sticksstones11 - Mathbox for metakunt

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Theorem sticksstones11 39781

Description: Establish bijective mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 6-Oct-2024.)

**Hypotheses**
Ref	Expression
sticksstones11.1	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
sticksstones11.2	⊢ (𝜑 → 𝐾 = 0)
sticksstones11.3	⊢ 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))))
sticksstones11.4	⊢ 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))))))
sticksstones11.5	⊢ 𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}
sticksstones11.6	⊢ 𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}

**Assertion**
Ref	Expression
sticksstones11	⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)

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

Proof of Theorem sticksstones11 Dummy variables 𝑐 𝑑 𝑢 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sticksstones11.1	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
2		sticksstones11.2	. . . 4 ⊢ (𝜑 → 𝐾 = 0)
3		0nn0 12070	. . . . 5 ⊢ 0 ∈ ℕ₀
4	3	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ ℕ₀)
5	2, 4	eqeltrd 2831	. . 3 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
6		sticksstones11.3	. . 3 ⊢ 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))))
7		sticksstones11.5	. . 3 ⊢ 𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}
8		sticksstones11.6	. . 3 ⊢ 𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}
9	1, 5, 6, 7, 8	sticksstones8 39778	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
10		sticksstones11.4	. . 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))))))
11	1, 2, 10, 7, 8	sticksstones9 39779	. 2 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
12	7	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)})
13		nfv 1922	. . . . . . . . . . 11 ⊢ Ⅎ𝑢𝜑
14		nfcv 2897	. . . . . . . . . . 11 ⊢ Ⅎ𝑢{𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}
15		nfcv 2897	. . . . . . . . . . 11 ⊢ Ⅎ𝑢{{⟨1, 𝑁⟩}}
16		ffn 6523	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢:{1}⟶ℕ₀ → 𝑢 Fn {1})
17	16	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → 𝑢 Fn {1})
18		1nn 11806	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℕ
19	18	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → 1 ∈ ℕ)
20	1	adantr 484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → 𝑁 ∈ ℕ₀)
21		fnsng 6410	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → {⟨1, 𝑁⟩} Fn {1})
22	19, 20, 21	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → {⟨1, 𝑁⟩} Fn {1})
23		elsni 4544	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑝 ∈ {1} → 𝑝 = 1)
24	23	adantl 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) → 𝑝 = 1)
25		simpr 488	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) ∧ 𝑝 = 1) → 𝑝 = 1)
26	25	fveq2d 6699	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) ∧ 𝑝 = 1) → (𝑢‘𝑝) = (𝑢‘1))
27		simprl 771	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → 𝑢:{1}⟶ℕ₀)
28		1ex 10794	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ 1 ∈ V
29	28	snid 4563	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ 1 ∈ {1}
30	29	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → 1 ∈ {1})
31	27, 30	ffvelrnd 6883	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → (𝑢‘1) ∈ ℕ₀)
32	31	nn0cnd 12117	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → (𝑢‘1) ∈ ℂ)
33		fveq2 6695	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑖 = 1 → (𝑢‘𝑖) = (𝑢‘1))
34	33	sumsn 15273	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((1 ∈ ℕ ∧ (𝑢‘1) ∈ ℂ) → Σ𝑖 ∈ {1} (𝑢‘𝑖) = (𝑢‘1))
35	19, 32, 34	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → Σ𝑖 ∈ {1} (𝑢‘𝑖) = (𝑢‘1))
36	35	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = (𝑢‘1)) → Σ𝑖 ∈ {1} (𝑢‘𝑖) = (𝑢‘1))
