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Mirrors > Home > MPE Home > Th. List > mnd1 | Structured version Visualization version GIF version |
Description: The (smallest) structure representing a trivial monoid consists of one element. (Contributed by AV, 28-Apr-2019.) (Proof shortened by AV, 11-Feb-2020.) |
Ref | Expression |
---|---|
mnd1.m | ⊢ 𝑀 = {〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉} |
Ref | Expression |
---|---|
mnd1 | ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Mnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnd1.m | . . 3 ⊢ 𝑀 = {〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉} | |
2 | 1 | sgrp1 17761 | . 2 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ SGrp) |
3 | df-ov 6979 | . . . . 5 ⊢ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) | |
4 | opex 5213 | . . . . . 6 ⊢ 〈𝐼, 𝐼〉 ∈ V | |
5 | fvsng 6765 | . . . . . 6 ⊢ ((〈𝐼, 𝐼〉 ∈ V ∧ 𝐼 ∈ 𝑉) → ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) = 𝐼) | |
6 | 4, 5 | mpan 677 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) = 𝐼) |
7 | 3, 6 | syl5eq 2827 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼) |
8 | oveq2 6984 | . . . . . . 7 ⊢ (𝑦 = 𝐼 → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼)) | |
9 | id 22 | . . . . . . 7 ⊢ (𝑦 = 𝐼 → 𝑦 = 𝐼) | |
10 | 8, 9 | eqeq12d 2794 | . . . . . 6 ⊢ (𝑦 = 𝐼 → ((𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼)) |
11 | oveq1 6983 | . . . . . . 7 ⊢ (𝑦 = 𝐼 → (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼)) | |
12 | 11, 9 | eqeq12d 2794 | . . . . . 6 ⊢ (𝑦 = 𝐼 → ((𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝑦 ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼)) |
13 | 10, 12 | anbi12d 621 | . . . . 5 ⊢ (𝑦 = 𝐼 → (((𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ∧ (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝑦) ↔ ((𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼 ∧ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼))) |
14 | 13 | ralsng 4487 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (∀𝑦 ∈ {𝐼} ((𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ∧ (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝑦) ↔ ((𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼 ∧ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼))) |
15 | 7, 7, 14 | mpbir2and 700 | . . 3 ⊢ (𝐼 ∈ 𝑉 → ∀𝑦 ∈ {𝐼} ((𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ∧ (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝑦)) |
16 | oveq1 6983 | . . . . . 6 ⊢ (𝑥 = 𝐼 → (𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦)) | |
17 | 16 | eqeq1d 2781 | . . . . 5 ⊢ (𝑥 = 𝐼 → ((𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦)) |
18 | 17 | ovanraleqv 7000 | . . . 4 ⊢ (𝑥 = 𝐼 → (∀𝑦 ∈ {𝐼} ((𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ∧ (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝑥) = 𝑦) ↔ ∀𝑦 ∈ {𝐼} ((𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ∧ (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝑦))) |
19 | 18 | rexsng 4488 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (∃𝑥 ∈ {𝐼}∀𝑦 ∈ {𝐼} ((𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ∧ (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝑥) = 𝑦) ↔ ∀𝑦 ∈ {𝐼} ((𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ∧ (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝑦))) |
20 | 15, 19 | mpbird 249 | . 2 ⊢ (𝐼 ∈ 𝑉 → ∃𝑥 ∈ {𝐼}∀𝑦 ∈ {𝐼} ((𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ∧ (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝑥) = 𝑦)) |
21 | snex 5188 | . . . 4 ⊢ {𝐼} ∈ V | |
22 | 1 | grpbase 16466 | . . . 4 ⊢ ({𝐼} ∈ V → {𝐼} = (Base‘𝑀)) |
23 | 21, 22 | ax-mp 5 | . . 3 ⊢ {𝐼} = (Base‘𝑀) |
24 | snex 5188 | . . . 4 ⊢ {〈〈𝐼, 𝐼〉, 𝐼〉} ∈ V | |
25 | 1 | grpplusg 16467 | . . . 4 ⊢ ({〈〈𝐼, 𝐼〉, 𝐼〉} ∈ V → {〈〈𝐼, 𝐼〉, 𝐼〉} = (+g‘𝑀)) |
26 | 24, 25 | ax-mp 5 | . . 3 ⊢ {〈〈𝐼, 𝐼〉, 𝐼〉} = (+g‘𝑀) |
27 | 23, 26 | ismnddef 17764 | . 2 ⊢ (𝑀 ∈ Mnd ↔ (𝑀 ∈ SGrp ∧ ∃𝑥 ∈ {𝐼}∀𝑦 ∈ {𝐼} ((𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ∧ (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝑥) = 𝑦))) |
28 | 2, 20, 27 | sylanbrc 575 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ∀wral 3089 ∃wrex 3090 Vcvv 3416 {csn 4441 {cpr 4443 〈cop 4447 ‘cfv 6188 (class class class)co 6976 ndxcnx 16336 Basecbs 16339 +gcplusg 16421 SGrpcsgrp 17751 Mndcmnd 17762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-n0 11708 df-z 11794 df-uz 12059 df-fz 12709 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-plusg 16434 df-mgm 17710 df-sgrp 17752 df-mnd 17763 |
This theorem is referenced by: grp1 17993 ring1 19075 |
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