Theorem sticksstones17 42120

Description: Extend sticks and stones to finite sets, bijective builder. (Contributed by metakunt, 23-Oct-2024.)

Hypotheses

Ref	Expression
sticksstones17.1	⊢ (𝜑 → 𝑁 ∈ ℕ0)
sticksstones17.2	⊢ (𝜑 → 𝐾 ∈ ℕ0)
sticksstones17.3	⊢ 𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}
sticksstones17.4	⊢ 𝐵 = {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)}
sticksstones17.5	⊢ (𝜑 → 𝑍:(1...𝐾)–1-1-onto→𝑆)
sticksstones17.6	⊢ 𝐺 = (𝑏 ∈ 𝐵 ↦ (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))))

Assertion

Ref	Expression
sticksstones17	⊢ (𝜑 → 𝐺:𝐵⟶𝐴)

Distinct variable groups:   𝐴,𝑏   𝐵,𝑏,𝑖,𝑦   𝑔,𝐾,𝑖,𝑦   𝑔,𝑁   ℎ,𝑁   𝑆,ℎ,𝑖   𝑔,𝑍,𝑖,𝑦   𝑔,𝑏   ℎ,𝑏   𝜑,𝑏,𝑖,𝑦

Allowed substitution hints:   𝜑(𝑔,ℎ)   𝐴(𝑦,𝑔,ℎ,𝑖)   𝐵(𝑔,ℎ)   𝑆(𝑦,𝑔,𝑏)   𝐺(𝑦,𝑔,ℎ,𝑖,𝑏)   𝐾(ℎ,𝑏)   𝑁(𝑦,𝑖,𝑏)   𝑍(ℎ,𝑏)

