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Mirrors > Home > MPE Home > Th. List > Mathboxes > sticksstones13 | Structured version Visualization version GIF version |
Description: Establish bijective mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 6-Oct-2024.) |
Ref | Expression |
---|---|
sticksstones13.1 | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
sticksstones13.2 | ⊢ (𝜑 → 𝐾 ∈ ℕ0) |
sticksstones13.3 | ⊢ 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))) |
sticksstones13.4 | ⊢ 𝐺 = (𝑏 ∈ 𝐵 ↦ if(𝐾 = 0, {〈1, 𝑁〉}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1)))))) |
sticksstones13.5 | ⊢ 𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} |
sticksstones13.6 | ⊢ 𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))} |
Ref | Expression |
---|---|
sticksstones13 | ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sticksstones13.1 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
2 | 1 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐾 = 0) → 𝑁 ∈ ℕ0) |
3 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐾 = 0) → 𝐾 = 0) | |
4 | sticksstones13.3 | . . 3 ⊢ 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))) | |
5 | sticksstones13.4 | . . 3 ⊢ 𝐺 = (𝑏 ∈ 𝐵 ↦ if(𝐾 = 0, {〈1, 𝑁〉}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1)))))) | |
6 | sticksstones13.5 | . . 3 ⊢ 𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} | |
7 | sticksstones13.6 | . . 3 ⊢ 𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))} | |
8 | 2, 3, 4, 5, 6, 7 | sticksstones11 42137 | . 2 ⊢ ((𝜑 ∧ 𝐾 = 0) → 𝐹:𝐴–1-1-onto→𝐵) |
9 | 1 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ) → 𝑁 ∈ ℕ0) |
10 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ) → 𝐾 ∈ ℕ) | |
11 | 9, 10, 4, 5, 6, 7 | sticksstones12 42139 | . 2 ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ) → 𝐹:𝐴–1-1-onto→𝐵) |
12 | sticksstones13.2 | . . 3 ⊢ (𝜑 → 𝐾 ∈ ℕ0) | |
13 | elnn0 12525 | . . . . 5 ⊢ (𝐾 ∈ ℕ0 ↔ (𝐾 ∈ ℕ ∨ 𝐾 = 0)) | |
14 | 13 | biimpi 216 | . . . 4 ⊢ (𝐾 ∈ ℕ0 → (𝐾 ∈ ℕ ∨ 𝐾 = 0)) |
15 | 14 | orcomd 871 | . . 3 ⊢ (𝐾 ∈ ℕ0 → (𝐾 = 0 ∨ 𝐾 ∈ ℕ)) |
16 | 12, 15 | syl 17 | . 2 ⊢ (𝜑 → (𝐾 = 0 ∨ 𝐾 ∈ ℕ)) |
17 | 8, 11, 16 | mpjaodan 960 | 1 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1536 ∈ wcel 2105 {cab 2711 ∀wral 3058 ifcif 4530 {csn 4630 〈cop 4636 class class class wbr 5147 ↦ cmpt 5230 ⟶wf 6558 –1-1-onto→wf1o 6561 ‘cfv 6562 (class class class)co 7430 0cc0 11152 1c1 11153 + caddc 11155 < clt 11292 − cmin 11489 ℕcn 12263 ℕ0cn0 12523 ...cfz 13543 Σcsu 15718 |
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-inf2 9678 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 ax-pre-sup 11230 |
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-op 4637 df-uni 4912 df-int 4951 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-se 5641 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-isom 6571 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-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-oi 9547 df-card 9976 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-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-ico 13389 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-sum 15719 |
This theorem is referenced by: sticksstones14 42141 |
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