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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 7162 | . . . . 5 ⊢ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) | |
2 | opex 5359 | . . . . . 6 ⊢ 〈𝐼, 𝐼〉 ∈ V | |
3 | fvsng 6945 | . . . . . 6 ⊢ ((〈𝐼, 𝐼〉 ∈ V ∧ 𝐼 ∈ 𝑉) → ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) = 𝐼) | |
4 | 2, 3 | mpan 688 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) = 𝐼) |
5 | 1, 4 | syl5eq 2871 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼) |
6 | snidg 4602 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ {𝐼}) | |
7 | 5, 6 | eqeltrd 2916 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) ∈ {𝐼}) |
8 | oveq1 7166 | . . . . . . 7 ⊢ (𝑥 = 𝐼 → (𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦)) | |
9 | 8 | eleq1d 2900 | . . . . . 6 ⊢ (𝑥 = 𝐼 → ((𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼} ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼})) |
10 | 9 | ralbidv 3200 | . . . . 5 ⊢ (𝑥 = 𝐼 → (∀𝑦 ∈ {𝐼} (𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼} ↔ ∀𝑦 ∈ {𝐼} (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼})) |
11 | 10 | ralsng 4616 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (∀𝑥 ∈ {𝐼}∀𝑦 ∈ {𝐼} (𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼} ↔ ∀𝑦 ∈ {𝐼} (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼})) |
12 | oveq2 7167 | . . . . . 6 ⊢ (𝑦 = 𝐼 → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼)) | |
13 | 12 | eleq1d 2900 | . . . . 5 ⊢ (𝑦 = 𝐼 → ((𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼} ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) ∈ {𝐼})) |
14 | 13 | ralsng 4616 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (∀𝑦 ∈ {𝐼} (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼} ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) ∈ {𝐼})) |
15 | 11, 14 | bitrd 281 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (∀𝑥 ∈ {𝐼}∀𝑦 ∈ {𝐼} (𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼} ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) ∈ {𝐼})) |
16 | 7, 15 | mpbird 259 | . 2 ⊢ (𝐼 ∈ 𝑉 → ∀𝑥 ∈ {𝐼}∀𝑦 ∈ {𝐼} (𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼}) |
17 | snex 5335 | . . . . 5 ⊢ {𝐼} ∈ V | |
18 | mgm1.m | . . . . . 6 ⊢ 𝑀 = {〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉} | |
19 | 18 | grpbase 16613 | . . . . 5 ⊢ ({𝐼} ∈ V → {𝐼} = (Base‘𝑀)) |
20 | 17, 19 | ax-mp 5 | . . . 4 ⊢ {𝐼} = (Base‘𝑀) |
21 | snex 5335 | . . . . 5 ⊢ {〈〈𝐼, 𝐼〉, 𝐼〉} ∈ V | |
22 | 18 | grpplusg 16614 | . . . . 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 17857 | . . 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 259 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 ∈ wcel 2113 ∀wral 3141 Vcvv 3497 {csn 4570 {cpr 4572 〈cop 4576 ‘cfv 6358 (class class class)co 7159 ndxcnx 16483 Basecbs 16486 +gcplusg 16568 Mgmcmgm 17853 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-plusg 16581 df-mgm 17855 |
This theorem is referenced by: sgrp1 17913 |
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