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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 13763 | . 2 ⊢ (♯‘{0, 1}) = 2 | |
2 | eqid 2823 | . . . 4 ⊢ {0, 1} = {0, 1} | |
3 | prex 5335 | . . . . . 6 ⊢ {0, 1} ∈ V | |
4 | eqeq1 2827 | . . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (𝑥 = 0 ↔ 𝑢 = 0)) | |
5 | 4 | ifbid 4491 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → if(𝑥 = 0, 0, 1) = if(𝑢 = 0, 0, 1)) |
6 | eqidd 2824 | . . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → if(𝑢 = 0, 0, 1) = if(𝑢 = 0, 0, 1)) | |
7 | 5, 6 | cbvmpov 7251 | . . . . . . . . 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 4809 | . . . . . . . 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 4675 | . . . . . . 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 16612 | . . . . . 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 2832 | . . . 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 7779 | . . . . . 6 ⊢ (𝑢 ∈ {0, 1}, 𝑣 ∈ {0, 1} ↦ if(𝑢 = 0, 0, 1)) ∈ V |
14 | 9 | grpplusg 16613 | . . . . . 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 2832 | . . . 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 18095 | . . 3 ⊢ ((♯‘{0, 1}) = 2 → {〈(Base‘ndx), {0, 1}〉, 〈(+g‘ndx), (𝑥 ∈ {0, 1}, 𝑦 ∈ {0, 1} ↦ if(𝑥 = 0, 0, 1))〉} ∈ Smgrp) |
18 | neleq1 3130 | . . . 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 484 | . . 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 18096 | . . 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 3628 | . 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 208 = wceq 1537 ∈ wcel 2114 ∉ wnel 3125 ∃wrex 3141 Vcvv 3496 ifcif 4469 {cpr 4571 〈cop 4575 ‘cfv 6357 ∈ cmpo 7160 0cc0 10539 1c1 10540 2c2 11695 ♯chash 13693 ndxcnx 16482 Basecbs 16485 +gcplusg 16567 Smgrpcsgrp 17902 Mndcmnd 17913 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-hash 13694 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-plusg 16580 df-mgm 17854 df-sgrp 17903 df-mnd 17914 |
This theorem is referenced by: mndsssgrp 18101 |
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