Theorem sticksstones17 39796

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 3949	. . . . . . . . . . . . . . 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 3890	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑏 ∈ 𝐵 → 𝑏 ∈ {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)}))
5	4	imp 410	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)})
6		vex 3405	. . . . . . . . . . . . 13 ⊢ 𝑏 ∈ V
7		feq1 6515	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑏 → (ℎ:𝑆⟶ℕ0 ↔ 𝑏:𝑆⟶ℕ0))
8		simpl 486	. . . . . . . . . . . . . . . . 17 ⊢ ((ℎ = 𝑏 ∧ 𝑖 ∈ 𝑆) → ℎ = 𝑏)
9	8	fveq1d 6708	. . . . . . . . . . . . . . . 16 ⊢ ((ℎ = 𝑏 ∧ 𝑖 ∈ 𝑆) → (ℎ‘𝑖) = (𝑏‘𝑖))
10	9	sumeq2dv 15250	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝑏 → Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = Σ𝑖 ∈ 𝑆 (𝑏‘𝑖))
11	10	eqeq1d 2736	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑏 → (Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁 ↔ Σ𝑖 ∈ 𝑆 (𝑏‘𝑖) = 𝑁))
12	7, 11	anbi12d 634	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑏 → ((ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁) ↔ (𝑏:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (𝑏‘𝑖) = 𝑁)))
13	6, 12	elab 3580	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)} ↔ (𝑏:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (𝑏‘𝑖) = 𝑁))
14	5, 13	sylib 221	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (𝑏‘𝑖) = 𝑁))
15	14	simpld 498	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏:𝑆⟶ℕ0)
16	15	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (1...𝐾)) → 𝑏:𝑆⟶ℕ0)
17	16	3impa 1112	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑦 ∈ (1...𝐾)) → 𝑏:𝑆⟶ℕ0)
18		sticksstones17.5	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍:(1...𝐾)–1-1-onto→𝑆)
19		f1of 6650	. . . . . . . . . . . . 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 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑍:(1...𝐾)⟶𝑆)
22	21	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (1...𝐾)) → 𝑍:(1...𝐾)⟶𝑆)
23	22	3impa 1112	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑦 ∈ (1...𝐾)) → 𝑍:(1...𝐾)⟶𝑆)
24		simp3 1140	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑦 ∈ (1...𝐾)) → 𝑦 ∈ (1...𝐾))
25	23, 24	ffvelrnd 6894	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑦 ∈ (1...𝐾)) → (𝑍‘𝑦) ∈ 𝑆)
26	17, 25	ffvelrnd 6894	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑦 ∈ (1...𝐾)) → (𝑏‘(𝑍‘𝑦)) ∈ ℕ0)
27	26	3expa 1120	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (1...𝐾)) → (𝑏‘(𝑍‘𝑦)) ∈ ℕ0)
28	27	fmpttd 6921	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))):(1...𝐾)⟶ℕ0)
29		eqidd 2735	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) → (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))))
30		simpr 488	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) ∧ 𝑦 = 𝑖) → 𝑦 = 𝑖)
31	30	fveq2d 6710	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) ∧ 𝑦 = 𝑖) → (𝑍‘𝑦) = (𝑍‘𝑖))
32	31	fveq2d 6710	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) ∧ 𝑦 = 𝑖) → (𝑏‘(𝑍‘𝑦)) = (𝑏‘(𝑍‘𝑖)))
33		simpr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) → 𝑖 ∈ (1...𝐾))
34		fvexd 6721	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) → (𝑏‘(𝑍‘𝑖)) ∈ V)
35	29, 32, 33, 34	fvmptd 6814	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) → ((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖) = (𝑏‘(𝑍‘𝑖)))
36	35	sumeq2dv 15250	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖) = Σ𝑖 ∈ (1...𝐾)(𝑏‘(𝑍‘𝑖)))
37		fveq2 6706	. . . . . . . . 9 ⊢ (𝑠 = (𝑍‘𝑖) → (𝑏‘𝑠) = (𝑏‘(𝑍‘𝑖)))
38		fzfi 13528	. . . . . . . . . 10 ⊢ (1...𝐾) ∈ Fin
39	38	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (1...𝐾) ∈ Fin)
40	18	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑍:(1...𝐾)–1-1-onto→𝑆)
41		eqidd 2735	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) → (𝑍‘𝑖) = (𝑍‘𝑖))
42		nn0sscn 12078	. . . . . . . . . . . 12 ⊢ ℕ0 ⊆ ℂ
43	42	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ℕ0 ⊆ ℂ)
44		fss 6551	. . . . . . . . . . 11 ⊢ ((𝑏:𝑆⟶ℕ0 ∧ ℕ0 ⊆ ℂ) → 𝑏:𝑆⟶ℂ)
45	15, 43, 44	syl2anc 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏:𝑆⟶ℂ)
46	45	ffvelrnda 6893	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → (𝑏‘𝑠) ∈ ℂ)
47	37, 39, 40, 41, 46	fsumf1o 15270	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑠 ∈ 𝑆 (𝑏‘𝑠) = Σ𝑖 ∈ (1...𝐾)(𝑏‘(𝑍‘𝑖)))
48	47	eqcomd 2740	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑖 ∈ (1...𝐾)(𝑏‘(𝑍‘𝑖)) = Σ𝑠 ∈ 𝑆 (𝑏‘𝑠))
49		fveq2 6706	. . . . . . . . . 10 ⊢ (𝑠 = 𝑖 → (𝑏‘𝑠) = (𝑏‘𝑖))
50	49	cbvsumv 15243	. . . . . . . . 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 499	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑖 ∈ 𝑆 (𝑏‘𝑖) = 𝑁)
53	51, 52	eqtrd 2774	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑠 ∈ 𝑆 (𝑏‘𝑠) = 𝑁)
54	48, 53	eqtrd 2774	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑖 ∈ (1...𝐾)(𝑏‘(𝑍‘𝑖)) = 𝑁)
55	36, 54	eqtrd 2774	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖) = 𝑁)
56	28, 55	jca 515	. . . 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 13529	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (1...𝐾) ∈ Fin)
58	57	mptexd 7029	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) ∈ V)
59		feq1 6515	. . . . . . 7 ⊢ (𝑔 = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) → (𝑔:(1...𝐾)⟶ℕ0 ↔ (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))):(1...𝐾)⟶ℕ0))
60		simpl 486	. . . . . . . . . 10 ⊢ ((𝑔 = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) ∧ 𝑖 ∈ (1...𝐾)) → 𝑔 = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))))
61	60	fveq1d 6708	. . . . . . . . 9 ⊢ ((𝑔 = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) ∧ 𝑖 ∈ (1...𝐾)) → (𝑔‘𝑖) = ((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖))
62	61	sumeq2dv 15250	. . . . . . . 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 2736	. . . . . . 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 634	. . . . . 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 3578	. . . . 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 260	. . 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 2837	. 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 6920	1 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  {cab 2712  Vcvv 3401   ⊆ wss 3857   ↦ cmpt 5124  ⟶wf 6365  –1-1-onto→wf1o 6368  ‘cfv 6369  (class class class)co 7202  Fincfn 8615  ℂcc 10710  1c1 10713  ℕ₀cn0 12073  ...cfz 13078  Σcsu 15232

