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Mirrors > Home > MPE Home > Th. List > smndex1id | Structured version Visualization version GIF version |
Description: The modulo function 𝐼 is the identity of the monoid of endofunctions on ℕ0 restricted to the modulo function 𝐼 and the constant functions (𝐺‘𝐾). (Contributed by AV, 16-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 |
---|---|
smndex1id | ⊢ 𝐼 = (0g‘𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smndex1ibas.i | . . . . . 6 ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) | |
2 | nn0ex 11904 | . . . . . . 7 ⊢ ℕ0 ∈ V | |
3 | 2 | mptex 6986 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) ∈ V |
4 | 1, 3 | eqeltri 2909 | . . . . 5 ⊢ 𝐼 ∈ V |
5 | 4 | snid 4601 | . . . 4 ⊢ 𝐼 ∈ {𝐼} |
6 | elun1 4152 | . . . 4 ⊢ (𝐼 ∈ {𝐼} → 𝐼 ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})) | |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ 𝐼 ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) |
8 | smndex1mgm.b | . . 3 ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) | |
9 | 7, 8 | eleqtrri 2912 | . 2 ⊢ 𝐼 ∈ 𝐵 |
10 | smndex1ibas.m | . . . . . 6 ⊢ 𝑀 = (EndoFMnd‘ℕ0) | |
11 | smndex1ibas.n | . . . . . 6 ⊢ 𝑁 ∈ ℕ | |
12 | smndex1ibas.g | . . . . . 6 ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛)) | |
13 | smndex1mgm.s | . . . . . 6 ⊢ 𝑆 = (𝑀 ↾s 𝐵) | |
14 | 10, 11, 1, 12, 8, 13 | smndex1bas 18071 | . . . . 5 ⊢ (Base‘𝑆) = 𝐵 |
15 | 14 | eqcomi 2830 | . . . 4 ⊢ 𝐵 = (Base‘𝑆) |
16 | 15 | a1i 11 | . . 3 ⊢ (𝐼 ∈ 𝐵 → 𝐵 = (Base‘𝑆)) |
17 | snex 5332 | . . . . . 6 ⊢ {𝐼} ∈ V | |
18 | ovex 7189 | . . . . . . 7 ⊢ (0..^𝑁) ∈ V | |
19 | snex 5332 | . . . . . . 7 ⊢ {(𝐺‘𝑛)} ∈ V | |
20 | 18, 19 | iunex 7669 | . . . . . 6 ⊢ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)} ∈ V |
21 | 17, 20 | unex 7469 | . . . . 5 ⊢ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) ∈ V |
22 | 8, 21 | eqeltri 2909 | . . . 4 ⊢ 𝐵 ∈ V |
23 | eqid 2821 | . . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
24 | 13, 23 | ressplusg 16612 | . . . 4 ⊢ (𝐵 ∈ V → (+g‘𝑀) = (+g‘𝑆)) |
25 | 22, 24 | mp1i 13 | . . 3 ⊢ (𝐼 ∈ 𝐵 → (+g‘𝑀) = (+g‘𝑆)) |
26 | id 22 | . . 3 ⊢ (𝐼 ∈ 𝐵 → 𝐼 ∈ 𝐵) | |
27 | 10, 11, 1 | smndex1ibas 18065 | . . . . . 6 ⊢ 𝐼 ∈ (Base‘𝑀) |
28 | 27 | a1i 11 | . . . . 5 ⊢ (𝐼 ∈ 𝐵 → 𝐼 ∈ (Base‘𝑀)) |
29 | 10, 11, 1, 12, 8 | smndex1basss 18070 | . . . . . 6 ⊢ 𝐵 ⊆ (Base‘𝑀) |
30 | 29 | sseli 3963 | . . . . 5 ⊢ (𝑎 ∈ 𝐵 → 𝑎 ∈ (Base‘𝑀)) |
31 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
32 | 10, 31, 23 | efmndov 18046 | . . . . 5 ⊢ ((𝐼 ∈ (Base‘𝑀) ∧ 𝑎 ∈ (Base‘𝑀)) → (𝐼(+g‘𝑀)𝑎) = (𝐼 ∘ 𝑎)) |
33 | 28, 30, 32 | syl2an 597 | . . . 4 ⊢ ((𝐼 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) → (𝐼(+g‘𝑀)𝑎) = (𝐼 ∘ 𝑎)) |
34 | 10, 11, 1, 12, 8, 13 | smndex1mndlem 18074 | . . . . . 6 ⊢ (𝑎 ∈ 𝐵 → ((𝐼 ∘ 𝑎) = 𝑎 ∧ (𝑎 ∘ 𝐼) = 𝑎)) |
35 | 34 | simpld 497 | . . . . 5 ⊢ (𝑎 ∈ 𝐵 → (𝐼 ∘ 𝑎) = 𝑎) |
36 | 35 | adantl 484 | . . . 4 ⊢ ((𝐼 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) → (𝐼 ∘ 𝑎) = 𝑎) |
37 | 33, 36 | eqtrd 2856 | . . 3 ⊢ ((𝐼 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) → (𝐼(+g‘𝑀)𝑎) = 𝑎) |
38 | 10, 31, 23 | efmndov 18046 | . . . . 5 ⊢ ((𝑎 ∈ (Base‘𝑀) ∧ 𝐼 ∈ (Base‘𝑀)) → (𝑎(+g‘𝑀)𝐼) = (𝑎 ∘ 𝐼)) |
39 | 30, 28, 38 | syl2anr 598 | . . . 4 ⊢ ((𝐼 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) → (𝑎(+g‘𝑀)𝐼) = (𝑎 ∘ 𝐼)) |
40 | 34 | simprd 498 | . . . . 5 ⊢ (𝑎 ∈ 𝐵 → (𝑎 ∘ 𝐼) = 𝑎) |
41 | 40 | adantl 484 | . . . 4 ⊢ ((𝐼 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) → (𝑎 ∘ 𝐼) = 𝑎) |
42 | 39, 41 | eqtrd 2856 | . . 3 ⊢ ((𝐼 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) → (𝑎(+g‘𝑀)𝐼) = 𝑎) |
43 | 16, 25, 26, 37, 42 | grpidd 17881 | . 2 ⊢ (𝐼 ∈ 𝐵 → 𝐼 = (0g‘𝑆)) |
44 | 9, 43 | ax-mp 5 | 1 ⊢ 𝐼 = (0g‘𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∪ cun 3934 {csn 4567 ∪ ciun 4919 ↦ cmpt 5146 ∘ ccom 5559 ‘cfv 6355 (class class class)co 7156 0cc0 10537 ℕcn 11638 ℕ0cn0 11898 ..^cfzo 13034 mod cmo 13238 Basecbs 16483 ↾s cress 16484 +gcplusg 16565 0gc0g 16713 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-0g 16715 df-efmnd 18034 |
This theorem is referenced by: (None) |
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