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Mirrors > Home > MPE Home > Th. List > nsmndex1 | Structured version Visualization version GIF version |
Description: The base set 𝐵 of the constructed monoid 𝑆 is not a submonoid of the monoid 𝑀 of endofunctions on set ℕ0, although 𝑀 ∈ Mnd and 𝑆 ∈ Mnd and 𝐵 ⊆ (Base‘𝑀) hold. (Contributed by AV, 17-Feb-2024.) |
Ref | Expression |
---|---|
smndex1ibas.m | ⊢ 𝑀 = (EndoFMnd‘ℕ0) |
smndex1ibas.n | ⊢ 𝑁 ∈ ℕ |
smndex1ibas.i | ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) |
smndex1ibas.g | ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛)) |
smndex1mgm.b | ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) |
smndex1mgm.s | ⊢ 𝑆 = (𝑀 ↾s 𝐵) |
Ref | Expression |
---|---|
nsmndex1 | ⊢ 𝐵 ∉ (SubMnd‘𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smndex1ibas.m | . . . . . . 7 ⊢ 𝑀 = (EndoFMnd‘ℕ0) | |
2 | smndex1ibas.n | . . . . . . 7 ⊢ 𝑁 ∈ ℕ | |
3 | smndex1ibas.i | . . . . . . 7 ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) | |
4 | smndex1ibas.g | . . . . . . 7 ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛)) | |
5 | smndex1mgm.b | . . . . . . 7 ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) | |
6 | smndex1mgm.s | . . . . . . 7 ⊢ 𝑆 = (𝑀 ↾s 𝐵) | |
7 | 1, 2, 3, 4, 5, 6 | smndex1n0mnd 18195 | . . . . . 6 ⊢ (0g‘𝑀) ∉ 𝐵 |
8 | 7 | neli 3040 | . . . . 5 ⊢ ¬ (0g‘𝑀) ∈ 𝐵 |
9 | 8 | intnan 490 | . . . 4 ⊢ ¬ (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵) |
10 | 9 | intnan 490 | . . 3 ⊢ ¬ ((𝑀 ∈ Mnd ∧ (𝑀 ↾s 𝐵) ∈ Mnd) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵)) |
11 | eqid 2738 | . . . 4 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
12 | eqid 2738 | . . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
13 | 11, 12 | issubmndb 18088 | . . 3 ⊢ (𝐵 ∈ (SubMnd‘𝑀) ↔ ((𝑀 ∈ Mnd ∧ (𝑀 ↾s 𝐵) ∈ Mnd) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵))) |
14 | 10, 13 | mtbir 326 | . 2 ⊢ ¬ 𝐵 ∈ (SubMnd‘𝑀) |
15 | 14 | nelir 3041 | 1 ⊢ 𝐵 ∉ (SubMnd‘𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∉ wnel 3038 ∪ cun 3841 ⊆ wss 3843 {csn 4516 ∪ ciun 4881 ↦ cmpt 5110 ‘cfv 6339 (class class class)co 7172 0cc0 10617 ℕcn 11718 ℕ0cn0 11978 ..^cfzo 13126 mod cmo 13330 Basecbs 16588 ↾s cress 16589 0gc0g 16818 Mndcmnd 18029 SubMndcsubmnd 18073 EndoFMndcefmnd 18151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 ax-pre-sup 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-om 7602 df-1st 7716 df-2nd 7717 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-1o 8133 df-er 8322 df-map 8441 df-en 8558 df-dom 8559 df-sdom 8560 df-fin 8561 df-sup 8981 df-inf 8982 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-div 11378 df-nn 11719 df-2 11781 df-3 11782 df-4 11783 df-5 11784 df-6 11785 df-7 11786 df-8 11787 df-9 11788 df-n0 11979 df-z 12065 df-uz 12327 df-rp 12475 df-fz 12984 df-fzo 13127 df-fl 13255 df-mod 13331 df-struct 16590 df-ndx 16591 df-slot 16592 df-base 16594 df-sets 16595 df-ress 16596 df-plusg 16683 df-tset 16689 df-0g 16820 df-mgm 17970 df-sgrp 18019 df-mnd 18030 df-submnd 18075 df-efmnd 18152 |
This theorem is referenced by: (None) |
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