Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > smndex1gid | Structured version Visualization version GIF version |
Description: The composition of a constant function (𝐺‘𝐾) with another endofunction on ℕ0 results in (𝐺‘𝐾) itself. (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)) |
Ref | Expression |
---|---|
smndex1gid | ⊢ ((𝐹 ∈ (Base‘𝑀) ∧ 𝐾 ∈ (0..^𝑁)) → ((𝐺‘𝐾) ∘ 𝐹) = (𝐺‘𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smndex1ibas.g | . . . . . . . 8 ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛)) | |
2 | 1 | a1i 11 | . . . . . . 7 ⊢ (𝐾 ∈ (0..^𝑁) → 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))) |
3 | id 22 | . . . . . . . . 9 ⊢ (𝑛 = 𝐾 → 𝑛 = 𝐾) | |
4 | 3 | mpteq2dv 5155 | . . . . . . . 8 ⊢ (𝑛 = 𝐾 → (𝑥 ∈ ℕ0 ↦ 𝑛) = (𝑥 ∈ ℕ0 ↦ 𝐾)) |
5 | 4 | adantl 484 | . . . . . . 7 ⊢ ((𝐾 ∈ (0..^𝑁) ∧ 𝑛 = 𝐾) → (𝑥 ∈ ℕ0 ↦ 𝑛) = (𝑥 ∈ ℕ0 ↦ 𝐾)) |
6 | id 22 | . . . . . . 7 ⊢ (𝐾 ∈ (0..^𝑁) → 𝐾 ∈ (0..^𝑁)) | |
7 | nn0ex 11897 | . . . . . . . . 9 ⊢ ℕ0 ∈ V | |
8 | 7 | mptex 6979 | . . . . . . . 8 ⊢ (𝑥 ∈ ℕ0 ↦ 𝐾) ∈ V |
9 | 8 | a1i 11 | . . . . . . 7 ⊢ (𝐾 ∈ (0..^𝑁) → (𝑥 ∈ ℕ0 ↦ 𝐾) ∈ V) |
10 | 2, 5, 6, 9 | fvmptd 6768 | . . . . . 6 ⊢ (𝐾 ∈ (0..^𝑁) → (𝐺‘𝐾) = (𝑥 ∈ ℕ0 ↦ 𝐾)) |
11 | 10 | adantl 484 | . . . . 5 ⊢ ((𝐹 ∈ (Base‘𝑀) ∧ 𝐾 ∈ (0..^𝑁)) → (𝐺‘𝐾) = (𝑥 ∈ ℕ0 ↦ 𝐾)) |
12 | 11 | adantr 483 | . . . 4 ⊢ (((𝐹 ∈ (Base‘𝑀) ∧ 𝐾 ∈ (0..^𝑁)) ∧ 𝑦 ∈ ℕ0) → (𝐺‘𝐾) = (𝑥 ∈ ℕ0 ↦ 𝐾)) |
13 | eqidd 2821 | . . . 4 ⊢ ((((𝐹 ∈ (Base‘𝑀) ∧ 𝐾 ∈ (0..^𝑁)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 = (𝐹‘𝑦)) → 𝐾 = 𝐾) | |
14 | smndex1ibas.m | . . . . . . . 8 ⊢ 𝑀 = (EndoFMnd‘ℕ0) | |
15 | eqid 2820 | . . . . . . . 8 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
16 | 14, 15 | efmndbasf 18033 | . . . . . . 7 ⊢ (𝐹 ∈ (Base‘𝑀) → 𝐹:ℕ0⟶ℕ0) |
17 | ffvelrn 6842 | . . . . . . . 8 ⊢ ((𝐹:ℕ0⟶ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝐹‘𝑦) ∈ ℕ0) | |
18 | 17 | ex 415 | . . . . . . 7 ⊢ (𝐹:ℕ0⟶ℕ0 → (𝑦 ∈ ℕ0 → (𝐹‘𝑦) ∈ ℕ0)) |
19 | 16, 18 | syl 17 | . . . . . 6 ⊢ (𝐹 ∈ (Base‘𝑀) → (𝑦 ∈ ℕ0 → (𝐹‘𝑦) ∈ ℕ0)) |
20 | 19 | adantr 483 | . . . . 5 ⊢ ((𝐹 ∈ (Base‘𝑀) ∧ 𝐾 ∈ (0..^𝑁)) → (𝑦 ∈ ℕ0 → (𝐹‘𝑦) ∈ ℕ0)) |
21 | 20 | imp 409 | . . . 4 ⊢ (((𝐹 ∈ (Base‘𝑀) ∧ 𝐾 ∈ (0..^𝑁)) ∧ 𝑦 ∈ ℕ0) → (𝐹‘𝑦) ∈ ℕ0) |
22 | simplr 767 | . . . 4 ⊢ (((𝐹 ∈ (Base‘𝑀) ∧ 𝐾 ∈ (0..^𝑁)) ∧ 𝑦 ∈ ℕ0) → 𝐾 ∈ (0..^𝑁)) | |
