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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 12540 | . . 3 ⊢ 2 ∈ ℕ0 | |
2 | nn0mulcl 12559 | . . . 4 ⊢ ((2 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (2 · 𝑦) ∈ ℕ0) | |
3 | smndex2dbas.d | . . . . . 6 ⊢ 𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥)) | |
4 | oveq2 7438 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (2 · 𝑥) = (2 · 𝑦)) | |
5 | 4 | cbvmptv 5260 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 ↦ (2 · 𝑥)) = (𝑦 ∈ ℕ0 ↦ (2 · 𝑦)) |
6 | 3, 5 | eqtri 2762 | . . . . 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 5151 | . . . . 5 ⊢ (𝑥 = (2 · 𝑦) → (2 ∥ 𝑥 ↔ 2 ∥ (2 · 𝑦))) | |
11 | oveq1 7437 | . . . . 5 ⊢ (𝑥 = (2 · 𝑦) → (𝑥 / 2) = ((2 · 𝑦) / 2)) | |
12 | 10, 11 | ifbieq1d 4554 | . . . 4 ⊢ (𝑥 = (2 · 𝑦) → if(2 ∥ 𝑥, (𝑥 / 2), 𝑁) = if(2 ∥ (2 · 𝑦), ((2 · 𝑦) / 2), 𝑁)) |
13 | 2, 7, 9, 12 | fmptco 7148 | . . 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 12635 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℤ) | |
16 | eqidd 2735 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → (2 · 𝑦) = (2 · 𝑦)) | |
17 | 2teven 16388 | . . . . . 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 4538 | . . . 4 ⊢ (𝑦 ∈ ℕ0 → if(2 ∥ (2 · 𝑦), ((2 · 𝑦) / 2), 𝑁) = ((2 · 𝑦) / 2)) |
20 | 19 | mpteq2ia 5250 | . . 3 ⊢ (𝑦 ∈ ℕ0 ↦ if(2 ∥ (2 · 𝑦), ((2 · 𝑦) / 2), 𝑁)) = (𝑦 ∈ ℕ0 ↦ ((2 · 𝑦) / 2)) |
21 | nn0cn 12533 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℂ) | |
22 | 2cnd 12341 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → 2 ∈ ℂ) | |
23 | 2ne0 12367 | . . . . . . 7 ⊢ 2 ≠ 0 | |
24 | 23 | a1i 11 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → 2 ≠ 0) |
25 | 21, 22, 24 | divcan3d 12045 | . . . . 5 ⊢ (𝑦 ∈ ℕ0 → ((2 · 𝑦) / 2) = 𝑦) |
26 | 25 | mpteq2ia 5250 | . . . 4 ⊢ (𝑦 ∈ ℕ0 ↦ ((2 · 𝑦) / 2)) = (𝑦 ∈ ℕ0 ↦ 𝑦) |
27 | smndex2dbas.0 | . . . . 5 ⊢ 0 = (0g‘𝑀) | |
28 | nn0ex 12529 | . . . . . 6 ⊢ ℕ0 ∈ V | |
29 | smndex2dbas.m | . . . . . . 7 ⊢ 𝑀 = (EndoFMnd‘ℕ0) | |
30 | 29 | efmndid 18913 | . . . . . 6 ⊢ (ℕ0 ∈ V → ( I ↾ ℕ0) = (0g‘𝑀)) |
31 | 28, 30 | ax-mp 5 | . . . . 5 ⊢ ( I ↾ ℕ0) = (0g‘𝑀) |
32 | mptresid 6070 | . . . . 5 ⊢ ( I ↾ ℕ0) = (𝑦 ∈ ℕ0 ↦ 𝑦) | |
33 | 27, 31, 32 | 3eqtr2ri 2769 | . . . 4 ⊢ (𝑦 ∈ ℕ0 ↦ 𝑦) = 0 |
34 | 26, 33 | eqtri 2762 | . . 3 ⊢ (𝑦 ∈ ℕ0 ↦ ((2 · 𝑦) / 2)) = 0 |
35 | 20, 34 | eqtri 2762 | . 2 ⊢ (𝑦 ∈ ℕ0 ↦ if(2 ∥ (2 · 𝑦), ((2 · 𝑦) / 2), 𝑁)) = 0 |
36 | 14, 35 | eqtri 2762 | 1 ⊢ (𝐻 ∘ 𝐷) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2105 ≠ wne 2937 Vcvv 3477 ifcif 4530 class class class wbr 5147 ↦ cmpt 5230 I cid 5581 ↾ cres 5690 ∘ ccom 5692 ‘cfv 6562 (class class class)co 7430 0cc0 11152 · cmul 11157 / cdiv 11917 2c2 12318 ℕ0cn0 12523 ℤcz 12610 ∥ cdvds 16286 Basecbs 17244 0gc0g 17485 EndoFMndcefmnd 18893 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-dvds 16287 df-struct 17180 df-slot 17215 df-ndx 17227 df-base 17245 df-plusg 17310 df-tset 17316 df-0g 17487 df-efmnd 18894 |
This theorem is referenced by: (None) |
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