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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 12476 | . . 3 ⊢ 2 ∈ ℕ0 | |
2 | nn0mulcl 12495 | . . . 4 ⊢ ((2 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (2 · 𝑦) ∈ ℕ0) | |
3 | smndex2dbas.d | . . . . . 6 ⊢ 𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥)) | |
4 | oveq2 7404 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (2 · 𝑥) = (2 · 𝑦)) | |
5 | 4 | cbvmptv 5257 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 ↦ (2 · 𝑥)) = (𝑦 ∈ ℕ0 ↦ (2 · 𝑦)) |
6 | 3, 5 | eqtri 2761 | . . . . 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 5148 | . . . . 5 ⊢ (𝑥 = (2 · 𝑦) → (2 ∥ 𝑥 ↔ 2 ∥ (2 · 𝑦))) | |
11 | oveq1 7403 | . . . . 5 ⊢ (𝑥 = (2 · 𝑦) → (𝑥 / 2) = ((2 · 𝑦) / 2)) | |
12 | 10, 11 | ifbieq1d 4548 | . . . 4 ⊢ (𝑥 = (2 · 𝑦) → if(2 ∥ 𝑥, (𝑥 / 2), 𝑁) = if(2 ∥ (2 · 𝑦), ((2 · 𝑦) / 2), 𝑁)) |
13 | 2, 7, 9, 12 | fmptco 7114 | . . 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 12570 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℤ) | |
16 | eqidd 2734 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → (2 · 𝑦) = (2 · 𝑦)) | |
17 | 2teven 16285 | . . . . . 6 ⊢ ((𝑦 ∈ ℤ ∧ (2 · 𝑦) = (2 · 𝑦)) → 2 ∥ (2 · 𝑦)) | |
18 | 15, 16, 17 | syl2anc 585 | . . . . 5 ⊢ (𝑦 ∈ ℕ0 → 2 ∥ (2 · 𝑦)) |
19 | 18 | iftrued 4532 | . . . 4 ⊢ (𝑦 ∈ ℕ0 → if(2 ∥ (2 · 𝑦), ((2 · 𝑦) / 2), 𝑁) = ((2 · 𝑦) / 2)) |
20 | 19 | mpteq2ia 5247 | . . 3 ⊢ (𝑦 ∈ ℕ0 ↦ if(2 ∥ (2 · 𝑦), ((2 · 𝑦) / 2), 𝑁)) = (𝑦 ∈ ℕ0 ↦ ((2 · 𝑦) / 2)) |
21 | nn0cn 12469 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℂ) | |
22 | 2cnd 12277 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → 2 ∈ ℂ) | |
23 | 2ne0 12303 | . . . . . . 7 ⊢ 2 ≠ 0 | |
24 | 23 | a1i 11 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → 2 ≠ 0) |
25 | 21, 22, 24 | divcan3d 11982 | . . . . 5 ⊢ (𝑦 ∈ ℕ0 → ((2 · 𝑦) / 2) = 𝑦) |
26 | 25 | mpteq2ia 5247 | . . . 4 ⊢ (𝑦 ∈ ℕ0 ↦ ((2 · 𝑦) / 2)) = (𝑦 ∈ ℕ0 ↦ 𝑦) |
27 | smndex2dbas.0 | . . . . 5 ⊢ 0 = (0g‘𝑀) | |
28 | nn0ex 12465 | . . . . . 6 ⊢ ℕ0 ∈ V | |
29 | smndex2dbas.m | . . . . . . 7 ⊢ 𝑀 = (EndoFMnd‘ℕ0) | |
30 | 29 | efmndid 18756 | . . . . . 6 ⊢ (ℕ0 ∈ V → ( I ↾ ℕ0) = (0g‘𝑀)) |
31 | 28, 30 | ax-mp 5 | . . . . 5 ⊢ ( I ↾ ℕ0) = (0g‘𝑀) |
32 | mptresid 6043 | . . . . 5 ⊢ ( I ↾ ℕ0) = (𝑦 ∈ ℕ0 ↦ 𝑦) | |
33 | 27, 31, 32 | 3eqtr2ri 2768 | . . . 4 ⊢ (𝑦 ∈ ℕ0 ↦ 𝑦) = 0 |
34 | 26, 33 | eqtri 2761 | . . 3 ⊢ (𝑦 ∈ ℕ0 ↦ ((2 · 𝑦) / 2)) = 0 |
35 | 20, 34 | eqtri 2761 | . 2 ⊢ (𝑦 ∈ ℕ0 ↦ if(2 ∥ (2 · 𝑦), ((2 · 𝑦) / 2), 𝑁)) = 0 |
36 | 14, 35 | eqtri 2761 | 1 ⊢ (𝐻 ∘ 𝐷) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 ≠ wne 2941 Vcvv 3475 ifcif 4524 class class class wbr 5144 ↦ cmpt 5227 I cid 5569 ↾ cres 5674 ∘ ccom 5676 ‘cfv 6535 (class class class)co 7396 0cc0 11097 · cmul 11102 / cdiv 11858 2c2 12254 ℕ0cn0 12459 ℤcz 12545 ∥ cdvds 16184 Basecbs 17131 0gc0g 17372 EndoFMndcefmnd 18736 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8691 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-div 11859 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-9 12269 df-n0 12460 df-z 12546 df-uz 12810 df-fz 13472 df-dvds 16185 df-struct 17067 df-slot 17102 df-ndx 17114 df-base 17132 df-plusg 17197 df-tset 17203 df-0g 17374 df-efmnd 18737 |
This theorem is referenced by: (None) |
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