Theorem sticksstones17 40571

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 4002	. . . . . . . . . . . . . . 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 3943	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑏 ∈ 𝐵 → 𝑏 ∈ {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)}))
5	4	imp 407	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)})
6		vex 3449	. . . . . . . . . . . . 13 ⊢ 𝑏 ∈ V
7		feq1 6649	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑏 → (ℎ:𝑆⟶ℕ0 ↔ 𝑏:𝑆⟶ℕ0))
8		simpl 483	. . . . . . . . . . . . . . . . 17 ⊢ ((ℎ = 𝑏 ∧ 𝑖 ∈ 𝑆) → ℎ = 𝑏)
9	8	fveq1d 6844	. . . . . . . . . . . . . . . 16 ⊢ ((ℎ = 𝑏 ∧ 𝑖 ∈ 𝑆) → (ℎ‘𝑖) = (𝑏‘𝑖))
10	9	sumeq2dv 15588	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝑏 → Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = Σ𝑖 ∈ 𝑆 (𝑏‘𝑖))
11	10	eqeq1d 2738	. . . . . . . . . . . . . 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 3630	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)} ↔ (𝑏:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (𝑏‘𝑖) = 𝑁))
14	5, 13	sylib 217	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (𝑏‘𝑖) = 𝑁))
15	14	simpld 495	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏:𝑆⟶ℕ0)
16	15	adantr 481	. . . . . . . . 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 6784	. . . . . . . . . . . . 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 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑍:(1...𝐾)⟶𝑆)
22	21	adantr 481	. . . . . . . . . 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 7036	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑦 ∈ (1...𝐾)) → (𝑍‘𝑦) ∈ 𝑆)
26	17, 25	ffvelcdmd 7036	. . . . . . 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 7063	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))):(1...𝐾)⟶ℕ0)
29		eqidd 2737	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) → (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))))
30		simpr 485	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) ∧ 𝑦 = 𝑖) → 𝑦 = 𝑖)
31	30	fveq2d 6846	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) ∧ 𝑦 = 𝑖) → (𝑍‘𝑦) = (𝑍‘𝑖))
32	31	fveq2d 6846	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) ∧ 𝑦 = 𝑖) → (𝑏‘(𝑍‘𝑦)) = (𝑏‘(𝑍‘𝑖)))
33		simpr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) → 𝑖 ∈ (1...𝐾))
34		fvexd 6857	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) → (𝑏‘(𝑍‘𝑖)) ∈ V)
35	29, 32, 33, 34	fvmptd 6955	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) → ((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖) = (𝑏‘(𝑍‘𝑖)))
36	35	sumeq2dv 15588	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖) = Σ𝑖 ∈ (1...𝐾)(𝑏‘(𝑍‘𝑖)))
37		fveq2 6842	. . . . . . . . 9 ⊢ (𝑠 = (𝑍‘𝑖) → (𝑏‘𝑠) = (𝑏‘(𝑍‘𝑖)))
38		fzfi 13877	. . . . . . . . . 10 ⊢ (1...𝐾) ∈ Fin
39	38	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (1...𝐾) ∈ Fin)
40	18	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑍:(1...𝐾)–1-1-onto→𝑆)
41		eqidd 2737	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) → (𝑍‘𝑖) = (𝑍‘𝑖))
42		nn0sscn 12418	. . . . . . . . . . . 12 ⊢ ℕ0 ⊆ ℂ
43	42	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ℕ0 ⊆ ℂ)
44		fss 6685	. . . . . . . . . . 11 ⊢ ((𝑏:𝑆⟶ℕ0 ∧ ℕ0 ⊆ ℂ) → 𝑏:𝑆⟶ℂ)
45	15, 43, 44	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏:𝑆⟶ℂ)
46	45	ffvelcdmda 7035	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → (𝑏‘𝑠) ∈ ℂ)
47	37, 39, 40, 41, 46	fsumf1o 15608	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑠 ∈ 𝑆 (𝑏‘𝑠) = Σ𝑖 ∈ (1...𝐾)(𝑏‘(𝑍‘𝑖)))
48	47	eqcomd 2742	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑖 ∈ (1...𝐾)(𝑏‘(𝑍‘𝑖)) = Σ𝑠 ∈ 𝑆 (𝑏‘𝑠))
49		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑠 = 𝑖 → (𝑏‘𝑠) = (𝑏‘𝑖))
50	49	cbvsumv 15581	. . . . . . . . 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 496	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑖 ∈ 𝑆 (𝑏‘𝑖) = 𝑁)
53	51, 52	eqtrd 2776	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑠 ∈ 𝑆 (𝑏‘𝑠) = 𝑁)
54	48, 53	eqtrd 2776	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑖 ∈ (1...𝐾)(𝑏‘(𝑍‘𝑖)) = 𝑁)
55	36, 54	eqtrd 2776	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖) = 𝑁)
56	28, 55	jca 512	. . . 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 13878	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (1...𝐾) ∈ Fin)
58	57	mptexd 7174	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) ∈ V)
59		feq1 6649	. . . . . . 7 ⊢ (𝑔 = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) → (𝑔:(1...𝐾)⟶ℕ0 ↔ (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))):(1...𝐾)⟶ℕ0))
60		simpl 483	. . . . . . . . . 10 ⊢ ((𝑔 = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) ∧ 𝑖 ∈ (1...𝐾)) → 𝑔 = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))))
61	60	fveq1d 6844	. . . . . . . . 9 ⊢ ((𝑔 = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) ∧ 𝑖 ∈ (1...𝐾)) → (𝑔‘𝑖) = ((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖))
62	61	sumeq2dv 15588	. . . . . . . 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 2738	. . . . . . 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 3628	. . . . 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 256	. . 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 2841	. 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 7062	1 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {cab 2713  Vcvv 3445   ⊆ wss 3910   ↦ cmpt 5188  ⟶wf 6492  –1-1-onto→wf1o 6495  ‘cfv 6496  (class class class)co 7357  Fincfn 8883  ℂcc 11049  1c1 11052  ℕ₀cn0 12413  ...cfz 13424  Σcsu 15570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571

This theorem is referenced by:  sticksstones19  40573