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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 11996 | . . 3 ⊢ 2 ∈ ℕ0 | |
2 | nn0mulcl 12015 | . . . 4 ⊢ ((2 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (2 · 𝑦) ∈ ℕ0) | |
3 | smndex2dbas.d | . . . . . 6 ⊢ 𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥)) | |
4 | oveq2 7181 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (2 · 𝑥) = (2 · 𝑦)) | |
5 | 4 | cbvmptv 5134 | . . . . . 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 5035 | . . . . 5 ⊢ (𝑥 = (2 · 𝑦) → (2 ∥ 𝑥 ↔ 2 ∥ (2 · 𝑦))) | |
11 | oveq1 7180 | . . . . 5 ⊢ (𝑥 = (2 · 𝑦) → (𝑥 / 2) = ((2 · 𝑦) / 2)) | |
12 | 10, 11 | ifbieq1d 4439 | . . . 4 ⊢ (𝑥 = (2 · 𝑦) → if(2 ∥ 𝑥, (𝑥 / 2), 𝑁) = if(2 ∥ (2 · 𝑦), ((2 · 𝑦) / 2), 𝑁)) |
13 | 2, 7, 9, 12 | fmptco 6904 | . . 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 12089 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℤ) | |
16 | eqidd 2740 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → (2 · 𝑦) = (2 · 𝑦)) | |
17 | 2teven 15803 | . . . . . 6 ⊢ ((𝑦 ∈ ℤ ∧ (2 · 𝑦) = (2 · 𝑦)) → 2 ∥ (2 · 𝑦)) | |
18 | 15, 16, 17 | syl2anc 587 | . . . . 5 ⊢ (𝑦 ∈ ℕ0 → 2 ∥ (2 · 𝑦)) |
19 | 18 | iftrued 4423 | . . . 4 ⊢ (𝑦 ∈ ℕ0 → if(2 ∥ (2 · 𝑦), ((2 · 𝑦) / 2), 𝑁) = ((2 · 𝑦) / 2)) |
20 | 19 | mpteq2ia 5122 | . . 3 ⊢ (𝑦 ∈ ℕ0 ↦ if(2 ∥ (2 · 𝑦), ((2 · 𝑦) / 2), 𝑁)) = (𝑦 ∈ ℕ0 ↦ ((2 · 𝑦) / 2)) |
21 | nn0cn 11989 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℂ) | |
22 | 2cnd 11797 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → 2 ∈ ℂ) | |
23 | 2ne0 11823 | . . . . . . 7 ⊢ 2 ≠ 0 | |
24 | 23 | a1i 11 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → 2 ≠ 0) |
25 | 21, 22, 24 | divcan3d 11502 | . . . . 5 ⊢ (𝑦 ∈ ℕ0 → ((2 · 𝑦) / 2) = 𝑦) |
26 | 25 | mpteq2ia 5122 | . . . 4 ⊢ (𝑦 ∈ ℕ0 ↦ ((2 · 𝑦) / 2)) = (𝑦 ∈ ℕ0 ↦ 𝑦) |
27 | smndex2dbas.0 | . . . . 5 ⊢ 0 = (0g‘𝑀) | |
28 | nn0ex 11985 | . . . . . 6 ⊢ ℕ0 ∈ V | |
29 | smndex2dbas.m | . . . . . . 7 ⊢ 𝑀 = (EndoFMnd‘ℕ0) | |
30 | 29 | efmndid 18172 | . . . . . 6 ⊢ (ℕ0 ∈ V → ( I ↾ ℕ0) = (0g‘𝑀)) |
31 | 28, 30 | ax-mp 5 | . . . . 5 ⊢ ( I ↾ ℕ0) = (0g‘𝑀) |
32 | mptresid 5893 | . . . . 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 1542 ∈ wcel 2114 ≠ wne 2935 Vcvv 3399 ifcif 4415 class class class wbr 5031 ↦ cmpt 5111 I cid 5429 ↾ cres 5528 ∘ ccom 5530 ‘cfv 6340 (class class class)co 7173 0cc0 10618 · cmul 10623 / cdiv 11378 2c2 11774 ℕ0cn0 11979 ℤcz 12065 ∥ cdvds 15702 Basecbs 16589 0gc0g 16819 EndoFMndcefmnd 18152 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-om 7603 df-1st 7717 df-2nd 7718 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-er 8323 df-map 8442 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-div 11379 df-nn 11720 df-2 11782 df-3 11783 df-4 11784 df-5 11785 df-6 11786 df-7 11787 df-8 11788 df-9 11789 df-n0 11980 df-z 12066 df-uz 12328 df-fz 12985 df-dvds 15703 df-struct 16591 df-ndx 16592 df-slot 16593 df-base 16595 df-plusg 16684 df-tset 16690 df-0g 16821 df-efmnd 18153 |
This theorem is referenced by: (None) |
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