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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 18334 | . 2 ⊢ 𝑆 ∈ Mgm |
8 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
9 | eqid 2737 | . . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
10 | 8, 9 | mgmcl 18117 | . . . . 5 ⊢ ((𝑆 ∈ Mgm ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆)) |
11 | 7, 10 | mp3an1 1450 | . . . 4 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆)) |
12 | snex 5324 | . . . . . . . . . 10 ⊢ {𝐼} ∈ V | |
13 | ovex 7246 | . . . . . . . . . . 11 ⊢ (0..^𝑁) ∈ V | |
14 | snex 5324 | . . . . . . . . . . 11 ⊢ {(𝐺‘𝑛)} ∈ V | |
15 | 13, 14 | iunex 7741 | . . . . . . . . . 10 ⊢ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)} ∈ V |
16 | 12, 15 | unex 7531 | . . . . . . . . 9 ⊢ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) ∈ V |
17 | 5, 16 | eqeltri 2834 | . . . . . . . 8 ⊢ 𝐵 ∈ V |
18 | eqid 2737 | . . . . . . . . 9 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
19 | 6, 18 | ressplusg 16834 | . . . . . . . 8 ⊢ (𝐵 ∈ V → (+g‘𝑀) = (+g‘𝑆)) |
20 | 17, 19 | ax-mp 5 | . . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑆) |
21 | 20 | eqcomi 2746 | . . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑀) |
22 | 21 | oveqi 7226 | . . . . 5 ⊢ (𝑥(+g‘𝑆)𝑦) = (𝑥(+g‘𝑀)𝑦) |
23 | 1, 2, 3, 4, 5, 6 | smndex1bas 18333 | . . . . . . . 8 ⊢ (Base‘𝑆) = 𝐵 |
24 | 1, 2, 3, 4, 5 | smndex1basss 18332 | . . . . . . . 8 ⊢ 𝐵 ⊆ (Base‘𝑀) |
25 | 23, 24 | eqsstri 3935 | . . . . . . 7 ⊢ (Base‘𝑆) ⊆ (Base‘𝑀) |
26 | ssel 3893 | . . . . . . . 8 ⊢ ((Base‘𝑆) ⊆ (Base‘𝑀) → (𝑥 ∈ (Base‘𝑆) → 𝑥 ∈ (Base‘𝑀))) | |
27 | ssel 3893 | . . . . . . . 8 ⊢ ((Base‘𝑆) ⊆ (Base‘𝑀) → (𝑦 ∈ (Base‘𝑆) → 𝑦 ∈ (Base‘𝑀))) | |
28 | 26, 27 | anim12d 612 | . . . . . . 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 2737 | . . . . . . 7 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
31 | 1, 30, 18 | efmndov 18308 | . . . . . 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 2790 | . . . 4 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g‘𝑆)𝑦) = (𝑥 ∘ 𝑦)) |
34 | 11, 33 | symggrplem 18311 | . . 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 3125 | . 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 18164 | . 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 711 | 1 ⊢ 𝑆 ∈ Smgrp |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 Vcvv 3408 ∪ cun 3864 ⊆ wss 3866 {csn 4541 ∪ ciun 4904 ↦ cmpt 5135 ∘ ccom 5555 ‘cfv 6380 (class class class)co 7213 0cc0 10729 ℕcn 11830 ℕ0cn0 12090 ..^cfzo 13238 mod cmo 13442 Basecbs 16760 ↾s cress 16784 +gcplusg 16802 Mgmcmgm 18112 Smgrpcsgrp 18162 EndoFMndcefmnd 18295 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-inf 9059 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-uz 12439 df-rp 12587 df-fz 13096 df-fzo 13239 df-fl 13367 df-mod 13443 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-tset 16821 df-mgm 18114 df-sgrp 18163 df-efmnd 18296 |
This theorem is referenced by: smndex1mnd 18337 |
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