37	36	eqcomd 2742	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = (𝑢‘1)) → (𝑢‘1) = Σ𝑖 ∈ {1} (𝑢‘𝑖))
38		simplrr 778	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = (𝑢‘1)) → Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)
39	37, 38	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = (𝑢‘1)) → (𝑢‘1) = 𝑁)
40	39	ex 416	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → (Σ𝑖 ∈ {1} (𝑢‘𝑖) = (𝑢‘1) → (𝑢‘1) = 𝑁))
41	35, 40	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → (𝑢‘1) = 𝑁)
42	41	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) → (𝑢‘1) = 𝑁)
43	18	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) → 1 ∈ ℕ)
44	20	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) → 𝑁 ∈ ℕ₀)
45		fvsng 6973	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((1 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → ({⟨1, 𝑁⟩}‘1) = 𝑁)
46	43, 44, 45	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) → ({⟨1, 𝑁⟩}‘1) = 𝑁)
47	46	eqcomd 2742	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) → 𝑁 = ({⟨1, 𝑁⟩}‘1))
48	42, 47	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) → (𝑢‘1) = ({⟨1, 𝑁⟩}‘1))
49	48	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) ∧ 𝑝 = 1) → (𝑢‘1) = ({⟨1, 𝑁⟩}‘1))
50	25	eqcomd 2742	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) ∧ 𝑝 = 1) → 1 = 𝑝)
51	50	fveq2d 6699	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) ∧ 𝑝 = 1) → ({⟨1, 𝑁⟩}‘1) = ({⟨1, 𝑁⟩}‘𝑝))
52	26, 49, 51	3eqtrd 2775	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) ∧ 𝑝 = 1) → (𝑢‘𝑝) = ({⟨1, 𝑁⟩}‘𝑝))
53	24, 52	mpdan 687	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) ∧ 𝑝 ∈ {1}) → (𝑢‘𝑝) = ({⟨1, 𝑁⟩}‘𝑝))
54	17, 22, 53	eqfnfvd 6833	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → 𝑢 = {⟨1, 𝑁⟩})
55		fsng 6930	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (𝑢:{1}⟶{𝑁} ↔ 𝑢 = {⟨1, 𝑁⟩}))
56	19, 20, 55	syl2anc 587	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → (𝑢:{1}⟶{𝑁} ↔ 𝑢 = {⟨1, 𝑁⟩}))
57	54, 56	mpbird 260	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → 𝑢:{1}⟶{𝑁})
58		ssidd 3910	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → {𝑁} ⊆ {𝑁})
59		fss 6540	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑢:{1}⟶{𝑁} ∧ {𝑁} ⊆ {𝑁}) → 𝑢:{1}⟶{𝑁})
60	57, 58, 59	syl2anc 587	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → 𝑢:{1}⟶{𝑁})
61	60, 58, 59	syl2anc 587	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → 𝑢:{1}⟶{𝑁})
62	61, 56	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → 𝑢 = {⟨1, 𝑁⟩})
63		vex 3402	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑢 ∈ V
64	63	elsn 4542	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ {{⟨1, 𝑁⟩}} ↔ 𝑢 = {⟨1, 𝑁⟩})
65	62, 64	sylibr 237	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)) → 𝑢 ∈ {{⟨1, 𝑁⟩}})
66	65	ex 416	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁) → 𝑢 ∈ {{⟨1, 𝑁⟩}}))
67		1zzd 12173	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 1 ∈ ℤ)
68		fzsn 13119	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 ∈ ℤ → (1...1) = {1})
69	67, 68	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (1...1) = {1})
70	69	eqcomd 2742	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → {1} = (1...1))
71		1e0p1 12300	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 = (0 + 1)
72	71	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 1 = (0 + 1))
73	72	oveq2d 7207	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (1...1) = (1...(0 + 1)))
74	70, 73	eqtrd 2771	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → {1} = (1...(0 + 1)))
75	2	eqcomd 2742	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 0 = 𝐾)
76	75	oveq1d 7206	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (0 + 1) = (𝐾 + 1))
77	76	oveq2d 7207	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (1...(0 + 1)) = (1...(𝐾 + 1)))
