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Mirrors > Home > MPE Home > Th. List > abl1 | Structured version Visualization version GIF version |
Description: The (smallest) structure representing a trivial abelian group. (Contributed by AV, 28-Apr-2019.) |
Ref | Expression |
---|---|
abl1.m | ⊢ 𝑀 = {〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉} |
Ref | Expression |
---|---|
abl1 | ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Abel) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abl1.m | . . 3 ⊢ 𝑀 = {〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉} | |
2 | 1 | grp1 17522 | . 2 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Grp) |
3 | eqidd 2623 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼)) | |
4 | oveq1 6657 | . . . . . . 7 ⊢ (𝑎 = 𝐼 → (𝑎{〈〈𝐼, 𝐼〉, 𝐼〉}𝑏) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑏)) | |
5 | oveq2 6658 | . . . . . . 7 ⊢ (𝑎 = 𝐼 → (𝑏{〈〈𝐼, 𝐼〉, 𝐼〉}𝑎) = (𝑏{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼)) | |
6 | 4, 5 | eqeq12d 2637 | . . . . . 6 ⊢ (𝑎 = 𝐼 → ((𝑎{〈〈𝐼, 𝐼〉, 𝐼〉}𝑏) = (𝑏{〈〈𝐼, 𝐼〉, 𝐼〉}𝑎) ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑏) = (𝑏{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼))) |
7 | 6 | ralbidv 2986 | . . . . 5 ⊢ (𝑎 = 𝐼 → (∀𝑏 ∈ {𝐼} (𝑎{〈〈𝐼, 𝐼〉, 𝐼〉}𝑏) = (𝑏{〈〈𝐼, 𝐼〉, 𝐼〉}𝑎) ↔ ∀𝑏 ∈ {𝐼} (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑏) = (𝑏{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼))) |
8 | 7 | ralsng 4218 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (∀𝑎 ∈ {𝐼}∀𝑏 ∈ {𝐼} (𝑎{〈〈𝐼, 𝐼〉, 𝐼〉}𝑏) = (𝑏{〈〈𝐼, 𝐼〉, 𝐼〉}𝑎) ↔ ∀𝑏 ∈ {𝐼} (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑏) = (𝑏{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼))) |
9 | oveq2 6658 | . . . . . 6 ⊢ (𝑏 = 𝐼 → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑏) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼)) | |
10 | oveq1 6657 | . . . . . 6 ⊢ (𝑏 = 𝐼 → (𝑏{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼)) | |
11 | 9, 10 | eqeq12d 2637 | . . . . 5 ⊢ (𝑏 = 𝐼 → ((𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑏) = (𝑏{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼))) |
12 | 11 | ralsng 4218 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (∀𝑏 ∈ {𝐼} (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑏) = (𝑏{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼))) |
13 | 8, 12 | bitrd 268 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (∀𝑎 ∈ {𝐼}∀𝑏 ∈ {𝐼} (𝑎{〈〈𝐼, 𝐼〉, 𝐼〉}𝑏) = (𝑏{〈〈𝐼, 𝐼〉, 𝐼〉}𝑎) ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼))) |
14 | 3, 13 | mpbird 247 | . 2 ⊢ (𝐼 ∈ 𝑉 → ∀𝑎 ∈ {𝐼}∀𝑏 ∈ {𝐼} (𝑎{〈〈𝐼, 𝐼〉, 𝐼〉}𝑏) = (𝑏{〈〈𝐼, 𝐼〉, 𝐼〉}𝑎)) |
15 | snex 4908 | . . . 4 ⊢ {𝐼} ∈ V | |
16 | 1 | grpbase 15991 | . . . 4 ⊢ ({𝐼} ∈ V → {𝐼} = (Base‘𝑀)) |
17 | 15, 16 | ax-mp 5 | . . 3 ⊢ {𝐼} = (Base‘𝑀) |
18 | snex 4908 | . . . 4 ⊢ {〈〈𝐼, 𝐼〉, 𝐼〉} ∈ V | |
19 | 1 | grpplusg 15992 | . . . 4 ⊢ ({〈〈𝐼, 𝐼〉, 𝐼〉} ∈ V → {〈〈𝐼, 𝐼〉, 𝐼〉} = (+g‘𝑀)) |
20 | 18, 19 | ax-mp 5 | . . 3 ⊢ {〈〈𝐼, 𝐼〉, 𝐼〉} = (+g‘𝑀) |
21 | 17, 20 | isabl2 18201 | . 2 ⊢ (𝑀 ∈ Abel ↔ (𝑀 ∈ Grp ∧ ∀𝑎 ∈ {𝐼}∀𝑏 ∈ {𝐼} (𝑎{〈〈𝐼, 𝐼〉, 𝐼〉}𝑏) = (𝑏{〈〈𝐼, 𝐼〉, 𝐼〉}𝑎))) |
22 | 2, 14, 21 | sylanbrc 698 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 {csn 4177 {cpr 4179 〈cop 4183 ‘cfv 5888 (class class class)co 6650 ndxcnx 15854 Basecbs 15857 +gcplusg 15941 Grpcgrp 17422 Abelcabl 18194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-cmn 18195 df-abl 18196 |
This theorem is referenced by: abln0 18270 |
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