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| 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 7434 | . . . . 5 ⊢ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) | |
| 2 | opex 5469 | . . . . . 6 ⊢ 〈𝐼, 𝐼〉 ∈ V | |
| 3 | fvsng 7200 | . . . . . 6 ⊢ ((〈𝐼, 𝐼〉 ∈ V ∧ 𝐼 ∈ 𝑉) → ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) = 𝐼) | |
| 4 | 2, 3 | mpan 690 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) = 𝐼) | 
| 5 | 1, 4 | eqtrid 2789 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼) | 
| 6 | snidg 4660 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ {𝐼}) | |
| 7 | 5, 6 | eqeltrd 2841 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) ∈ {𝐼}) | 
| 8 | oveq1 7438 | . . . . . . 7 ⊢ (𝑥 = 𝐼 → (𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦)) | |
| 9 | 8 | eleq1d 2826 | . . . . . 6 ⊢ (𝑥 = 𝐼 → ((𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼} ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼})) | 
| 10 | 9 | ralbidv 3178 | . . . . 5 ⊢ (𝑥 = 𝐼 → (∀𝑦 ∈ {𝐼} (𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼} ↔ ∀𝑦 ∈ {𝐼} (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼})) | 
| 11 | 10 | ralsng 4675 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (∀𝑥 ∈ {𝐼}∀𝑦 ∈ {𝐼} (𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼} ↔ ∀𝑦 ∈ {𝐼} (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼})) | 
| 12 | oveq2 7439 | . . . . . 6 ⊢ (𝑦 = 𝐼 → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼)) | |
| 13 | 12 | eleq1d 2826 | . . . . 5 ⊢ (𝑦 = 𝐼 → ((𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼} ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) ∈ {𝐼})) | 
| 14 | 13 | ralsng 4675 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (∀𝑦 ∈ {𝐼} (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼} ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) ∈ {𝐼})) | 
| 15 | 11, 14 | bitrd 279 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (∀𝑥 ∈ {𝐼}∀𝑦 ∈ {𝐼} (𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼} ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) ∈ {𝐼})) | 
| 16 | 7, 15 | mpbird 257 | . 2 ⊢ (𝐼 ∈ 𝑉 → ∀𝑥 ∈ {𝐼}∀𝑦 ∈ {𝐼} (𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) ∈ {𝐼}) | 
| 17 | snex 5436 | . . . . 5 ⊢ {𝐼} ∈ V | |
| 18 | mgm1.m | . . . . . 6 ⊢ 𝑀 = {〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉} | |
| 19 | 18 | grpbase 17330 | . . . . 5 ⊢ ({𝐼} ∈ V → {𝐼} = (Base‘𝑀)) | 
| 20 | 17, 19 | ax-mp 5 | . . . 4 ⊢ {𝐼} = (Base‘𝑀) | 
| 21 | snex 5436 | . . . . 5 ⊢ {〈〈𝐼, 𝐼〉, 𝐼〉} ∈ V | |
| 22 | 18 | grpplusg 17332 | . . . . 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 18655 | . . 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 257 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Mgm) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 {csn 4626 {cpr 4628 〈cop 4632 ‘cfv 6561 (class class class)co 7431 ndxcnx 17230 Basecbs 17247 +gcplusg 17297 Mgmcmgm 18651 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mgm 18653 | 
| This theorem is referenced by: sgrp1 18742 | 
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