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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 18925 | . . . . . 6 ⊢ (0g‘𝑀) ∉ 𝐵 | 
| 8 | 7 | neli 3048 | . . . . 5 ⊢ ¬ (0g‘𝑀) ∈ 𝐵 | 
| 9 | 8 | intnan 486 | . . . 4 ⊢ ¬ (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵) | 
| 10 | 9 | intnan 486 | . . 3 ⊢ ¬ ((𝑀 ∈ Mnd ∧ (𝑀 ↾s 𝐵) ∈ Mnd) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵)) | 
| 11 | eqid 2737 | . . . 4 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
| 12 | eqid 2737 | . . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
| 13 | 11, 12 | issubmndb 18818 | . . 3 ⊢ (𝐵 ∈ (SubMnd‘𝑀) ↔ ((𝑀 ∈ Mnd ∧ (𝑀 ↾s 𝐵) ∈ Mnd) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵))) | 
| 14 | 10, 13 | mtbir 323 | . 2 ⊢ ¬ 𝐵 ∈ (SubMnd‘𝑀) | 
| 15 | 14 | nelir 3049 | 1 ⊢ 𝐵 ∉ (SubMnd‘𝑀) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∉ wnel 3046 ∪ cun 3949 ⊆ wss 3951 {csn 4626 ∪ ciun 4991 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 0cc0 11155 ℕcn 12266 ℕ0cn0 12526 ..^cfzo 13694 mod cmo 13909 Basecbs 17247 ↾s cress 17274 0gc0g 17484 Mndcmnd 18747 SubMndcsubmnd 18795 EndoFMndcefmnd 18881 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-tset 17316 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-efmnd 18882 | 
| This theorem is referenced by: (None) | 
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