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Mirrors > Home > MPE Home > Th. List > smndex2dlinvh | Structured version Visualization version GIF version |
Description: The halving functions 𝐻 are left inverses of the doubling function 𝐷. (Contributed by AV, 18-Feb-2024.) |
Ref | Expression |
---|---|
smndex2dbas.m | ⊢ 𝑀 = (EndoFMnd‘ℕ0) |
smndex2dbas.b | ⊢ 𝐵 = (Base‘𝑀) |
smndex2dbas.0 | ⊢ 0 = (0g‘𝑀) |
smndex2dbas.d | ⊢ 𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥)) |
smndex2hbas.n | ⊢ 𝑁 ∈ ℕ0 |
smndex2hbas.h | ⊢ 𝐻 = (𝑥 ∈ ℕ0 ↦ if(2 ∥ 𝑥, (𝑥 / 2), 𝑁)) |
Ref | Expression |
---|---|
smndex2dlinvh | ⊢ (𝐻 ∘ 𝐷) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn0 12250 | . . 3 ⊢ 2 ∈ ℕ0 | |
2 | nn0mulcl 12269 | . . . 4 ⊢ ((2 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (2 · 𝑦) ∈ ℕ0) | |
3 | smndex2dbas.d | . . . . . 6 ⊢ 𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥)) | |
4 | oveq2 7283 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (2 · 𝑥) = (2 · 𝑦)) | |
5 | 4 | cbvmptv 5187 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 ↦ (2 · 𝑥)) = (𝑦 ∈ ℕ0 ↦ (2 · 𝑦)) |
6 | 3, 5 | eqtri 2766 | . . . . 5 ⊢ 𝐷 = (𝑦 ∈ ℕ0 ↦ (2 · 𝑦)) |
7 | 6 | a1i 11 | . . . 4 ⊢ (2 ∈ ℕ0 → 𝐷 = (𝑦 ∈ ℕ0 ↦ (2 · 𝑦))) |
8 | smndex2hbas.h | . . . . 5 ⊢ 𝐻 = (𝑥 ∈ ℕ0 ↦ if(2 ∥ 𝑥, (𝑥 / 2), 𝑁)) | |
9 | 8 | a1i 11 | . . . 4 ⊢ (2 ∈ ℕ0 → 𝐻 = (𝑥 ∈ ℕ0 ↦ if(2 ∥ 𝑥, (𝑥 / 2), 𝑁))) |
10 | breq2 5078 | . . . . 5 ⊢ (𝑥 = (2 · 𝑦) → (2 ∥ 𝑥 ↔ 2 ∥ (2 · 𝑦))) | |
11 | oveq1 7282 | . . . . 5 ⊢ (𝑥 = (2 · 𝑦) → (𝑥 / 2) = ((2 · 𝑦) / 2)) | |
12 | 10, 11 | ifbieq1d 4483 | . . . 4 ⊢ (𝑥 = (2 · 𝑦) → if(2 ∥ 𝑥, (𝑥 / 2), 𝑁) = if(2 ∥ (2 · 𝑦), ((2 · 𝑦) / 2), 𝑁)) |
13 | 2, 7, 9, 12 | fmptco 7001 | . . 3 ⊢ (2 ∈ ℕ0 → (𝐻 ∘ 𝐷) = (𝑦 ∈ ℕ0 ↦ if(2 ∥ (2 · 𝑦), ((2 · 𝑦) / 2), 𝑁))) |
14 | 1, 13 | ax-mp 5 | . 2 ⊢ (𝐻 ∘ 𝐷) = (𝑦 ∈ ℕ0 ↦ if(2 ∥ (2 · 𝑦), ((2 · 𝑦) / 2), 𝑁)) |
15 | nn0z 12343 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℤ) | |
16 | eqidd 2739 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → (2 · 𝑦) = (2 · 𝑦)) | |
