Theorem sticksstones17 42785

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 3998	. . . . . . . . . . . . . . 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 3937	. . . . . . . . . . . . 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 3460	. . . . . . . . . . . . 13 ⊢ 𝑏 ∈ V
7		feq1 6671	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑏 → (ℎ:𝑆⟶ℕ0 ↔ 𝑏:𝑆⟶ℕ0))
8		simpl 486	. . . . . . . . . . . . . . . . 17 ⊢ ((ℎ = 𝑏 ∧ 𝑖 ∈ 𝑆) → ℎ = 𝑏)
9	8	fveq1d 6871	. . . . . . . . . . . . . . . 16 ⊢ ((ℎ = 𝑏 ∧ 𝑖 ∈ 𝑆) → (ℎ‘𝑖) = (𝑏‘𝑖))
10	9	sumeq2dv 15731	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝑏 → Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = Σ𝑖 ∈ 𝑆 (𝑏‘𝑖))
11	10	eqeq1d 2766	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑏 → (Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁 ↔ Σ𝑖 ∈ 𝑆 (𝑏‘𝑖) = 𝑁))
12	7, 11	anbi12d 641	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑏 → ((ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁) ↔ (𝑏:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (𝑏‘𝑖) = 𝑁)))
13	6, 12	elab 3640	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)} ↔ (𝑏:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (𝑏‘𝑖) = 𝑁))
14	5, 13	sylib 220	. . . . . . . . . . 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 1123	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑦 ∈ (1...𝐾)) → 𝑏:𝑆⟶ℕ0)
18		sticksstones17.5	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍:(1...𝐾)–1-1-onto→𝑆)
19		f1of 6808	. . . . . . . . . . . . 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 1123	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑦 ∈ (1...𝐾)) → 𝑍:(1...𝐾)⟶𝑆)
24		simp3 1152	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑦 ∈ (1...𝐾)) → 𝑦 ∈ (1...𝐾))
25	23, 24	ffvelcdmd 7068	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑦 ∈ (1...𝐾)) → (𝑍‘𝑦) ∈ 𝑆)
26	17, 25	ffvelcdmd 7068	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑦 ∈ (1...𝐾)) → (𝑏‘(𝑍‘𝑦)) ∈ ℕ0)
27	26	3expa 1132	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (1...𝐾)) → (𝑏‘(𝑍‘𝑦)) ∈ ℕ0)
28	27	fmpttd 7098	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))):(1...𝐾)⟶ℕ0)
29		eqidd 2765	. . . . . . . 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 6873	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) ∧ 𝑦 = 𝑖) → (𝑍‘𝑦) = (𝑍‘𝑖))
32	31	fveq2d 6873	. . . . . . . 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 6884	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) → (𝑏‘(𝑍‘𝑖)) ∈ V)
35	29, 32, 33, 34	fvmptd 6985	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) → ((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖) = (𝑏‘(𝑍‘𝑖)))
36	35	sumeq2dv 15731	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖) = Σ𝑖 ∈ (1...𝐾)(𝑏‘(𝑍‘𝑖)))
37		fveq2 6869	. . . . . . . . 9 ⊢ (𝑠 = (𝑍‘𝑖) → (𝑏‘𝑠) = (𝑏‘(𝑍‘𝑖)))
38		fzfi 13987	. . . . . . . . . 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 2765	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) → (𝑍‘𝑖) = (𝑍‘𝑖))
42		nn0sscn 12488	. . . . . . . . . . . 12 ⊢ ℕ0 ⊆ ℂ
43	42	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ℕ0 ⊆ ℂ)
44		fss 6710	. . . . . . . . . . 11 ⊢ ((𝑏:𝑆⟶ℕ0 ∧ ℕ0 ⊆ ℂ) → 𝑏:𝑆⟶ℂ)
45	15, 43, 44	syl2anc 593	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏:𝑆⟶ℂ)
46	45	ffvelcdmda 7067	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → (𝑏‘𝑠) ∈ ℂ)
47	37, 39, 40, 41, 46	fsumf1o 15752	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑠 ∈ 𝑆 (𝑏‘𝑠) = Σ𝑖 ∈ (1...𝐾)(𝑏‘(𝑍‘𝑖)))
48	47	eqcomd 2770	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑖 ∈ (1...𝐾)(𝑏‘(𝑍‘𝑖)) = Σ𝑠 ∈ 𝑆 (𝑏‘𝑠))
49		fveq2 6869	. . . . . . . . . 10 ⊢ (𝑠 = 𝑖 → (𝑏‘𝑠) = (𝑏‘𝑖))
50	49	cbvsumv 15725	. . . . . . . . 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 2799	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑠 ∈ 𝑆 (𝑏‘𝑠) = 𝑁)
54	48, 53	eqtrd 2799	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑖 ∈ (1...𝐾)(𝑏‘(𝑍‘𝑖)) = 𝑁)
55	36, 54	eqtrd 2799	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖) = 𝑁)
56	28, 55	jca 519	. . . 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 13988	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (1...𝐾) ∈ Fin)
58	57	mptexd 7210	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) ∈ V)
59		feq1 6671	. . . . . . 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 6871	. . . . . . . . 9 ⊢ ((𝑔 = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) ∧ 𝑖 ∈ (1...𝐾)) → (𝑔‘𝑖) = ((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖))
62	61	sumeq2dv 15731	. . . . . . . 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 2766	. . . . . . 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 641	. . . . . 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 3637	. . . . 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 259	. . 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 2867	. 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 7097	1 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  {cab 2742  Vcvv 3456   ⊆ wss 3906   ↦ cmpt 5183  ⟶wf 6519  –1-1-onto→wf1o 6522  ‘cfv 6523  (class class class)co 7398  Fincfn 8929  ℂcc 11073  1c1 11076  ℕ₀cn0 12483  ...cfz 13514  Σcsu 15715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-inf2 9598  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-se 5603  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-isom 6532  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-er 8680  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-sup 9390  df-oi 9460  df-card 9899  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-div 11847  df-nn 12213  df-2 12282  df-3 12283  df-n0 12484  df-z 12571  df-uz 12842  df-rp 12996  df-fz 13515  df-fzo 13662  df-seq 14017  df-exp 14077  df-hash 14346  df-cj 15128  df-re 15129  df-im 15130  df-sqrt 15264  df-abs 15265  df-clim 15517  df-sum 15716

This theorem is referenced by:  sticksstones19  42787