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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 ∈ SGrp 𝑚 ∉ Mnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prhash2ex 13187 | . 2 ⊢ (#‘{0, 1}) = 2 | |
2 | eqid 2622 | . . . 4 ⊢ {0, 1} = {0, 1} | |
3 | prex 4909 | . . . . . 6 ⊢ {0, 1} ∈ V | |
4 | eqeq1 2626 | . . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (𝑥 = 0 ↔ 𝑢 = 0)) | |
5 | 4 | ifbid 4108 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → if(𝑥 = 0, 0, 1) = if(𝑢 = 0, 0, 1)) |
6 | eqidd 2623 | . . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → if(𝑢 = 0, 0, 1) = if(𝑢 = 0, 0, 1)) | |
7 | 5, 6 | cbvmpt2v 6735 | . . . . . . . . 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 4406 | . . . . . . . 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 4272 | . . . . . . 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 15991 | . . . . . 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 2631 | . . . 4 ⊢ (Base‘{〈(Base‘ndx), {0, 1}〉, 〈(+g‘ndx), (𝑥 ∈ {0, 1}, 𝑦 ∈ {0, 1} ↦ if(𝑥 = 0, 0, 1))〉}) = {0, 1} |
13 | 3, 3 | mpt2ex 7247 | . . . . . 6 ⊢ (𝑢 ∈ {0, 1}, 𝑣 ∈ {0, 1} ↦ if(𝑢 = 0, 0, 1)) ∈ V |
14 | 9 | grpplusg 15992 | . . . . . 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 2631 | . . . 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 17415 | . . 3 ⊢ ((#‘{0, 1}) = 2 → {〈(Base‘ndx), {0, 1}〉, 〈(+g‘ndx), (𝑥 ∈ {0, 1}, 𝑦 ∈ {0, 1} ↦ if(𝑥 = 0, 0, 1))〉} ∈ SGrp) |
18 | neleq1 2902 | . . . 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 482 | . . 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 17416 | . . 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 3317 | . 2 ⊢ ((#‘{0, 1}) = 2 → ∃𝑚 ∈ SGrp 𝑚 ∉ Mnd) |
22 | 1, 21 | ax-mp 5 | 1 ⊢ ∃𝑚 ∈ SGrp 𝑚 ∉ Mnd |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 ∈ wcel 1990 ∉ wnel 2897 ∃wrex 2913 Vcvv 3200 ifcif 4086 {cpr 4179 〈cop 4183 ‘cfv 5888 ↦ cmpt2 6652 0cc0 9936 1c1 9937 2c2 11070 #chash 13117 ndxcnx 15854 Basecbs 15857 +gcplusg 15941 SGrpcsgrp 17283 Mndcmnd 17294 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-mgm 17242 df-sgrp 17284 df-mnd 17295 |
This theorem is referenced by: mndsssgrp 17421 |
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