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 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-rep 5168  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512  ax-inf2 9245  ax-cnex 10768  ax-resscn 10769  ax-1cn 10770  ax-icn 10771  ax-addcl 10772  ax-addrcl 10773  ax-mulcl 10774  ax-mulrcl 10775  ax-mulcom 10776  ax-addass 10777  ax-mulass 10778  ax-distr 10779  ax-i2m1 10780  ax-1ne0 10781  ax-1rid 10782  ax-rnegex 10783  ax-rrecex 10784  ax-cnre 10785  ax-pre-lttri 10786  ax-pre-lttrn 10787  ax-pre-ltadd 10788  ax-pre-mulgt0 10789  ax-pre-sup 10790

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 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-int 4850  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-se 5499  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-ord 6205  df-on 6206  df-lim 6207  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-isom 6378  df-riota 7159  df-ov 7205  df-oprab 7206  df-mpo 7207  df-om 7634  df-1st 7750  df-2nd 7751  df-wrecs 8036  df-recs 8097  df-rdg 8135  df-1o 8191  df-er 8380  df-en 8616  df-dom 8617  df-sdom 8618  df-fin 8619  df-sup 9047  df-oi 9115  df-card 9538  df-pnf 10852  df-mnf 10853  df-xr 10854  df-ltxr 10855  df-le 10856  df-sub 11047  df-neg 11048  df-div 11473  df-nn 11814  df-2 11876  df-3 11877  df-n0 12074  df-z 12160  df-uz 12422  df-rp 12570  df-fz 13079  df-fzo 13222  df-seq 13558  df-exp 13619  df-hash 13880  df-cj 14645  df-re 14646  df-im 14647  df-sqrt 14781  df-abs 14782  df-clim 15032  df-sum 15233

This theorem is referenced by:  sticksstones19  39798