78	74, 77	eqtrd 2771	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → {1} = (1...(𝐾 + 1)))
79	78	feq2d 6509	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑢:{1}⟶ℕ₀ ↔ 𝑢:(1...(𝐾 + 1))⟶ℕ₀))
80	78	sumeq1d 15230	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → Σ𝑖 ∈ {1} (𝑢‘𝑖) = Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖))
81	80	eqeq1d 2738	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁 ↔ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = 𝑁))
82	79, 81	anbi12d 634	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁) ↔ (𝑢:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = 𝑁)))
83	82	imbi1d 345	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((𝑢:{1}⟶ℕ₀ ∧ Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁) → 𝑢 ∈ {{⟨1, 𝑁⟩}}) ↔ ((𝑢:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = 𝑁) → 𝑢 ∈ {{⟨1, 𝑁⟩}})))
84	66, 83	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑢:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = 𝑁) → 𝑢 ∈ {{⟨1, 𝑁⟩}}))
85		feq1 6504	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = 𝑢 → (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ↔ 𝑢:(1...(𝐾 + 1))⟶ℕ₀))
86		simpl 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔 = 𝑢 ∧ 𝑖 ∈ (1...(𝐾 + 1))) → 𝑔 = 𝑢)
87	86	fveq1d 6697	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔 = 𝑢 ∧ 𝑖 ∈ (1...(𝐾 + 1))) → (𝑔‘𝑖) = (𝑢‘𝑖))
88	87	sumeq2dv 15232	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = 𝑢 → Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖))
89	88	eqeq1d 2738	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = 𝑢 → (Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁 ↔ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = 𝑁))
90	85, 89	anbi12d 634	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = 𝑢 → ((𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁) ↔ (𝑢:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = 𝑁)))
91	63, 90	elab 3576	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} ↔ (𝑢:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = 𝑁))
92	91	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑢 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} ↔ (𝑢:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = 𝑁)))
93	92	imbi1d 345	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑢 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} → 𝑢 ∈ {{⟨1, 𝑁⟩}}) ↔ ((𝑢:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = 𝑁) → 𝑢 ∈ {{⟨1, 𝑁⟩}})))
94	84, 93	mpbird 260	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑢 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} → 𝑢 ∈ {{⟨1, 𝑁⟩}}))
95	94	imp 410	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}) → 𝑢 ∈ {{⟨1, 𝑁⟩}})
96	95	ex 416	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} → 𝑢 ∈ {{⟨1, 𝑁⟩}}))
97	13, 14, 15, 96	ssrd 3892	. . . . . . . . . 10 ⊢ (𝜑 → {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} ⊆ {{⟨1, 𝑁⟩}})
98	18	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 1 ∈ ℕ)
99	98, 1, 55	syl2anc 587	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑢:{1}⟶{𝑁} ↔ 𝑢 = {⟨1, 𝑁⟩}))
100	99	bicomd 226	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑢 = {⟨1, 𝑁⟩} ↔ 𝑢:{1}⟶{𝑁}))
101	100	biimpd 232	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑢 = {⟨1, 𝑁⟩} → 𝑢:{1}⟶{𝑁}))
102		elsni 4544	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ {{⟨1, 𝑁⟩}} → 𝑢 = {⟨1, 𝑁⟩})
103	101, 102	impel 509	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}}) → 𝑢:{1}⟶{𝑁})
104	1	snssd 4708	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → {𝑁} ⊆ ℕ₀)
105	104	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}}) → {𝑁} ⊆ ℕ₀)
106		fss 6540	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢:{1}⟶{𝑁} ∧ {𝑁} ⊆ ℕ₀) → 𝑢:{1}⟶ℕ₀)
107	103, 105, 106	syl2anc 587	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}}) → 𝑢:{1}⟶ℕ₀)
