sticksstones14 - Mathbox for metakunt

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Theorem sticksstones14 43013

Description: Sticks and stones with definitions as hypotheses. (Contributed by metakunt, 7-Oct-2024.)

**Hypotheses**
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)))}

**Assertion**
Ref	Expression
sticksstones14	⊢ (𝜑 → (♯‘𝐴) = ((𝑁 + 𝐾)C𝐾))

Distinct variable groups: 𝐴,𝑎,𝑖,𝑘,𝑙 𝐴,𝑏,𝑖,𝑘,𝑙 𝐴,𝑗,𝑥,𝑦,𝑎,𝑘,𝑙 𝐵,𝑎,𝑓,𝑗,𝑙 𝐵,𝑏,𝑓,𝑗 𝐵,𝑖,𝑘 𝐹,𝑏,𝑖,𝑘 𝐾,𝑎,𝑓,𝑗,𝑙,𝑥,𝑦 𝐾,𝑏,𝑥,𝑦 𝑔,𝐾,𝑖,𝑘,𝑎 𝑁,𝑎,𝑓,𝑗,𝑙,𝑥,𝑦 𝑁,𝑏,𝑔,𝑖,𝑘 𝜑,𝑎,𝑓,𝑗,𝑙,𝑥,𝑦 𝑔,𝑏,𝜑,𝑖,𝑘 Allowed substitution hints: 𝐴(𝑓, 𝑔) 𝐵(𝑥, 𝑦, 𝑔) 𝐹(𝑥, 𝑦, 𝑓, 𝑔, 𝑗, 𝑎, 𝑙) 𝐺(𝑥, 𝑦, 𝑓, 𝑔, 𝑖, 𝑗, 𝑘, 𝑎, 𝑏, 𝑙)

**Proof of Theorem sticksstones14**
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 488	. . . . . . 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 4016	. . . . 5 ⊢ (𝜑 → {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} ⊆ {𝑔 ∣ 𝑔:(1...(𝐾 + 1))⟶ℕ₀})
6		fzfid 14039	. . . . . 6 ⊢ (𝜑 → (1...(𝐾 + 1)) ∈ Fin)
7		nn0ex 12537	. . . . . . 7 ⊢ ℕ₀ ∈ V
8	7	a1i 11	. . . . . 6 ⊢ (𝜑 → ℕ₀ ∈ V)
9		mapex 7940	. . . . . 6 ⊢ (((1...(𝐾 + 1)) ∈ Fin ∧ ℕ₀ ∈ V) → {𝑔 ∣ 𝑔:(1...(𝐾 + 1))⟶ℕ₀} ∈ V)
10	6, 8, 9	syl2anc 596	. . . . 5 ⊢ (𝜑 → {𝑔 ∣ 𝑔:(1...(𝐾 + 1))⟶ℕ₀} ∈ V)
11		ssexg 5288	. . . . 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 596	. . . 4 ⊢ (𝜑 → {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} ∈ V)
13	2, 12	eqeltrd 2862	. . 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 43012	. . 3 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
20	13, 19	hasheqf1od 14419	. 2 ⊢ (𝜑 → (♯‘𝐴) = (♯‘𝐵))
21	14, 15	nn0addcld 12596	. . 3 ⊢ (𝜑 → (𝑁 + 𝐾) ∈ ℕ₀)
22	21, 15, 18	sticksstones5 43003	. 2 ⊢ (𝜑 → (♯‘𝐵) = ((𝑁 + 𝐾)C𝐾))
23	20, 22	eqtrd 2797	1 ⊢ (𝜑 → (♯‘𝐴) = ((𝑁 + 𝐾)C𝐾))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 {cab 2740 ∀wral 3078 Vcvv 3453 ⊆ wss 3902 ifcif 4485 {csn 4587 ⟨cop 4593 class class class wbr 5107 ↦ cmpt 5190 ⟶wf 6533 ‘cfv 6537 (class class class)co 7416 Fincfn 8955 0cc0 11127 1c1 11128 + caddc 11130 < clt 11270 − cmin 11468 ℕ₀cn0 12531 ...cfz 13563 Ccbc 14368 ♯chash 14396 Σcsu 15775

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-inf2 9623 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-oadd 8462 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9415 df-inf 9416 df-oi 9485 df-dju 9909 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-n0 12532 df-z 12619 df-uz 12891 df-rp 13045 df-ico 13406 df-fz 13564 df-fzo 13712 df-seq 14068 df-exp 14128 df-fac 14340 df-bc 14369 df-hash 14397 df-cj 15188 df-re 15189 df-im 15190 df-sqrt 15324 df-abs 15325 df-clim 15577 df-sum 15776

This theorem is used by: sticksstones15 43014

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