sticksstones14 - Mathbox for metakunt

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Theorem sticksstones14 43401

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 3993	. . . . 5 ⊢ (𝜑 → {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} ⊆ {𝑔 ∣ 𝑔:(1...(𝐾 + 1))⟶ℕ₀})
6		fzfid 14140	. . . . . 6 ⊢ (𝜑 → (1...(𝐾 + 1)) ∈ Fin)
7		nn0ex 12636	. . . . . . 7 ⊢ ℕ₀ ∈ V
8	7	a1i 11	. . . . . 6 ⊢ (𝜑 → ℕ₀ ∈ V)
9		mapex 7973	. . . . . 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 5285	. . . . 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 2864	. . 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 43400	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶1-1onto𝐵)
20	13, 19	hasheqf1od 14521	. 2 ⊢ (𝜑 → (♯‘𝐴) = (♯‘𝐵))
21	14, 15	nn0addcld 12695	. . 3 ⊢ (𝜑 → (𝑁 + 𝐾) ∈ ℕ₀)
22	21, 15, 18	sticksstones5 43391	. 2 ⊢ (𝜑 → (♯‘𝐵) = ((𝑁 + 𝐾)C𝐾))
23	20, 22	eqtrd 2798	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 2147 {cab 2741 ∀wral 3080 Vcvv 3454 ⊆ wss 3900 ifcif 4485 {csn 4588 ⟨cop 4838 class class class wbr 5106 ↦ cmpt 5189 ⟶wf 6551 ‘cfv 6555 (class class class)co 7440 Fincfn 8998 0cc0 11224 1c1 11225 + caddc 11227 < clt 11367 − cmin 11565 ℕ₀cn0 12630 ...cfz 13663 Ccbc 14470 ♯chash 14498 Σcsu 15879

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-inf2 9666 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301 ax-pre-sup 11302

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-isom 6564 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-om 7899 df-1st 8022 df-2nd 8023 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-1o 8501 df-oadd 8505 df-er 8742 df-en 8999 df-dom 9000 df-sdom 9001 df-fin 9002 df-sup 9458 df-inf 9459 df-oi 9528 df-dju 10006 df-card 10044 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-div 11998 df-nn 12360 df-2 12429 df-3 12430 df-n0 12631 df-z 12718 df-uz 12990 df-rp 13145 df-ico 13506 df-fz 13664 df-fzo 13813 df-seq 14169 df-exp 14229 df-fac 14442 df-bc 14471 df-hash 14499 df-cj 15292 df-re 15293 df-im 15294 df-sqrt 15428 df-abs 15429 df-clim 15681 df-sum 15880

This theorem is used by: sticksstones15 43402

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