17 | 2teven 16064 | . . . . . 6 ⊢ ((𝑦 ∈ ℤ ∧ (2 · 𝑦) = (2 · 𝑦)) → 2 ∥ (2 · 𝑦)) | |
18 | 15, 16, 17 | syl2anc 584 | . . . . 5 ⊢ (𝑦 ∈ ℕ0 → 2 ∥ (2 · 𝑦)) |
19 | 18 | iftrued 4467 | . . . 4 ⊢ (𝑦 ∈ ℕ0 → if(2 ∥ (2 · 𝑦), ((2 · 𝑦) / 2), 𝑁) = ((2 · 𝑦) / 2)) |
20 | 19 | mpteq2ia 5177 | . . 3 ⊢ (𝑦 ∈ ℕ0 ↦ if(2 ∥ (2 · 𝑦), ((2 · 𝑦) / 2), 𝑁)) = (𝑦 ∈ ℕ0 ↦ ((2 · 𝑦) / 2)) |
21 | nn0cn 12243 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℂ) | |
22 | 2cnd 12051 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → 2 ∈ ℂ) | |
23 | 2ne0 12077 | . . . . . . 7 ⊢ 2 ≠ 0 | |
24 | 23 | a1i 11 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → 2 ≠ 0) |
25 | 21, 22, 24 | divcan3d 11756 | . . . . 5 ⊢ (𝑦 ∈ ℕ0 → ((2 · 𝑦) / 2) = 𝑦) |
26 | 25 | mpteq2ia 5177 | . . . 4 ⊢ (𝑦 ∈ ℕ0 ↦ ((2 · 𝑦) / 2)) = (𝑦 ∈ ℕ0 ↦ 𝑦) |
27 | smndex2dbas.0 | . . . . 5 ⊢ 0 = (0g‘𝑀) | |
28 | nn0ex 12239 | . . . . . 6 ⊢ ℕ0 ∈ V | |
29 | smndex2dbas.m | . . . . . . 7 ⊢ 𝑀 = (EndoFMnd‘ℕ0) | |
30 | 29 | efmndid 18527 | . . . . . 6 ⊢ (ℕ0 ∈ V → ( I ↾ ℕ0) = (0g‘𝑀)) |
31 | 28, 30 | ax-mp 5 | . . . . 5 ⊢ ( I ↾ ℕ0) = (0g‘𝑀) |
32 | mptresid 5958 | . . . . 5 ⊢ ( I ↾ ℕ0) = (𝑦 ∈ ℕ0 ↦ 𝑦) | |
33 | 27, 31, 32 | 3eqtr2ri 2773 | . . . 4 ⊢ (𝑦 ∈ ℕ0 ↦ 𝑦) = 0 |
34 | 26, 33 | eqtri 2766 | . . 3 ⊢ (𝑦 ∈ ℕ0 ↦ ((2 · 𝑦) / 2)) = 0 |
35 | 20, 34 | eqtri 2766 | . 2 ⊢ (𝑦 ∈ ℕ0 ↦ if(2 ∥ (2 · 𝑦), ((2 · 𝑦) / 2), 𝑁)) = 0 |
36 | 14, 35 | eqtri 2766 | 1 ⊢ (𝐻 ∘ 𝐷) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ∈ wcel 2106 ≠ wne 2943 Vcvv 3432 ifcif 4459 class class class wbr 5074 ↦ cmpt 5157 I cid 5488 ↾ cres 5591 ∘ ccom 5593 ‘cfv 6433 (class class class)co 7275 0cc0 10871 · cmul 10876 / cdiv 11632 2c2 12028 ℕ0cn0 12233 ℤcz 12319 ∥ cdvds 15963 Basecbs 16912 0gc0g 17150 EndoFMndcefmnd 18507 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-dvds 15964 df-struct 16848 df-slot 16883 df-ndx 16895 df-base 16913 df-plusg 16975 df-tset 16981 df-0g 17152 df-efmnd 18508 |
This theorem is referenced by: (None) |
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