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 18126 | . 2 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Smgrp) |
3 | df-ov 7194 | . . . . 5 ⊢ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) | |
4 | opex 5333 | . . . . . 6 ⊢ 〈𝐼, 𝐼〉 ∈ V | |
5 | fvsng 6973 | . . . . . 6 ⊢ ((〈𝐼, 𝐼〉 ∈ V ∧ 𝐼 ∈ 𝑉) → ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) = 𝐼) | |
6 | 4, 5 | mpan 690 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) = 𝐼) |
7 | 3, 6 | syl5eq 2783 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼) |
8 | oveq2 7199 | . . . . . . 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 2752 | . . . . . 6 ⊢ (𝑦 = 𝐼 → ((𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼)) |
11 | oveq1 7198 | . . . . . . 7 ⊢ (𝑦 = 𝐼 → (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼)) | |
12 | 11, 9 | eqeq12d 2752 | . . . . . 6 ⊢ (𝑦 = 𝐼 → ((𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝑦 ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼)) |
13 | 10, 12 | anbi12d 634 | . . . . 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 4575 | . . . 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 713 | . . 3 ⊢ (𝐼 ∈ 𝑉 → ∀𝑦 ∈ {𝐼} ((𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ∧ (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝑦)) |
16 | oveq1 7198 | . . . . . 6 ⊢ (𝑥 = 𝐼 → (𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦)) | |
17 | 16 | eqeq1d 2738 | . . . . 5 ⊢ (𝑥 = 𝐼 → ((𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦)) |
18 | 17 | ovanraleqv 7215 | . . . 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 4576 | . . 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 260 | . 2 ⊢ (𝐼 ∈ 𝑉 → ∃𝑥 ∈ {𝐼}∀𝑦 ∈ {𝐼} ((𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ∧ (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝑥) = 𝑦)) |
21 | snex 5309 | . . . 4 ⊢ {𝐼} ∈ V | |
22 | 1 | grpbase 16794 | . . . 4 ⊢ ({𝐼} ∈ V → {𝐼} = (Base‘𝑀)) |
23 | 21, 22 | ax-mp 5 | . . 3 ⊢ {𝐼} = (Base‘𝑀) |
24 | snex 5309 | . . . 4 ⊢ {〈〈𝐼, 𝐼〉, 𝐼〉} ∈ V | |
25 | 1 | grpplusg 16795 | . . . 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 18129 | . 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 586 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ∃wrex 3052 Vcvv 3398 {csn 4527 {cpr 4529 〈cop 4533 ‘cfv 6358 (class class class)co 7191 ndxcnx 16663 Basecbs 16666 +gcplusg 16749 Smgrpcsgrp 18116 Mndcmnd 18127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-plusg 16762 df-mgm 18068 df-sgrp 18117 df-mnd 18128 |
This theorem is referenced by: grp1 18424 ring1 19574 |
Copyright terms: Public domain | W3C validator |