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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 11897 | . . . . . . 7 ⊢ ℕ0 ∈ V | |
3 | 2 | mptex 6979 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) ∈ V |
4 | 1, 3 | eqeltri 2908 | . . . . 5 ⊢ 𝐼 ∈ V |
5 | 4 | snid 4594 | . . . 4 ⊢ 𝐼 ∈ {𝐼} |
6 | elun1 4145 | . . . 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 2911 | . 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 18064 | . . . . 5 ⊢ (Base‘𝑆) = 𝐵 |
15 | 14 | eqcomi 2829 | . . . 4 ⊢ 𝐵 = (Base‘𝑆) |
16 | 15 | a1i 11 | . . 3 ⊢ (𝐼 ∈ 𝐵 → 𝐵 = (Base‘𝑆)) |
17 | snex 5325 | . . . . . 6 ⊢ {𝐼} ∈ V | |
18 | ovex 7182 | . . . . . . 7 ⊢ (0..^𝑁) ∈ V | |
19 | snex 5325 | . . . . . . 7 ⊢ {(𝐺‘𝑛)} ∈ V | |
20 | 18, 19 | iunex 7662 | . . . . . 6 ⊢ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)} ∈ V |
21 | 17, 20 | unex 7462 | . . . . 5 ⊢ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) ∈ V |
22 | 8, 21 | eqeltri 2908 | . . . 4 ⊢ 𝐵 ∈ V |
23 | eqid 2820 | . . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
24 | 13, 23 | ressplusg 16605 | . . . 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 18058 | . . . . . 6 ⊢ 𝐼 ∈ (Base‘𝑀) |
28 | 27 | a1i 11 | . . . . 5 ⊢ (𝐼 ∈ 𝐵 → 𝐼 ∈ (Base‘𝑀)) |
29 | 10, 11, 1, 12, 8 | smndex1basss 18063 | . . . . . 6 ⊢ 𝐵 ⊆ (Base‘𝑀) |
30 | 29 | sseli 3956 | . . . . 5 ⊢ (𝑎 ∈ 𝐵 → 𝑎 ∈ (Base‘𝑀)) |
31 | eqid 2820 | . . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
32 | 10, 31, 23 | efmndov 18039 | . . . . 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 18067 | . . . . . 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 2855 | . . 3 ⊢ ((𝐼 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) → (𝐼(+g‘𝑀)𝑎) = 𝑎) |
38 | 10, 31, 23 | efmndov 18039 | . . . . 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 2855 | . . 3 ⊢ ((𝐼 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) → (𝑎(+g‘𝑀)𝐼) = 𝑎) |
43 | 16, 25, 26, 37, 42 | grpidd 17874 | . 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 1536 ∈ wcel 2113 Vcvv 3491 ∪ cun 3927 {csn 4560 ∪ ciun 4912 ↦ cmpt 5139 ∘ ccom 5552 ‘cfv 6348 (class class class)co 7149 0cc0 10530 ℕcn 11631 ℕ0cn0 11891 ..^cfzo 13030 mod cmo 13234 Basecbs 16476 ↾s cress 16477 +gcplusg 16558 0gc0g 16706 EndoFMndcefmnd 18026 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-sup 8899 df-inf 8900 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12890 df-fzo 13031 df-fl 13159 df-mod 13235 df-struct 16478 df-ndx 16479 df-slot 16480 df-base 16482 df-sets 16483 df-ress 16484 df-plusg 16571 df-tset 16577 df-0g 16708 df-efmnd 18027 |
This theorem is referenced by: (None) |
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