Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 12349 | . . . . . . 7 ⊢ ℕ0 ∈ V | |
3 | 2 | mptex 7164 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) ∈ V |
4 | 1, 3 | eqeltri 2834 | . . . . 5 ⊢ 𝐼 ∈ V |
5 | 4 | snid 4617 | . . . 4 ⊢ 𝐼 ∈ {𝐼} |
6 | elun1 4131 | . . . 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 2837 | . 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 18646 | . . . . 5 ⊢ (Base‘𝑆) = 𝐵 |
15 | 14 | eqcomi 2746 | . . . 4 ⊢ 𝐵 = (Base‘𝑆) |
16 | 15 | a1i 11 | . . 3 ⊢ (𝐼 ∈ 𝐵 → 𝐵 = (Base‘𝑆)) |
17 | snex 5383 | . . . . . 6 ⊢ {𝐼} ∈ V | |
18 | ovex 7379 | . . . . . . 7 ⊢ (0..^𝑁) ∈ V | |
19 | snex 5383 | . . . . . . 7 ⊢ {(𝐺‘𝑛)} ∈ V | |
20 | 18, 19 | iunex 7888 | . . . . . 6 ⊢ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)} ∈ V |
21 | 17, 20 | unex 7667 | . . . . 5 ⊢ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) ∈ V |
22 | 8, 21 | eqeltri 2834 | . . . 4 ⊢ 𝐵 ∈ V |
23 | eqid 2737 | . . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
24 | 13, 23 | ressplusg 17102 | . . . 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 18640 | . . . . . 6 ⊢ 𝐼 ∈ (Base‘𝑀) |
28 | 27 | a1i 11 | . . . . 5 ⊢ (𝐼 ∈ 𝐵 → 𝐼 ∈ (Base‘𝑀)) |
29 | 10, 11, 1, 12, 8 | smndex1basss 18645 | . . . . . 6 ⊢ 𝐵 ⊆ (Base‘𝑀) |
30 | 29 | sseli 3935 | . . . . 5 ⊢ (𝑎 ∈ 𝐵 → 𝑎 ∈ (Base‘𝑀)) |
31 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
32 | 10, 31, 23 | efmndov 18621 | . . . . 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 18649 | . . . . . 6 ⊢ (𝑎 ∈ 𝐵 → ((𝐼 ∘ 𝑎) = 𝑎 ∧ (𝑎 ∘ 𝐼) = 𝑎)) |
35 | 34 | simpld 496 | . . . . 5 ⊢ (𝑎 ∈ 𝐵 → (𝐼 ∘ 𝑎) = 𝑎) |
36 | 35 | adantl 483 | . . . 4 ⊢ ((𝐼 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) → (𝐼 ∘ 𝑎) = 𝑎) |
37 | 33, 36 | eqtrd 2777 | . . 3 ⊢ ((𝐼 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) → (𝐼(+g‘𝑀)𝑎) = 𝑎) |
38 | 10, 31, 23 | efmndov 18621 | . . . . 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 497 | . . . . 5 ⊢ (𝑎 ∈ 𝐵 → (𝑎 ∘ 𝐼) = 𝑎) |
41 | 40 | adantl 483 | . . . 4 ⊢ ((𝐼 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) → (𝑎 ∘ 𝐼) = 𝑎) |
42 | 39, 41 | eqtrd 2777 | . . 3 ⊢ ((𝐼 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) → (𝑎(+g‘𝑀)𝐼) = 𝑎) |
43 | 16, 25, 26, 37, 42 | grpidd 18457 | . 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 397 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ∪ cun 3903 {csn 4581 ∪ ciun 4949 ↦ cmpt 5183 ∘ ccom 5631 ‘cfv 6488 (class class class)co 7346 0cc0 10981 ℕcn 12083 ℕ0cn0 12343 ..^cfzo 13492 mod cmo 13699 Basecbs 17014 ↾s cress 17043 +gcplusg 17064 0gc0g 17252 EndoFMndcefmnd 18608 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5237 ax-sep 5251 ax-nul 5258 ax-pow 5315 ax-pr 5379 ax-un 7659 ax-cnex 11037 ax-resscn 11038 ax-1cn 11039 ax-icn 11040 ax-addcl 11041 ax-addrcl 11042 ax-mulcl 11043 ax-mulrcl 11044 ax-mulcom 11045 ax-addass 11046 ax-mulass 11047 ax-distr 11048 ax-i2m1 11049 ax-1ne0 11050 ax-1rid 11051 ax-rnegex 11052 ax-rrecex 11053 ax-cnre 11054 ax-pre-lttri 11055 ax-pre-lttrn 11056 ax-pre-ltadd 11057 ax-pre-mulgt0 11058 ax-pre-sup 11059 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3735 df-csb 3851 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3924 df-nul 4278 df-if 4482 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4861 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5184 df-tr 5218 df-id 5525 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5582 df-we 5584 df-xp 5633 df-rel 5634 df-cnv 5635 df-co 5636 df-dm 5637 df-rn 5638 df-res 5639 df-ima 5640 df-pred 6246 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6440 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-riota 7302 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7790 df-1st 7908 df-2nd 7909 df-frecs 8176 df-wrecs 8207 df-recs 8281 df-rdg 8320 df-1o 8376 df-er 8578 df-map 8697 df-en 8814 df-dom 8815 df-sdom 8816 df-fin 8817 df-sup 9308 df-inf 9309 df-pnf 11121 df-mnf 11122 df-xr 11123 df-ltxr 11124 df-le 11125 df-sub 11317 df-neg 11318 df-div 11743 df-nn 12084 df-2 12146 df-3 12147 df-4 12148 df-5 12149 df-6 12150 df-7 12151 df-8 12152 df-9 12153 df-n0 12344 df-z 12430 df-uz 12693 df-rp 12841 df-fz 13350 df-fzo 13493 df-fl 13622 df-mod 13700 df-struct 16950 df-sets 16967 df-slot 16985 df-ndx 16997 df-base 17015 df-ress 17044 df-plusg 17077 df-tset 17083 df-0g 17254 df-efmnd 18609 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |