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Mirrors > Home > MPE Home > Th. List > sgrpnmndex | Structured version Visualization version GIF version |
Description: There is a semigroup which is not a monoid. (Contributed by AV, 29-Jan-2020.) |
Ref | Expression |
---|---|
sgrpnmndex | ⊢ ∃𝑚 ∈ Smgrp 𝑚 ∉ Mnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prhash2ex 13966 | . 2 ⊢ (♯‘{0, 1}) = 2 | |
2 | eqid 2737 | . . . 4 ⊢ {0, 1} = {0, 1} | |
3 | prex 5325 | . . . . . 6 ⊢ {0, 1} ∈ V | |
4 | eqeq1 2741 | . . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (𝑥 = 0 ↔ 𝑢 = 0)) | |
5 | 4 | ifbid 4462 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → if(𝑥 = 0, 0, 1) = if(𝑢 = 0, 0, 1)) |
6 | eqidd 2738 | . . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → if(𝑢 = 0, 0, 1) = if(𝑢 = 0, 0, 1)) | |
7 | 5, 6 | cbvmpov 7306 | . . . . . . . . 9 ⊢ (𝑥 ∈ {0, 1}, 𝑦 ∈ {0, 1} ↦ if(𝑥 = 0, 0, 1)) = (𝑢 ∈ {0, 1}, 𝑣 ∈ {0, 1} ↦ if(𝑢 = 0, 0, 1)) |
8 | 7 | opeq2i 4788 | . . . . . . . 8 ⊢ 〈(+g‘ndx), (𝑥 ∈ {0, 1}, 𝑦 ∈ {0, 1} ↦ if(𝑥 = 0, 0, 1))〉 = 〈(+g‘ndx), (𝑢 ∈ {0, 1}, 𝑣 ∈ {0, 1} ↦ if(𝑢 = 0, 0, 1))〉 |
9 | 8 | preq2i 4653 | . . . . . . 7 ⊢ {〈(Base‘ndx), {0, 1}〉, 〈(+g‘ndx), (𝑥 ∈ {0, 1}, 𝑦 ∈ {0, 1} ↦ if(𝑥 = 0, 0, 1))〉} = {〈(Base‘ndx), {0, 1}〉, 〈(+g‘ndx), (𝑢 ∈ {0, 1}, 𝑣 ∈ {0, 1} ↦ if(𝑢 = 0, 0, 1))〉} |
10 | 9 | grpbase 16832 | . . . . . 6 ⊢ ({0, 1} ∈ V → {0, 1} = (Base‘{〈(Base‘ndx), {0, 1}〉, 〈(+g‘ndx), (𝑥 ∈ {0, 1}, 𝑦 ∈ {0, 1} ↦ if(𝑥 = 0, 0, 1))〉})) |
11 | 3, 10 | ax-mp 5 | . . . . 5 ⊢ {0, 1} = (Base‘{〈(Base‘ndx), {0, 1}〉, 〈(+g‘ndx), (𝑥 ∈ {0, 1}, 𝑦 ∈ {0, 1} ↦ if(𝑥 = 0, 0, 1))〉}) |
12 | 11 | eqcomi 2746 | . . . 4 ⊢ (Base‘{〈(Base‘ndx), {0, 1}〉, 〈(+g‘ndx), (𝑥 ∈ {0, 1}, 𝑦 ∈ {0, 1} ↦ if(𝑥 = 0, 0, 1))〉}) = {0, 1} |
13 | 3, 3 | mpoex 7850 | . . . . . 6 ⊢ (𝑢 ∈ {0, 1}, 𝑣 ∈ {0, 1} ↦ if(𝑢 = 0, 0, 1)) ∈ V |
14 | 9 | grpplusg 16833 | . . . . . 6 ⊢ ((𝑢 ∈ {0, 1}, 𝑣 ∈ {0, 1} ↦ if(𝑢 = 0, 0, 1)) ∈ V → (𝑢 ∈ {0, 1}, 𝑣 ∈ {0, 1} ↦ if(𝑢 = 0, 0, 1)) = (+g‘{〈(Base‘ndx), {0, 1}〉, 〈(+g‘ndx), (𝑥 ∈ {0, 1}, 𝑦 ∈ {0, 1} ↦ if(𝑥 = 0, 0, 1))〉})) |
