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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 17154 | . 2 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Mnd) |
3 | df-ov 6552 | . . . . 5 ⊢ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) | |
4 | opex 4859 | . . . . . 6 ⊢ 〈𝐼, 𝐼〉 ∈ V | |
5 | fvsng 6352 | . . . . . 6 ⊢ ((〈𝐼, 𝐼〉 ∈ V ∧ 𝐼 ∈ 𝑉) → ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) = 𝐼) | |
6 | 4, 5 | mpan 702 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) = 𝐼) |
7 | 3, 6 | syl5eq 2656 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼) |
8 | 1 | mnd1id 17155 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (0g‘𝑀) = 𝐼) |
9 | 7, 8 | eqtr4d 2647 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (0g‘𝑀)) |
10 | oveq2 6557 | . . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑒{〈〈𝐼, 𝐼〉, 𝐼〉}𝑖) = (𝑒{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼)) | |
11 | 10 | eqeq1d 2612 | . . . . . 6 ⊢ (𝑖 = 𝐼 → ((𝑒{〈〈𝐼, 𝐼〉, 𝐼〉}𝑖) = (0g‘𝑀) ↔ (𝑒{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (0g‘𝑀))) |
12 | 11 | rexbidv 3034 | . . . . 5 ⊢ (𝑖 = 𝐼 → (∃𝑒 ∈ {𝐼} (𝑒{〈〈𝐼, 𝐼〉, 𝐼〉}𝑖) = (0g‘𝑀) ↔ ∃𝑒 ∈ {𝐼} (𝑒{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (0g‘𝑀))) |
13 | 12 | ralsng 4165 | . . . 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 6556 | . . . . . 6 ⊢ (𝑒 = 𝐼 → (𝑒{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼)) | |
15 | 14 | eqeq1d 2612 | . . . . 5 ⊢ (𝑒 = 𝐼 → ((𝑒{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (0g‘𝑀) ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (0g‘𝑀))) |
16 | 15 | rexsng 4166 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (∃𝑒 ∈ {𝐼} (𝑒{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (0g‘𝑀) ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (0g‘𝑀))) |
17 | 13, 16 | bitrd 267 | . . 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 246 | . 2 ⊢ (𝐼 ∈ 𝑉 → ∀𝑖 ∈ {𝐼}∃𝑒 ∈ {𝐼} (𝑒{〈〈𝐼, 𝐼〉, 𝐼〉}𝑖) = (0g‘𝑀)) |
19 | snex 4835 | . . . 4 ⊢ {𝐼} ∈ V | |
20 | 1 | grpbase 15816 | . . . 4 ⊢ ({𝐼} ∈ V → {𝐼} = (Base‘𝑀)) |
21 | 19, 20 | ax-mp 5 | . . 3 ⊢ {𝐼} = (Base‘𝑀) |
22 | snex 4835 | . . . 4 ⊢ {〈〈𝐼, 𝐼〉, 𝐼〉} ∈ V | |
23 | 1 | grpplusg 15817 | . . . 4 ⊢ ({〈〈𝐼, 𝐼〉, 𝐼〉} ∈ V → {〈〈𝐼, 𝐼〉, 𝐼〉} = (+g‘𝑀)) |
24 | 22, 23 | ax-mp 5 | . . 3 ⊢ {〈〈𝐼, 𝐼〉, 𝐼〉} = (+g‘𝑀) |
25 | eqid 2610 | . . 3 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
26 | 21, 24, 25 | isgrp 17251 | . 2 ⊢ (𝑀 ∈ Grp ↔ (𝑀 ∈ Mnd ∧ ∀𝑖 ∈ {𝐼}∃𝑒 ∈ {𝐼} (𝑒{〈〈𝐼, 𝐼〉, 𝐼〉}𝑖) = (0g‘𝑀))) |
27 | 2, 18, 26 | sylanbrc 695 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 Vcvv 3173 {csn 4125 {cpr 4127 〈cop 4131 ‘cfv 5804 (class class class)co 6549 ndxcnx 15692 Basecbs 15695 +gcplusg 15768 0gc0g 15923 Mndcmnd 17117 Grpcgrp 17245 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 |
This theorem is referenced by: grp1inv 17346 abl1 18092 ring1 18425 lmod1 42075 |
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