108	79	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}}) → (𝑢:{1}⟶ℕ₀ ↔ 𝑢:(1...(𝐾 + 1))⟶ℕ₀))
109	107, 108	mpbid 235	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}}) → 𝑢:(1...(𝐾 + 1))⟶ℕ₀)
110	102	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}}) → 𝑢 = {⟨1, 𝑁⟩})
111	78	3ad2ant1 1135	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → {1} = (1...(𝐾 + 1)))
112	111	eqcomd 2742	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → (1...(𝐾 + 1)) = {1})
113	112	sumeq1d 15230	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = Σ𝑖 ∈ {1} (𝑢‘𝑖))
114	18	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → 1 ∈ ℕ)
115	1	3ad2ant1 1135	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → 𝑁 ∈ ℕ₀)
116	115	nn0cnd 12117	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → 𝑁 ∈ ℂ)
117	114, 115, 45	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → ({⟨1, 𝑁⟩}‘1) = 𝑁)
118	117	eqcomd 2742	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → 𝑁 = ({⟨1, 𝑁⟩}‘1))
119	110	3adant3 1134	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → 𝑢 = {⟨1, 𝑁⟩})
120	119	fveq1d 6697	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → (𝑢‘1) = ({⟨1, 𝑁⟩}‘1))
121	118, 120	eqtr4d 2774	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → 𝑁 = (𝑢‘1))
122	121	eleq1d 2815	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → (𝑁 ∈ ℂ ↔ (𝑢‘1) ∈ ℂ))
123	116, 122	mpbid 235	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → (𝑢‘1) ∈ ℂ)
124	114, 123, 34	syl2anc 587	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → Σ𝑖 ∈ {1} (𝑢‘𝑖) = (𝑢‘1))
125	120, 117	eqtrd 2771	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → (𝑢‘1) = 𝑁)
126	124, 125	eqtrd 2771	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → Σ𝑖 ∈ {1} (𝑢‘𝑖) = 𝑁)
127	113, 126	eqtrd 2771	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}} ∧ 𝑢 = {⟨1, 𝑁⟩}) → Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = 𝑁)
128	127	3expa 1120	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}}) ∧ 𝑢 = {⟨1, 𝑁⟩}) → Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = 𝑁)
129	110, 128	mpdan 687	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}}) → Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = 𝑁)
130	109, 129	jca 515	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}}) → (𝑢:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑢‘𝑖) = 𝑁))
131	130, 91	sylibr 237	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ {{⟨1, 𝑁⟩}}) → 𝑢 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)})
132	131	ex 416	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑢 ∈ {{⟨1, 𝑁⟩}} → 𝑢 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}))
133	13, 15, 14, 132	ssrd 3892	. . . . . . . . . 10 ⊢ (𝜑 → {{⟨1, 𝑁⟩}} ⊆ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)})
134	97, 133	eqssd 3904	. . . . . . . . 9 ⊢ (𝜑 → {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} = {{⟨1, 𝑁⟩}})
135	12, 134	eqtrd 2771	. . . . . . . 8 ⊢ (𝜑 → 𝐴 = {{⟨1, 𝑁⟩}})
136		eqss 3902	. . . . . . . . . 10 ⊢ (𝐴 = {{⟨1, 𝑁⟩}} ↔ (𝐴 ⊆ {{⟨1, 𝑁⟩}} ∧ {{⟨1, 𝑁⟩}} ⊆ 𝐴))
137	136	biimpi 219	. . . . . . . . 9 ⊢ (𝐴 = {{⟨1, 𝑁⟩}} → (𝐴 ⊆ {{⟨1, 𝑁⟩}} ∧ {{⟨1, 𝑁⟩}} ⊆ 𝐴))
138	137	simpld 498	. . . . . . . 8 ⊢ (𝐴 = {{⟨1, 𝑁⟩}} → 𝐴 ⊆ {{⟨1, 𝑁⟩}})
139	135, 138	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ {{⟨1, 𝑁⟩}})
140		fss 6540	. . . . . . 7 ⊢ ((𝐺:𝐵⟶𝐴 ∧ 𝐴 ⊆ {{⟨1, 𝑁⟩}}) → 𝐺:𝐵⟶{{⟨1, 𝑁⟩}})
141	11, 139, 140	syl2anc 587	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐵⟶{{⟨1, 𝑁⟩}})
142	141	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝐺:𝐵⟶{{⟨1, 𝑁⟩}})
143	9	ffvelrnda 6882	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝐹‘𝑐) ∈ 𝐵)
144		fvconst 6957	. . . . 5 ⊢ ((𝐺:𝐵⟶{{⟨1, 𝑁⟩}} ∧ (𝐹‘𝑐) ∈ 𝐵) → (𝐺‘(𝐹‘𝑐)) = {⟨1, 𝑁⟩})
