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Mirrors > Home > MPE Home > Th. List > smndex1bas | Structured version Visualization version GIF version |
Description: The base set of the monoid of endofunctions on ℕ0 restricted to the modulo function 𝐼 and the constant functions (𝐺‘𝐾). (Contributed by AV, 12-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 |
---|---|
smndex1bas | ⊢ (Base‘𝑆) = 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smndex1ibas.m | . . . 4 ⊢ 𝑀 = (EndoFMnd‘ℕ0) | |
2 | smndex1ibas.n | . . . 4 ⊢ 𝑁 ∈ ℕ | |
3 | smndex1ibas.i | . . . 4 ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) | |
4 | smndex1ibas.g | . . . 4 ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛)) | |
5 | smndex1mgm.b | . . . 4 ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) | |
6 | 1, 2, 3, 4, 5 | smndex1basss 18070 | . . 3 ⊢ 𝐵 ⊆ (Base‘𝑀) |
7 | dfss 3953 | . . 3 ⊢ (𝐵 ⊆ (Base‘𝑀) ↔ 𝐵 = (𝐵 ∩ (Base‘𝑀))) | |
8 | 6, 7 | mpbi 232 | . 2 ⊢ 𝐵 = (𝐵 ∩ (Base‘𝑀)) |
9 | snex 5332 | . . . . 5 ⊢ {𝐼} ∈ V | |
10 | ovex 7189 | . . . . . 6 ⊢ (0..^𝑁) ∈ V | |
11 | snex 5332 | . . . . . 6 ⊢ {(𝐺‘𝑛)} ∈ V | |
12 | 10, 11 | iunex 7669 | . . . . 5 ⊢ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)} ∈ V |
13 | 9, 12 | unex 7469 | . . . 4 ⊢ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) ∈ V |
14 | 5, 13 | eqeltri 2909 | . . 3 ⊢ 𝐵 ∈ V |
15 | smndex1mgm.s | . . . 4 ⊢ 𝑆 = (𝑀 ↾s 𝐵) | |
16 | eqid 2821 | . . . 4 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
17 | 15, 16 | ressbas 16554 | . . 3 ⊢ (𝐵 ∈ V → (𝐵 ∩ (Base‘𝑀)) = (Base‘𝑆)) |
18 | 14, 17 | ax-mp 5 | . 2 ⊢ (𝐵 ∩ (Base‘𝑀)) = (Base‘𝑆) |
19 | 8, 18 | eqtr2i 2845 | 1 ⊢ (Base‘𝑆) = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∪ cun 3934 ∩ cin 3935 ⊆ wss 3936 {csn 4567 ∪ ciun 4919 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 0cc0 10537 ℕcn 11638 ℕ0cn0 11898 ..^cfzo 13034 mod cmo 13238 Basecbs 16483 ↾s cress 16484 EndoFMndcefmnd 18033 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-tset 16584 df-efmnd 18034 |
This theorem is referenced by: smndex1mgm 18072 smndex1sgrp 18073 smndex1mnd 18075 smndex1id 18076 |
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