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Mirrors > Home > MPE Home > Th. List > smndex1sgrp | Structured version Visualization version GIF version |
Description: The monoid of endofunctions on ℕ0 restricted to the modulo function 𝐼 and the constant functions (𝐺‘𝐾) is a semigroup. (Contributed by AV, 14-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 |
---|---|
smndex1sgrp | ⊢ 𝑆 ∈ Smgrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smndex1ibas.m | . . 3 ⊢ 𝑀 = (EndoFMnd‘ℕ0) | |
2 | smndex1ibas.n | . . 3 ⊢ 𝑁 ∈ ℕ | |
3 | smndex1ibas.i | . . 3 ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) | |
4 | smndex1ibas.g | . . 3 ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛)) | |
5 | smndex1mgm.b | . . 3 ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) | |
6 | smndex1mgm.s | . . 3 ⊢ 𝑆 = (𝑀 ↾s 𝐵) | |
7 | 1, 2, 3, 4, 5, 6 | smndex1mgm 18064 | . 2 ⊢ 𝑆 ∈ Mgm |
8 | eqid 2798 | . . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
9 | eqid 2798 | . . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
10 | 8, 9 | mgmcl 17847 | . . . . 5 ⊢ ((𝑆 ∈ Mgm ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆)) |
11 | 7, 10 | mp3an1 1445 | . . . 4 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆)) |
12 | snex 5297 | . . . . . . . . . 10 ⊢ {𝐼} ∈ V | |
13 | ovex 7168 | . . . . . . . . . . 11 ⊢ (0..^𝑁) ∈ V | |
14 | snex 5297 | . . . . . . . . . . 11 ⊢ {(𝐺‘𝑛)} ∈ V | |
15 | 13, 14 | iunex 7651 | . . . . . . . . . 10 ⊢ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)} ∈ V |
16 | 12, 15 | unex 7449 | . . . . . . . . 9 ⊢ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) ∈ V |
17 | 5, 16 | eqeltri 2886 | . . . . . . . 8 ⊢ 𝐵 ∈ V |
18 | eqid 2798 | . . . . . . . . 9 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
19 | 6, 18 | ressplusg 16604 | . . . . . . . 8 ⊢ (𝐵 ∈ V → (+g‘𝑀) = (+g‘𝑆)) |
20 | 17, 19 | ax-mp 5 | . . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑆) |
21 | 20 | eqcomi 2807 | . . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑀) |
22 | 21 | oveqi 7148 | . . . . 5 ⊢ (𝑥(+g‘𝑆)𝑦) = (𝑥(+g‘𝑀)𝑦) |
23 | 1, 2, 3, 4, 5, 6 | smndex1bas 18063 | . . . . . . . 8 ⊢ (Base‘𝑆) = 𝐵 |
24 | 1, 2, 3, 4, 5 | smndex1basss 18062 | . . . . . . . 8 ⊢ 𝐵 ⊆ (Base‘𝑀) |
25 | 23, 24 | eqsstri 3949 | . . . . . . 7 ⊢ (Base‘𝑆) ⊆ (Base‘𝑀) |
26 | ssel 3908 | . . . . . . . 8 ⊢ ((Base‘𝑆) ⊆ (Base‘𝑀) → (𝑥 ∈ (Base‘𝑆) → 𝑥 ∈ (Base‘𝑀))) | |
27 | ssel 3908 | . . . . . . . 8 ⊢ ((Base‘𝑆) ⊆ (Base‘𝑀) → (𝑦 ∈ (Base‘𝑆) → 𝑦 ∈ (Base‘𝑀))) | |
28 | 26, 27 | anim12d 611 | . . . . . . 7 ⊢ ((Base‘𝑆) ⊆ (Base‘𝑀) → ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)))) |
29 | 25, 28 | ax-mp 5 | . . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) |
30 | eqid 2798 | . . . . . . 7 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
31 | 1, 30, 18 | efmndov 18038 | . . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) = (𝑥 ∘ 𝑦)) |
32 | 29, 31 | syl 17 | . . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g‘𝑀)𝑦) = (𝑥 ∘ 𝑦)) |
33 | 22, 32 | syl5eq 2845 | . . . 4 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g‘𝑆)𝑦) = (𝑥 ∘ 𝑦)) |
34 | 11, 33 | symggrplem 18041 | . . 3 ⊢ ((𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑆)) → ((𝑎(+g‘𝑆)𝑏)(+g‘𝑆)𝑐) = (𝑎(+g‘𝑆)(𝑏(+g‘𝑆)𝑐))) |
35 | 34 | rgen3 3169 | . 2 ⊢ ∀𝑎 ∈ (Base‘𝑆)∀𝑏 ∈ (Base‘𝑆)∀𝑐 ∈ (Base‘𝑆)((𝑎(+g‘𝑆)𝑏)(+g‘𝑆)𝑐) = (𝑎(+g‘𝑆)(𝑏(+g‘𝑆)𝑐)) |
36 | 8, 9 | issgrp 17894 | . 2 ⊢ (𝑆 ∈ Smgrp ↔ (𝑆 ∈ Mgm ∧ ∀𝑎 ∈ (Base‘𝑆)∀𝑏 ∈ (Base‘𝑆)∀𝑐 ∈ (Base‘𝑆)((𝑎(+g‘𝑆)𝑏)(+g‘𝑆)𝑐) = (𝑎(+g‘𝑆)(𝑏(+g‘𝑆)𝑐)))) |
37 | 7, 35, 36 | mpbir2an 710 | 1 ⊢ 𝑆 ∈ Smgrp |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ∪ cun 3879 ⊆ wss 3881 {csn 4525 ∪ ciun 4881 ↦ cmpt 5110 ∘ ccom 5523 ‘cfv 6324 (class class class)co 7135 0cc0 10526 ℕcn 11625 ℕ0cn0 11885 ..^cfzo 13028 mod cmo 13232 Basecbs 16475 ↾s cress 16476 +gcplusg 16557 Mgmcmgm 17842 Smgrpcsgrp 17892 EndoFMndcefmnd 18025 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-tset 16576 df-mgm 17844 df-sgrp 17893 df-efmnd 18026 |
This theorem is referenced by: smndex1mnd 18067 |
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