145	142, 143, 144	syl2anc 587	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝐺‘(𝐹‘𝑐)) = {⟨1, 𝑁⟩})
146	135	eleq2d 2816	. . . . . . . 8 ⊢ (𝜑 → (𝑐 ∈ 𝐴 ↔ 𝑐 ∈ {{⟨1, 𝑁⟩}}))
147	146	biimpd 232	. . . . . . 7 ⊢ (𝜑 → (𝑐 ∈ 𝐴 → 𝑐 ∈ {{⟨1, 𝑁⟩}}))
148	147	imp 410	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝑐 ∈ {{⟨1, 𝑁⟩}})
149		vex 3402	. . . . . . 7 ⊢ 𝑐 ∈ V
150	149	elsn 4542	. . . . . 6 ⊢ (𝑐 ∈ {{⟨1, 𝑁⟩}} ↔ 𝑐 = {⟨1, 𝑁⟩})
151	148, 150	sylib 221	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝑐 = {⟨1, 𝑁⟩})
152	151	eqcomd 2742	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → {⟨1, 𝑁⟩} = 𝑐)
153	145, 152	eqtrd 2771	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝐺‘(𝐹‘𝑐)) = 𝑐)
154	153	ralrimiva 3095	. 2 ⊢ (𝜑 → ∀𝑐 ∈ 𝐴 (𝐺‘(𝐹‘𝑐)) = 𝑐)
155		simpr 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝑑 ∈ 𝐵)
156		nfv 1922	. . . . . . . . . . 11 ⊢ Ⅎ𝑑𝜑
157		nfcv 2897	. . . . . . . . . . 11 ⊢ Ⅎ𝑑𝐵
158		nfcv 2897	. . . . . . . . . . 11 ⊢ Ⅎ𝑑{∅}
159	8	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))})
160	159	eleq2d 2816	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝑑 ∈ 𝐵 ↔ 𝑑 ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}))
161	160	biimpd 232	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝑑 ∈ 𝐵 → 𝑑 ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}))
162	161	syldbl2 841	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝑑 ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))})
163		vex 3402	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑑 ∈ V
164		feq1 6504	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑑 → (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ↔ 𝑑:(1...𝐾)⟶(1...(𝑁 + 𝐾))))
165		fveq1 6694	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 = 𝑑 → (𝑓‘𝑥) = (𝑑‘𝑥))
166		fveq1 6694	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 = 𝑑 → (𝑓‘𝑦) = (𝑑‘𝑦))
167	165, 166	breq12d 5052	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = 𝑑 → ((𝑓‘𝑥) < (𝑓‘𝑦) ↔ (𝑑‘𝑥) < (𝑑‘𝑦)))
168	167	imbi2d 344	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = 𝑑 → ((𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)) ↔ (𝑥 < 𝑦 → (𝑑‘𝑥) < (𝑑‘𝑦))))
169	168	2ralbidv 3110	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑑 → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)) ↔ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑑‘𝑥) < (𝑑‘𝑦))))
170	164, 169	anbi12d 634	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = 𝑑 → ((𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦))) ↔ (𝑑:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑑‘𝑥) < (𝑑‘𝑦)))))
171	163, 170	elab 3576	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))} ↔ (𝑑:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑑‘𝑥) < (𝑑‘𝑦))))
172	162, 171	sylib 221	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝑑:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑑‘𝑥) < (𝑑‘𝑦))))
173	172	simpld 498	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝑑:(1...𝐾)⟶(1...(𝑁 + 𝐾)))
174		0lt1 11319	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 < 1
175	174	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 0 < 1)
176	2, 175	eqbrtrd 5061	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐾 < 1)
177	5	nn0zd 12245	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐾 ∈ ℤ)
178		fzn 13093	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐾 < 1 ↔ (1...𝐾) = ∅))
179	67, 177, 178	syl2anc 587	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐾 < 1 ↔ (1...𝐾) = ∅))
180	176, 179	mpbid 235	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1...𝐾) = ∅)
181	180	feq2d 6509	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑑:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ↔ 𝑑:∅⟶(1...(𝑁 + 𝐾))))
182	181	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝑑:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ↔ 𝑑:∅⟶(1...(𝑁 + 𝐾))))
