![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 18717 | . 2 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Smgrp) |
3 | df-ov 7426 | . . . . 5 ⊢ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) | |
4 | opex 5469 | . . . . . 6 ⊢ 〈𝐼, 𝐼〉 ∈ V | |
5 | fvsng 7193 | . . . . . 6 ⊢ ((〈𝐼, 𝐼〉 ∈ V ∧ 𝐼 ∈ 𝑉) → ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) = 𝐼) | |
6 | 4, 5 | mpan 688 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) = 𝐼) |
7 | 3, 6 | eqtrid 2777 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼) |
8 | oveq2 7431 | . . . . . . 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 2741 | . . . . . 6 ⊢ (𝑦 = 𝐼 → ((𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼)) |
11 | oveq1 7430 | . . . . . . 7 ⊢ (𝑦 = 𝐼 → (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼)) | |
12 | 11, 9 | eqeq12d 2741 | . . . . . 6 ⊢ (𝑦 = 𝐼 → ((𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝑦 ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼)) |
13 | 10, 12 | anbi12d 630 | . . . . 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 4681 | . . . 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 711 | . . 3 ⊢ (𝐼 ∈ 𝑉 → ∀𝑦 ∈ {𝐼} ((𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ∧ (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝑦)) |
16 | oveq1 7430 | . . . . . 6 ⊢ (𝑥 = 𝐼 → (𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦)) | |
17 | 16 | eqeq1d 2727 | . . . . 5 ⊢ (𝑥 = 𝐼 → ((𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦)) |
18 | 17 | ovanraleqv 7447 | . . . 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 4682 | . . 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 256 | . 2 ⊢ (𝐼 ∈ 𝑉 → ∃𝑥 ∈ {𝐼}∀𝑦 ∈ {𝐼} ((𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ∧ (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝑥) = 𝑦)) |
21 | snex 5436 | . . . 4 ⊢ {𝐼} ∈ V | |
22 | 1 | grpbase 17295 | . . . 4 ⊢ ({𝐼} ∈ V → {𝐼} = (Base‘𝑀)) |
23 | 21, 22 | ax-mp 5 | . . 3 ⊢ {𝐼} = (Base‘𝑀) |
24 | snex 5436 | . . . 4 ⊢ {〈〈𝐼, 𝐼〉, 𝐼〉} ∈ V | |
25 | 1 | grpplusg 17297 | . . . 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 18724 | . 2 ⊢ (𝑀 ∈ Mnd ↔ (𝑀 ∈ Smgrp ∧ ∃𝑥 ∈ {𝐼}∀𝑦 ∈ {𝐼} ((𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ∧ (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝑥) = 𝑦))) |
28 | 2, 20, 27 | sylanbrc 581 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ∃wrex 3059 Vcvv 3461 {csn 4632 {cpr 4634 〈cop 4638 ‘cfv 6553 (class class class)co 7423 ndxcnx 17190 Basecbs 17208 +gcplusg 17261 Smgrpcsgrp 18706 Mndcmnd 18722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-1st 8002 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-nn 12260 df-2 12322 df-n0 12520 df-z 12606 df-uz 12870 df-fz 13534 df-struct 17144 df-slot 17179 df-ndx 17191 df-base 17209 df-plusg 17274 df-mgm 18628 df-sgrp 18707 df-mnd 18723 |
This theorem is referenced by: grp1 19036 ring1 20284 |
Copyright terms: Public domain | W3C validator |