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Mirrors > Home > MPE Home > Th. List > grp1 | Structured version Visualization version GIF version |
Description: The (smallest) structure representing a trivial group. According to Wikipedia ("Trivial group", 28-Apr-2019, https://en.wikipedia.org/wiki/Trivial_group) "In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element". (Contributed by AV, 28-Apr-2019.) |
Ref | Expression |
---|---|
grp1.m | ⊢ 𝑀 = {〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉} |
Ref | Expression |
---|---|
grp1 | ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grp1.m | . . 3 ⊢ 𝑀 = {〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉} | |
2 | 1 | mnd1 17946 | . 2 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Mnd) |
3 | df-ov 7153 | . . . . 5 ⊢ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) | |
4 | opex 5348 | . . . . . 6 ⊢ 〈𝐼, 𝐼〉 ∈ V | |
5 | fvsng 6936 | . . . . . 6 ⊢ ((〈𝐼, 𝐼〉 ∈ V ∧ 𝐼 ∈ 𝑉) → ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) = 𝐼) | |
6 | 4, 5 | mpan 688 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) = 𝐼) |
7 | 3, 6 | syl5eq 2868 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼) |
8 | 1 | mnd1id 17947 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (0g‘𝑀) = 𝐼) |
9 | 7, 8 | eqtr4d 2859 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (0g‘𝑀)) |
10 | oveq2 7158 | . . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑒{〈〈𝐼, 𝐼〉, 𝐼〉}𝑖) = (𝑒{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼)) | |
11 | 10 | eqeq1d 2823 | . . . . . 6 ⊢ (𝑖 = 𝐼 → ((𝑒{〈〈𝐼, 𝐼〉, 𝐼〉}𝑖) = (0g‘𝑀) ↔ (𝑒{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (0g‘𝑀))) |
12 | 11 | rexbidv 3297 | . . . . 5 ⊢ (𝑖 = 𝐼 → (∃𝑒 ∈ {𝐼} (𝑒{〈〈𝐼, 𝐼〉, 𝐼〉}𝑖) = (0g‘𝑀) ↔ ∃𝑒 ∈ {𝐼} (𝑒{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (0g‘𝑀))) |
13 | 12 | ralsng 4606 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (∀𝑖 ∈ {𝐼}∃𝑒 ∈ {𝐼} (𝑒{〈〈𝐼, 𝐼〉, 𝐼〉}𝑖) = (0g‘𝑀) ↔ ∃𝑒 ∈ {𝐼} (𝑒{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (0g‘𝑀))) |
14 | oveq1 7157 | . . . . . 6 ⊢ (𝑒 = 𝐼 → (𝑒{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼)) | |
15 | 14 | eqeq1d 2823 | . . . . 5 ⊢ (𝑒 = 𝐼 → ((𝑒{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (0g‘𝑀) ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (0g‘𝑀))) |
16 | 15 | rexsng 4607 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (∃𝑒 ∈ {𝐼} (𝑒{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (0g‘𝑀) ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (0g‘𝑀))) |
17 | 13, 16 | bitrd 281 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (∀𝑖 ∈ {𝐼}∃𝑒 ∈ {𝐼} (𝑒{〈〈𝐼, 𝐼〉, 𝐼〉}𝑖) = (0g‘𝑀) ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (0g‘𝑀))) |
18 | 9, 17 | mpbird 259 | . 2 ⊢ (𝐼 ∈ 𝑉 → ∀𝑖 ∈ {𝐼}∃𝑒 ∈ {𝐼} (𝑒{〈〈𝐼, 𝐼〉, 𝐼〉}𝑖) = (0g‘𝑀)) |
19 | snex 5323 | . . . 4 ⊢ {𝐼} ∈ V | |
20 | 1 | grpbase 16604 | . . . 4 ⊢ ({𝐼} ∈ V → {𝐼} = (Base‘𝑀)) |
21 | 19, 20 | ax-mp 5 | . . 3 ⊢ {𝐼} = (Base‘𝑀) |
22 | snex 5323 | . . . 4 ⊢ {〈〈𝐼, 𝐼〉, 𝐼〉} ∈ V | |
23 | 1 | grpplusg 16605 | . . . 4 ⊢ ({〈〈𝐼, 𝐼〉, 𝐼〉} ∈ V → {〈〈𝐼, 𝐼〉, 𝐼〉} = (+g‘𝑀)) |
24 | 22, 23 | ax-mp 5 | . . 3 ⊢ {〈〈𝐼, 𝐼〉, 𝐼〉} = (+g‘𝑀) |
25 | eqid 2821 | . . 3 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
26 | 21, 24, 25 | isgrp 18103 | . 2 ⊢ (𝑀 ∈ Grp ↔ (𝑀 ∈ Mnd ∧ ∀𝑖 ∈ {𝐼}∃𝑒 ∈ {𝐼} (𝑒{〈〈𝐼, 𝐼〉, 𝐼〉}𝑖) = (0g‘𝑀))) |
27 | 2, 18, 26 | sylanbrc 585 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 Vcvv 3494 {csn 4560 {cpr 4562 〈cop 4566 ‘cfv 6349 (class class class)co 7150 ndxcnx 16474 Basecbs 16477 +gcplusg 16559 0gc0g 16707 Mndcmnd 17905 Grpcgrp 18097 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-plusg 16572 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 |
This theorem is referenced by: grp1inv 18201 abl1 18980 ring1 19346 lmod1 44541 |
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