↾ {𝐼}) = ({𝐼} × {𝐼}) | |
13 | xpsng 6903 | . . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ∈ 𝑉) → ({𝐼} × {𝐼}) = {〈𝐼, 𝐼〉}) | |
14 | 13 | anidms 569 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → ({𝐼} × {𝐼}) = {〈𝐼, 𝐼〉}) |
15 | 12, 14 | syl5req 2871 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → {〈𝐼, 𝐼〉} = ( I ↾ {𝐼})) |
16 | 15 | adantr 483 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (invg‘𝑀) = {〈𝐼, 𝐼〉}) → {〈𝐼, 𝐼〉} = ( I ↾ {𝐼})) |
17 | 11, 16 | eqtrd 2858 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (invg‘𝑀) = {〈𝐼, 𝐼〉}) → (invg‘𝑀) = ( I ↾ {𝐼})) |
18 | 17 | ex 415 | . . 3 ⊢ (𝐼 ∈ 𝑉 → ((invg‘𝑀) = {〈𝐼, 𝐼〉} → (invg‘𝑀) = ( I ↾ {𝐼}))) |
19 | 10, 18 | sylbid 242 | . 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 {csn 4569 {cpr 4571 〈cop 4575 I cid 5461 × cxp 5555 ↾ cres 5559 ⟶wf 6353 ‘cfv 6357 ndxcnx 16482 Basecbs 16485 +gcplusg 16567 Grpcgrp 18105 invgcminusg 18106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-plusg 16580 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 |
This theorem is referenced by: (None) |
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