23 | 12, 13, 21, 22 | fvmptd 6768 | . . 3 ⊢ (((𝐹 ∈ (Base‘𝑀) ∧ 𝐾 ∈ (0..^𝑁)) ∧ 𝑦 ∈ ℕ0) → ((𝐺‘𝐾)‘(𝐹‘𝑦)) = 𝐾) |
24 | 23 | mpteq2dva 5154 | . 2 ⊢ ((𝐹 ∈ (Base‘𝑀) ∧ 𝐾 ∈ (0..^𝑁)) → (𝑦 ∈ ℕ0 ↦ ((𝐺‘𝐾)‘(𝐹‘𝑦))) = (𝑦 ∈ ℕ0 ↦ 𝐾)) |
25 | smndex1ibas.n | . . . . 5 ⊢ 𝑁 ∈ ℕ | |
26 | smndex1ibas.i | . . . . 5 ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁)) | |
27 | 14, 25, 26, 1 | smndex1gbas 18060 | . . . 4 ⊢ (𝐾 ∈ (0..^𝑁) → (𝐺‘𝐾) ∈ (Base‘𝑀)) |
28 | 14, 15 | efmndbasf 18033 | . . . 4 ⊢ ((𝐺‘𝐾) ∈ (Base‘𝑀) → (𝐺‘𝐾):ℕ0⟶ℕ0) |
29 | 27, 28 | syl 17 | . . 3 ⊢ (𝐾 ∈ (0..^𝑁) → (𝐺‘𝐾):ℕ0⟶ℕ0) |
30 | fcompt 6888 | . . 3 ⊢ (((𝐺‘𝐾):ℕ0⟶ℕ0 ∧ 𝐹:ℕ0⟶ℕ0) → ((𝐺‘𝐾) ∘ 𝐹) = (𝑦 ∈ ℕ0 ↦ ((𝐺‘𝐾)‘(𝐹‘𝑦)))) | |
31 | 29, 16, 30 | syl2anr 598 | . 2 ⊢ ((𝐹 ∈ (Base‘𝑀) ∧ 𝐾 ∈ (0..^𝑁)) → ((𝐺‘𝐾) ∘ 𝐹) = (𝑦 ∈ ℕ0 ↦ ((𝐺‘𝐾)‘(𝐹‘𝑦)))) |
32 | eqidd 2821 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝐾 = 𝐾) | |
33 | 32 | cbvmptv 5162 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 ↦ 𝐾) = (𝑦 ∈ ℕ0 ↦ 𝐾) |
34 | 4, 33 | syl6eq 2871 | . . . . 5 ⊢ (𝑛 = 𝐾 → (𝑥 ∈ ℕ0 ↦ 𝑛) = (𝑦 ∈ ℕ0 ↦ 𝐾)) |
35 | 34 | adantl 484 | . . . 4 ⊢ ((𝐾 ∈ (0..^𝑁) ∧ 𝑛 = 𝐾) → (𝑥 ∈ ℕ0 ↦ 𝑛) = (𝑦 ∈ ℕ0 ↦ 𝐾)) |
36 | 7 | mptex 6979 | . . . . 5 ⊢ (𝑦 ∈ ℕ0 ↦ 𝐾) ∈ V |
37 | 36 | a1i 11 | . . . 4 ⊢ (𝐾 ∈ (0..^𝑁) → (𝑦 ∈ ℕ0 ↦ 𝐾) ∈ V) |
38 | 2, 35, 6, 37 | fvmptd 6768 | . . 3 ⊢ (𝐾 ∈ (0..^𝑁) → (𝐺‘𝐾) = (𝑦 ∈ ℕ0 ↦ 𝐾)) |
39 | 38 | adantl 484 | . 2 ⊢ ((𝐹 ∈ (Base‘𝑀) ∧ 𝐾 ∈ (0..^𝑁)) → (𝐺‘𝐾) = (𝑦 ∈ ℕ0 ↦ 𝐾)) |
40 | 24, 31, 39 | 3eqtr4d 2865 | 1 ⊢ ((𝐹 ∈ (Base‘𝑀) ∧ 𝐾 ∈ (0..^𝑁)) → ((𝐺‘𝐾) ∘ 𝐹) = (𝐺‘𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3491 ↦ cmpt 5139 ∘ ccom 5552 ⟶wf 6344 ‘cfv 6348 (class class class)co 7149 0cc0 10530 ℕcn 11631 ℕ0cn0 11891 ..^cfzo 13030 mod cmo 13234 Basecbs 16476 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 |
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-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-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 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-fz 12890 df-fzo 13031 df-struct 16478 df-ndx 16479 df-slot 16480 df-base 16482 df-plusg 16571 df-tset 16577 df-efmnd 18027 |
This theorem is referenced by: smndex1mgm 18065 smndex1mndlem 18067 |
Copyright terms: Public domain | W3C validator |