183	173, 182	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝑑:∅⟶(1...(𝑁 + 𝐾)))
184		f0bi 6580	. . . . . . . . . . . . . . 15 ⊢ (𝑑:∅⟶(1...(𝑁 + 𝐾)) ↔ 𝑑 = ∅)
185	183, 184	sylib 221	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝑑 = ∅)
186		velsn 4543	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ {∅} ↔ 𝑑 = ∅)
187	185, 186	sylibr 237	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝑑 ∈ {∅})
188	187	ex 416	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑑 ∈ 𝐵 → 𝑑 ∈ {∅}))
189		f0 6578	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∅:∅⟶(1...(𝑁 + 𝐾))
190	189	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∅:∅⟶(1...(𝑁 + 𝐾)))
191	180	feq2d 6509	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (∅:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ↔ ∅:∅⟶(1...(𝑁 + 𝐾))))
192	190, 191	mpbird 260	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∅:(1...𝐾)⟶(1...(𝑁 + 𝐾)))
193		ral0 4410	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (∅‘𝑥) < (∅‘𝑦))
194	193	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑥 ∈ ∅ ∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (∅‘𝑥) < (∅‘𝑦)))
195		biidd 265	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝐾)) → (∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (∅‘𝑥) < (∅‘𝑦)) ↔ ∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (∅‘𝑥) < (∅‘𝑦))))
196	180, 195	raleqbidva 3321	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (∅‘𝑥) < (∅‘𝑦)) ↔ ∀𝑥 ∈ ∅ ∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (∅‘𝑥) < (∅‘𝑦))))
197	194, 196	mpbird 260	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (∅‘𝑥) < (∅‘𝑦)))
198	192, 197	jca 515	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (∅:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (∅‘𝑥) < (∅‘𝑦))))
199		0ex 5185	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ∈ V
200		feq1 6504	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = ∅ → (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ↔ ∅:(1...𝐾)⟶(1...(𝑁 + 𝐾))))
201		fveq1 6694	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 = ∅ → (𝑓‘𝑥) = (∅‘𝑥))
202		fveq1 6694	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 = ∅ → (𝑓‘𝑦) = (∅‘𝑦))
203	201, 202	breq12d 5052	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = ∅ → ((𝑓‘𝑥) < (𝑓‘𝑦) ↔ (∅‘𝑥) < (∅‘𝑦)))
204	203	imbi2d 344	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = ∅ → ((𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)) ↔ (𝑥 < 𝑦 → (∅‘𝑥) < (∅‘𝑦))))
205	204	2ralbidv 3110	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = ∅ → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)) ↔ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (∅‘𝑥) < (∅‘𝑦))))
206	200, 205	anbi12d 634	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = ∅ → ((𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦))) ↔ (∅:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (∅‘𝑥) < (∅‘𝑦)))))
207	206	elabg 3574	. . . . . . . . . . . . . . . . . 18 ⊢ (∅ ∈ V → (∅ ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))} ↔ (∅:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (∅‘𝑥) < (∅‘𝑦)))))
208	199, 207	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))} ↔ (∅:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (∅‘𝑥) < (∅‘𝑦))))
209	198, 208	sylibr 237	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∅ ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))})
210	8	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))})
211	209, 210	eleqtrrd 2834	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∅ ∈ 𝐵)
212	211	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ {∅}) → ∅ ∈ 𝐵)
213		elsni 4544	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 ∈ {∅} → 𝑑 = ∅)
214	213	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑑 ∈ {∅}) → 𝑑 = ∅)
