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Mirrors > Home > MPE Home > Th. List > mgm1 | Structured version Visualization version GIF version |
Description: The structure with one element and the only closed internal operation for a singleton is a magma. (Contributed by AV, 10-Feb-2020.) |
Ref | Expression |
---|---|
mgm1.m | ⊢ 𝑀 = {〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉} |
Ref | Expression |
---|---|
mgm1 | ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Mgm) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 7278 | . . . . 5 ⊢ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) | |
2 | opex 5379 | . . . . . 6 ⊢ 〈𝐼, 𝐼〉 ∈ V | |
3 | fvsng 7052 | . . . . . 6 ⊢ ((〈𝐼, 𝐼〉 ∈ V ∧ 𝐼 ∈ 𝑉) → ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) = 𝐼) | |
4 | 2, 3 | mpan 687 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) = 𝐼) |
5 | 1, 4 | eqtrid 2790 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼) |
6 | snidg 4595 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ {𝐼}) | |
7 | 5, 6 | eqeltrd 2839 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) ∈ {𝐼}) |
8 | oveq1 7282 | . . . . . . 7 ⊢ (𝑥 = 𝐼 → (𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦)) | |
9 | 8 | eleq1d 2823 | . . . . . 6 ⊢ (𝑥 = 𝐼 → ((𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼} ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼})) |
10 | 9 | ralbidv 3112 | . . . . 5 ⊢ (𝑥 = 𝐼 → (∀𝑦 ∈ {𝐼} (𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼} ↔ ∀𝑦 ∈ {𝐼} (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼})) |
11 | 10 | ralsng 4609 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (∀𝑥 ∈ {𝐼}∀𝑦 ∈ {𝐼} (𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼} ↔ ∀𝑦 ∈ {𝐼} (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼})) |
12 | oveq2 7283 | . . . . . 6 ⊢ (𝑦 = 𝐼 → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼)) | |
13 | 12 | eleq1d 2823 | . . . . 5 ⊢ (𝑦 = 𝐼 → ((𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼} ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) ∈ {𝐼})) |
14 | 13 | ralsng 4609 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (∀𝑦 ∈ {𝐼} (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼} ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) ∈ {𝐼})) |
15 | 11, 14 | bitrd 278 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (∀𝑥 ∈ {𝐼}∀𝑦 ∈ {𝐼} (𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼} ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) ∈ {𝐼})) |
16 | 7, 15 | mpbird 256 | . 2 ⊢ (𝐼 ∈ 𝑉 → ∀𝑥 ∈ {𝐼}∀𝑦 ∈ {𝐼} (𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼}) |
17 | snex 5354 | . . . . 5 ⊢ {𝐼} ∈ V | |
18 | mgm1.m | . . . . . 6 ⊢ 𝑀 = {〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉} | |
19 | 18 | grpbase 16996 | . . . . 5 ⊢ ({𝐼} ∈ V → {𝐼} = (Base‘𝑀)) |
20 | 17, 19 | ax-mp 5 | . . . 4 ⊢ {𝐼} = (Base‘𝑀) |
21 | snex 5354 | . . . . 5 ⊢ {〈〈𝐼, 𝐼〉, 𝐼〉} ∈ V | |
22 | 18 | grpplusg 16998 | . . . . 5 ⊢ ({〈〈𝐼, 𝐼〉, 𝐼〉} ∈ V → {〈〈𝐼, 𝐼〉, 𝐼〉} = (+g‘𝑀)) |
23 | 21, 22 | ax-mp 5 | . . . 4 ⊢ {〈〈𝐼, 𝐼〉, 𝐼〉} = (+g‘𝑀) |
24 | 20, 23 | ismgmn0 18328 | . . 3 ⊢ (𝐼 ∈ {𝐼} → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ {𝐼}∀𝑦 ∈ {𝐼} (𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼})) |
25 | 6, 24 | syl 17 | . 2 ⊢ (𝐼 ∈ 𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ {𝐼}∀𝑦 ∈ {𝐼} (𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼})) |
26 | 16, 25 | mpbird 256 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2106 ∀wral 3064 Vcvv 3432 {csn 4561 {cpr 4563 〈cop 4567 ‘cfv 6433 (class class class)co 7275 ndxcnx 16894 Basecbs 16912 +gcplusg 16962 Mgmcmgm 18324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-struct 16848 df-slot 16883 df-ndx 16895 df-base 16913 df-plusg 16975 df-mgm 18326 |
This theorem is referenced by: sgrp1 18384 |
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