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Mirrors > Home > MPE Home > Th. List > Mathboxes > sticksstones14 | Structured version Visualization version GIF version |
Description: Sticks and stones with definitions as hypotheses. (Contributed by metakunt, 7-Oct-2024.) |
Ref | Expression |
---|---|
sticksstones14.1 | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
sticksstones14.2 | ⊢ (𝜑 → 𝐾 ∈ ℕ0) |
sticksstones14.3 | ⊢ 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))) |
sticksstones14.4 | ⊢ 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)))))) |
sticksstones14.5 | ⊢ 𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} |
sticksstones14.6 | ⊢ 𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))} |
Ref | Expression |
---|---|
sticksstones14 | ⊢ (𝜑 → (♯‘𝐴) = ((𝑁 + 𝐾)C𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sticksstones14.5 | . . . . 5 ⊢ 𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}) |
3 | simpl 486 | . . . . . . 7 ⊢ ((𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁) → 𝑔:(1...(𝐾 + 1))⟶ℕ0) | |
4 | 3 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ((𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁) → 𝑔:(1...(𝐾 + 1))⟶ℕ0)) |
5 | 4 | ss2abdv 3963 | . . . . 5 ⊢ (𝜑 → {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} ⊆ {𝑔 ∣ 𝑔:(1...(𝐾 + 1))⟶ℕ0}) |
6 | fzfid 13511 | . . . . . 6 ⊢ (𝜑 → (1...(𝐾 + 1)) ∈ Fin) | |
7 | nn0ex 12061 | . . . . . . 7 ⊢ ℕ0 ∈ V | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℕ0 ∈ V) |
9 | mapex 8492 | . . . . . 6 ⊢ (((1...(𝐾 + 1)) ∈ Fin ∧ ℕ0 ∈ V) → {𝑔 ∣ 𝑔:(1...(𝐾 + 1))⟶ℕ0} ∈ V) | |
10 | 6, 8, 9 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → {𝑔 ∣ 𝑔:(1...(𝐾 + 1))⟶ℕ0} ∈ V) |
11 | ssexg 5201 | . . . . 5 ⊢ (({𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} ⊆ {𝑔 ∣ 𝑔:(1...(𝐾 + 1))⟶ℕ0} ∧ {𝑔 ∣ 𝑔:(1...(𝐾 + 1))⟶ℕ0} ∈ V) → {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} ∈ V) | |
12 | 5, 10, 11 | syl2anc 587 | . . . 4 ⊢ (𝜑 → {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} ∈ V) |
13 | 2, 12 | eqeltrd 2831 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) |
14 | sticksstones14.1 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
15 | sticksstones14.2 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ ℕ0) | |
16 | sticksstones14.3 | . . . 4 ⊢ 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))) | |
17 | sticksstones14.4 | . . . 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)))))) | |
18 | sticksstones14.6 | . . . 4 ⊢ 𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))} | |
19 | 14, 15, 16, 17, 1, 18 | sticksstones13 39784 | . . 3 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) |
20 | 13, 19 | hasheqf1od 13885 | . 2 ⊢ (𝜑 → (♯‘𝐴) = (♯‘𝐵)) |
21 | 14, 15 | nn0addcld 12119 | . . 3 ⊢ (𝜑 → (𝑁 + 𝐾) ∈ ℕ0) |
22 | 21, 15, 18 | sticksstones5 39775 | . 2 ⊢ (𝜑 → (♯‘𝐵) = ((𝑁 + 𝐾)C𝐾)) |
23 | 20, 22 | eqtrd 2771 | 1 ⊢ (𝜑 → (♯‘𝐴) = ((𝑁 + 𝐾)C𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 {cab 2714 ∀wral 3051 Vcvv 3398 ⊆ wss 3853 ifcif 4425 {csn 4527 〈cop 4533 class class class wbr 5039 ↦ cmpt 5120 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 Fincfn 8604 0cc0 10694 1c1 10695 + caddc 10697 < clt 10832 − cmin 11027 ℕ0cn0 12055 ...cfz 13060 Ccbc 13833 ♯chash 13861 Σcsu 15214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-oadd 8184 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-sup 9036 df-inf 9037 df-oi 9104 df-dju 9482 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-n0 12056 df-z 12142 df-uz 12404 df-rp 12552 df-ico 12906 df-fz 13061 df-fzo 13204 df-seq 13540 df-exp 13601 df-fac 13805 df-bc 13834 df-hash 13862 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-clim 15014 df-sum 15215 |
This theorem is referenced by: sticksstones15 39786 |
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