215	214	eleq1d 2815	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑑 ∈ {∅}) → (𝑑 ∈ 𝐵 ↔ ∅ ∈ 𝐵))
216	212, 215	mpbird 260	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ {∅}) → 𝑑 ∈ 𝐵)
217	216	ex 416	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑑 ∈ {∅} → 𝑑 ∈ 𝐵))
218	188, 217	impbid 215	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑑 ∈ 𝐵 ↔ 𝑑 ∈ {∅}))
219	156, 157, 158, 218	eqrd 3906	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = {∅})
220	219	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝐵 = {∅})
221	155, 220	eleqtrd 2833	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝑑 ∈ {∅})
222	163	elsn 4542	. . . . . . . 8 ⊢ (𝑑 ∈ {∅} ↔ 𝑑 = ∅)
223	221, 222	sylib 221	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝑑 = ∅)
224	223	fveq2d 6699	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝐺‘𝑑) = (𝐺‘∅))
225	224	fveq2d 6699	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝐹‘(𝐺‘𝑑)) = (𝐹‘(𝐺‘∅)))
226	180	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (1...𝐾) = ∅)
227	226	mpteq1d 5129	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) = (𝑗 ∈ ∅ ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))))
228		mpt0 6498	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ∅ ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) = ∅
229	228	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑗 ∈ ∅ ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) = ∅)
230	227, 229	eqtrd 2771	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) = ∅)
231		fzfid 13511	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...𝐾) ∈ Fin)
232	231	mptexd 7018	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) ∈ V)
233		elsng 4541	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) ∈ V → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) ∈ {∅} ↔ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) = ∅))
234	232, 233	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) ∈ {∅} ↔ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) = ∅))
235	234	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) ∈ {∅} ↔ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) = ∅))
236	230, 235	mpbird 260	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) ∈ {∅})
237	236, 6	fmptd 6909	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶{∅})
238	237	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → 𝐹:𝐴⟶{∅})
239		ffvelrn 6880	. . . . . . . 8 ⊢ ((𝐺:𝐵⟶𝐴 ∧ ∅ ∈ 𝐵) → (𝐺‘∅) ∈ 𝐴)
240	11, 211, 239	syl2anc 587	. . . . . . 7 ⊢ (𝜑 → (𝐺‘∅) ∈ 𝐴)
241	240	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝐺‘∅) ∈ 𝐴)
242		fvconst 6957	. . . . . 6 ⊢ ((𝐹:𝐴⟶{∅} ∧ (𝐺‘∅) ∈ 𝐴) → (𝐹‘(𝐺‘∅)) = ∅)
243	238, 241, 242	syl2anc 587	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝐹‘(𝐺‘∅)) = ∅)
244	225, 243	eqtrd 2771	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝐹‘(𝐺‘𝑑)) = ∅)
245	223	eqcomd 2742	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → ∅ = 𝑑)
246	244, 245	eqtrd 2771	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ 𝐵) → (𝐹‘(𝐺‘𝑑)) = 𝑑)
247	246	ralrimiva 3095	. 2 ⊢ (𝜑 → ∀𝑑 ∈ 𝐵 (𝐹‘(𝐺‘𝑑)) = 𝑑)
248	9, 11, 154, 247	2fvidf1od 7086	1 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 {cab 2714 ∀wral 3051 Vcvv 3398 ⊆ wss 3853 ∅c0 4223 ifcif 4425 {csn 4527 ⟨cop 4533 class class class wbr 5039 ↦ cmpt 5120 Fn wfn 6353 ⟶wf 6354 –1-1-onto→wf1o 6357 ‘cfv 6358 (class class class)co 7191 Fincfn 8604 ℂcc 10692 0cc0 10694 1c1 10695 + caddc 10697 < clt 10832 − cmin 11027 ℕcn 11795 ℕ₀cn0 12055 ℤcz 12141 ...cfz 13060 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-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-sup 9036 df-oi 9104 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-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: sticksstones13 39784

W3C validator