15 | 13, 14 | ax-mp 5 | . . . . 5 ⊢ (𝑢 ∈ {0, 1}, 𝑣 ∈ {0, 1} ↦ if(𝑢 = 0, 0, 1)) = (+g‘{〈(Base‘ndx), {0, 1}〉, 〈(+g‘ndx), (𝑥 ∈ {0, 1}, 𝑦 ∈ {0, 1} ↦ if(𝑥 = 0, 0, 1))〉}) |
16 | 15 | eqcomi 2746 | . . . 4 ⊢ (+g‘{〈(Base‘ndx), {0, 1}〉, 〈(+g‘ndx), (𝑥 ∈ {0, 1}, 𝑦 ∈ {0, 1} ↦ if(𝑥 = 0, 0, 1))〉}) = (𝑢 ∈ {0, 1}, 𝑣 ∈ {0, 1} ↦ if(𝑢 = 0, 0, 1)) |
17 | 2, 12, 16 | sgrp2nmndlem4 18355 | . . 3 ⊢ ((♯‘{0, 1}) = 2 → {〈(Base‘ndx), {0, 1}〉, 〈(+g‘ndx), (𝑥 ∈ {0, 1}, 𝑦 ∈ {0, 1} ↦ if(𝑥 = 0, 0, 1))〉} ∈ Smgrp) |
18 | neleq1 3051 | . . . 4 ⊢ (𝑚 = {〈(Base‘ndx), {0, 1}〉, 〈(+g‘ndx), (𝑥 ∈ {0, 1}, 𝑦 ∈ {0, 1} ↦ if(𝑥 = 0, 0, 1))〉} → (𝑚 ∉ Mnd ↔ {〈(Base‘ndx), {0, 1}〉, 〈(+g‘ndx), (𝑥 ∈ {0, 1}, 𝑦 ∈ {0, 1} ↦ if(𝑥 = 0, 0, 1))〉} ∉ Mnd)) | |
19 | 18 | adantl 485 | . . 3 ⊢ (((♯‘{0, 1}) = 2 ∧ 𝑚 = {〈(Base‘ndx), {0, 1}〉, 〈(+g‘ndx), (𝑥 ∈ {0, 1}, 𝑦 ∈ {0, 1} ↦ if(𝑥 = 0, 0, 1))〉}) → (𝑚 ∉ Mnd ↔ {〈(Base‘ndx), {0, 1}〉, 〈(+g‘ndx), (𝑥 ∈ {0, 1}, 𝑦 ∈ {0, 1} ↦ if(𝑥 = 0, 0, 1))〉} ∉ Mnd)) |
20 | 2, 12, 16 | sgrp2nmndlem5 18356 | . . 3 ⊢ ((♯‘{0, 1}) = 2 → {〈(Base‘ndx), {0, 1}〉, 〈(+g‘ndx), (𝑥 ∈ {0, 1}, 𝑦 ∈ {0, 1} ↦ if(𝑥 = 0, 0, 1))〉} ∉ Mnd) |
21 | 17, 19, 20 | rspcedvd 3540 | . 2 ⊢ ((♯‘{0, 1}) = 2 → ∃𝑚 ∈ Smgrp 𝑚 ∉ Mnd) |
22 | 1, 21 | ax-mp 5 | 1 ⊢ ∃𝑚 ∈ Smgrp 𝑚 ∉ Mnd |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 ∈ wcel 2110 ∉ wnel 3046 ∃wrex 3062 Vcvv 3408 ifcif 4439 {cpr 4543 〈cop 4547 ‘cfv 6380 ∈ cmpo 7215 0cc0 10729 1c1 10730 2c2 11885 ♯chash 13896 ndxcnx 16744 Basecbs 16760 +gcplusg 16802 Smgrpcsgrp 18162 Mndcmnd 18173 |
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 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
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 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-oadd 8206 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-dju 9517 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-n0 12091 df-z 12177 df-uz 12439 df-fz 13096 df-hash 13897 df-struct 16700 df-slot 16735 df-ndx 16745 df-base 16761 df-plusg 16815 df-mgm 18114 df-sgrp 18163 df-mnd 18174 |
This theorem is referenced by: mndsssgrp 18361 |
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