Proof of Theorem sticksstones17

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sticksstones17.4	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 = {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)}
2	1	eqimssi 4069	. . . . . . . . . . . . . . 15 ⊢ 𝐵 ⊆ {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)}
3	2	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ⊆ {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)})
4	3	sseld 4007	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑏 ∈ 𝐵 → 𝑏 ∈ {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)}))
5	4	imp 406	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)})
6		vex 3492	. . . . . . . . . . . . 13 ⊢ 𝑏 ∈ V
7		feq1 6728	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑏 → (ℎ:𝑆⟶ℕ0 ↔ 𝑏:𝑆⟶ℕ0))
8		simpl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((ℎ = 𝑏 ∧ 𝑖 ∈ 𝑆) → ℎ = 𝑏)
9	8	fveq1d 6922	. . . . . . . . . . . . . . . 16 ⊢ ((ℎ = 𝑏 ∧ 𝑖 ∈ 𝑆) → (ℎ‘𝑖) = (𝑏‘𝑖))
10	9	sumeq2dv 15750	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝑏 → Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = Σ𝑖 ∈ 𝑆 (𝑏‘𝑖))
11	10	eqeq1d 2742	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑏 → (Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁 ↔ Σ𝑖 ∈ 𝑆 (𝑏‘𝑖) = 𝑁))
12	7, 11	anbi12d 631	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑏 → ((ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁) ↔ (𝑏:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (𝑏‘𝑖) = 𝑁)))
13	6, 12	elab 3694	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)} ↔ (𝑏:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (𝑏‘𝑖) = 𝑁))
14	5, 13	sylib 218	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (𝑏‘𝑖) = 𝑁))
15	14	simpld 494	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏:𝑆⟶ℕ0)
16	15	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (1...𝐾)) → 𝑏:𝑆⟶ℕ0)
17	16	3impa 1110	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑦 ∈ (1...𝐾)) → 𝑏:𝑆⟶ℕ0)
18		sticksstones17.5	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍:(1...𝐾)–1-1-onto→𝑆)
19		f1of 6862	. . . . . . . . . . . . 13 ⊢ (𝑍:(1...𝐾)–1-1-onto→𝑆 → 𝑍:(1...𝐾)⟶𝑆)
20	18, 19	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑍:(1...𝐾)⟶𝑆)
21	20	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑍:(1...𝐾)⟶𝑆)
22	21	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (1...𝐾)) → 𝑍:(1...𝐾)⟶𝑆)
23	22	3impa 1110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑦 ∈ (1...𝐾)) → 𝑍:(1...𝐾)⟶𝑆)
24		simp3 1138	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑦 ∈ (1...𝐾)) → 𝑦 ∈ (1...𝐾))
25	23, 24	ffvelcdmd 7119	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑦 ∈ (1...𝐾)) → (𝑍‘𝑦) ∈ 𝑆)
26	17, 25	ffvelcdmd 7119	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑦 ∈ (1...𝐾)) → (𝑏‘(𝑍‘𝑦)) ∈ ℕ0)
27	26	3expa 1118	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (1...𝐾)) → (𝑏‘(𝑍‘𝑦)) ∈ ℕ0)
28	27	fmpttd 7149	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))):(1...𝐾)⟶ℕ0)
29		eqidd 2741	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) → (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))))
30		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) ∧ 𝑦 = 𝑖) → 𝑦 = 𝑖)
31	30	fveq2d 6924	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) ∧ 𝑦 = 𝑖) → (𝑍‘𝑦) = (𝑍‘𝑖))
32	31	fveq2d 6924	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) ∧ 𝑦 = 𝑖) → (𝑏‘(𝑍‘𝑦)) = (𝑏‘(𝑍‘𝑖)))
33		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) → 𝑖 ∈ (1...𝐾))
34		fvexd 6935	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) → (𝑏‘(𝑍‘𝑖)) ∈ V)
35	29, 32, 33, 34	fvmptd 7036	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) → ((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖) = (𝑏‘(𝑍‘𝑖)))
36	35	sumeq2dv 15750	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖) = Σ𝑖 ∈ (1...𝐾)(𝑏‘(𝑍‘𝑖)))
37		fveq2 6920	. . . . . . . . 9 ⊢ (𝑠 = (𝑍‘𝑖) → (𝑏‘𝑠) = (𝑏‘(𝑍‘𝑖)))
38		fzfi 14023	. . . . . . . . . 10 ⊢ (1...𝐾) ∈ Fin
39	38	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (1...𝐾) ∈ Fin)
40	18	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑍:(1...𝐾)–1-1-onto→𝑆)
41		eqidd 2741	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) → (𝑍‘𝑖) = (𝑍‘𝑖))
42		nn0sscn 12558	. . . . . . . . . . . 12 ⊢ ℕ0 ⊆ ℂ
43	42	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ℕ0 ⊆ ℂ)
44		fss 6763	. . . . . . . . . . 11 ⊢ ((𝑏:𝑆⟶ℕ0 ∧ ℕ0 ⊆ ℂ) → 𝑏:𝑆⟶ℂ)
45	15, 43, 44	syl2anc 583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏:𝑆⟶ℂ)
46	45	ffvelcdmda 7118	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → (𝑏‘𝑠) ∈ ℂ)
47	37, 39, 40, 41, 46	fsumf1o 15771	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑠 ∈ 𝑆 (𝑏‘𝑠) = Σ𝑖 ∈ (1...𝐾)(𝑏‘(𝑍‘𝑖)))
48	47	eqcomd 2746	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑖 ∈ (1...𝐾)(𝑏‘(𝑍‘𝑖)) = Σ𝑠 ∈ 𝑆 (𝑏‘𝑠))
49		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑠 = 𝑖 → (𝑏‘𝑠) = (𝑏‘𝑖))
50	49	cbvsumv 15744	. . . . . . . . 9 ⊢ Σ𝑠 ∈ 𝑆 (𝑏‘𝑠) = Σ𝑖 ∈ 𝑆 (𝑏‘𝑖)
51	50	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑠 ∈ 𝑆 (𝑏‘𝑠) = Σ𝑖 ∈ 𝑆 (𝑏‘𝑖))
52	14	simprd 495	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑖 ∈ 𝑆 (𝑏‘𝑖) = 𝑁)
53	51, 52	eqtrd 2780	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑠 ∈ 𝑆 (𝑏‘𝑠) = 𝑁)
54	48, 53	eqtrd 2780	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑖 ∈ (1...𝐾)(𝑏‘(𝑍‘𝑖)) = 𝑁)
55	36, 54	eqtrd 2780	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖) = 𝑁)
56	28, 55	jca 511	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))):(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖) = 𝑁))
57		fzfid 14024	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (1...𝐾) ∈ Fin)
58	57	mptexd 7261	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) ∈ V)
59		feq1 6728	. . . . . . 7 ⊢ (𝑔 = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) → (𝑔:(1...𝐾)⟶ℕ0 ↔ (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))):(1...𝐾)⟶ℕ0))
60		simpl 482	. . . . . . . . . 10 ⊢ ((𝑔 = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) ∧ 𝑖 ∈ (1...𝐾)) → 𝑔 = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))))
61	60	fveq1d 6922	. . . . . . . . 9 ⊢ ((𝑔 = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) ∧ 𝑖 ∈ (1...𝐾)) → (𝑔‘𝑖) = ((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖))
62	61	sumeq2dv 15750	. . . . . . . 8 ⊢ (𝑔 = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) → Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖))
63	62	eqeq1d 2742	. . . . . . 7 ⊢ (𝑔 = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) → (Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁 ↔ Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖) = 𝑁))
64	59, 63	anbi12d 631	. . . . . 6 ⊢ (𝑔 = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) → ((𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁) ↔ ((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))):(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖) = 𝑁)))
65	64	elabg 3690	. . . . 5 ⊢ ((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) ∈ V → ((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)} ↔ ((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))):(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖) = 𝑁)))
66	58, 65	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)} ↔ ((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))):(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖) = 𝑁)))
67	56, 66	mpbird 257	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)})
68		sticksstones17.3	. . . 4 ⊢ 𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}
69	68	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)})
70	67, 69	eleqtrrd 2847	. 2 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) ∈ 𝐴)
71		sticksstones17.6	. 2 ⊢ 𝐺 = (𝑏 ∈ 𝐵 ↦ (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))))
72	70, 71	fmptd 7148	1 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  {cab 2717  Vcvv 3488   ⊆ wss 3976   ↦ cmpt 5249  ⟶wf 6569  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448  Fincfn 9003  ℂcc 11182  1c1 11185  ℕ₀cn0 12553  ...cfz 13567  Σcsu 15734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-rp 13058  df-fz 13568  df-fzo 13712  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-sum 15735

This theorem is referenced by:  sticksstones19  42122