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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 12543 | . . 3 ⊢ 2 ∈ ℕ0 | |
| 2 | nn0mulcl 12562 | . . . 4 ⊢ ((2 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (2 · 𝑦) ∈ ℕ0) | |
| 3 | smndex2dbas.d | . . . . . 6 ⊢ 𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥)) | |
| 4 | oveq2 7439 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (2 · 𝑥) = (2 · 𝑦)) | |
| 5 | 4 | cbvmptv 5255 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 ↦ (2 · 𝑥)) = (𝑦 ∈ ℕ0 ↦ (2 · 𝑦)) |
| 6 | 3, 5 | eqtri 2765 | . . . . 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 5147 | . . . . 5 ⊢ (𝑥 = (2 · 𝑦) → (2 ∥ 𝑥 ↔ 2 ∥ (2 · 𝑦))) | |
| 11 | oveq1 7438 | . . . . 5 ⊢ (𝑥 = (2 · 𝑦) → (𝑥 / 2) = ((2 · 𝑦) / 2)) | |
| 12 | 10, 11 | ifbieq1d 4550 | . . . 4 ⊢ (𝑥 = (2 · 𝑦) → if(2 ∥ 𝑥, (𝑥 / 2), 𝑁) = if(2 ∥ (2 · 𝑦), ((2 · 𝑦) / 2), 𝑁)) |
| 13 | 2, 7, 9, 12 | fmptco 7149 | . . 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 12638 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℤ) | |
| 16 | eqidd 2738 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → (2 · 𝑦) = (2 · 𝑦)) | |
| 17 | 2teven 16392 | . . . . . 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 4533 | . . . 4 ⊢ (𝑦 ∈ ℕ0 → if(2 ∥ (2 · 𝑦), ((2 · 𝑦) / 2), 𝑁) = ((2 · 𝑦) / 2)) |
| 20 | 19 | mpteq2ia 5245 | . . 3 ⊢ (𝑦 ∈ ℕ0 ↦ if(2 ∥ (2 · 𝑦), ((2 · 𝑦) / 2), 𝑁)) = (𝑦 ∈ ℕ0 ↦ ((2 · 𝑦) / 2)) |
| 21 | nn0cn 12536 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℂ) | |
| 22 | 2cnd 12344 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → 2 ∈ ℂ) | |
| 23 | 2ne0 12370 | . . . . . . 7 ⊢ 2 ≠ 0 | |
| 24 | 23 | a1i 11 | . . . . . 6 ⊢ (𝑦 ∈ ℕ0 → 2 ≠ 0) |
| 25 | 21, 22, 24 | divcan3d 12048 | . . . . 5 ⊢ (𝑦 ∈ ℕ0 → ((2 · 𝑦) / 2) = 𝑦) |
| 26 | 25 | mpteq2ia 5245 | . . . 4 ⊢ (𝑦 ∈ ℕ0 ↦ ((2 · 𝑦) / 2)) = (𝑦 ∈ ℕ0 ↦ 𝑦) |
| 27 | smndex2dbas.0 | . . . . 5 ⊢ 0 = (0g‘𝑀) | |
| 28 | nn0ex 12532 | . . . . . 6 ⊢ ℕ0 ∈ V | |
| 29 | smndex2dbas.m | . . . . . . 7 ⊢ 𝑀 = (EndoFMnd‘ℕ0) | |
| 30 | 29 | efmndid 18901 | . . . . . 6 ⊢ (ℕ0 ∈ V → ( I ↾ ℕ0) = (0g‘𝑀)) |
| 31 | 28, 30 | ax-mp 5 | . . . . 5 ⊢ ( I ↾ ℕ0) = (0g‘𝑀) |
| 32 | mptresid 6069 | . . . . 5 ⊢ ( I ↾ ℕ0) = (𝑦 ∈ ℕ0 ↦ 𝑦) | |
| 33 | 27, 31, 32 | 3eqtr2ri 2772 | . . . 4 ⊢ (𝑦 ∈ ℕ0 ↦ 𝑦) = 0 |
| 34 | 26, 33 | eqtri 2765 | . . 3 ⊢ (𝑦 ∈ ℕ0 ↦ ((2 · 𝑦) / 2)) = 0 |
| 35 | 20, 34 | eqtri 2765 | . 2 ⊢ (𝑦 ∈ ℕ0 ↦ if(2 ∥ (2 · 𝑦), ((2 · 𝑦) / 2), 𝑁)) = 0 |
| 36 | 14, 35 | eqtri 2765 | 1 ⊢ (𝐻 ∘ 𝐷) = 0 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ifcif 4525 class class class wbr 5143 ↦ cmpt 5225 I cid 5577 ↾ cres 5687 ∘ ccom 5689 ‘cfv 6561 (class class class)co 7431 0cc0 11155 · cmul 11160 / cdiv 11920 2c2 12321 ℕ0cn0 12526 ℤcz 12613 ∥ cdvds 16290 Basecbs 17247 0gc0g 17484 EndoFMndcefmnd 18881 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-dvds 16291 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-tset 17316 df-0g 17486 df-efmnd 18882 |
| This theorem is referenced by: (None) |
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