Theorem sticksstones18 42166

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

Hypotheses

Ref	Expression
sticksstones18.1	⊢ (𝜑 → 𝑁 ∈ ℕ0)
sticksstones18.2	⊢ (𝜑 → 𝐾 ∈ ℕ0)
sticksstones18.3	⊢ 𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}
sticksstones18.4	⊢ 𝐵 = {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)}
sticksstones18.5	⊢ (𝜑 → 𝑍:(1...𝐾)–1-1-onto→𝑆)
sticksstones18.6	⊢ 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))))

Assertion

Ref	Expression
sticksstones18	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

Distinct variable groups:   𝐴,𝑎,𝑖,𝑥   𝐵,𝑎   𝑔,𝐾,𝑖   𝑔,𝑁   ℎ,𝑁   𝑆,ℎ,𝑖,𝑥   ℎ,𝑍,𝑖,𝑥   𝑔,𝑎   ℎ,𝑎   𝜑,𝑎,𝑖,𝑥

Allowed substitution hints:   𝜑(𝑔,ℎ)   𝐴(𝑔,ℎ)   𝐵(𝑥,𝑔,ℎ,𝑖)   𝑆(𝑔,𝑎)   𝐹(𝑥,𝑔,ℎ,𝑖,𝑎)   𝐾(𝑥,ℎ,𝑎)   𝑁(𝑥,𝑖,𝑎)   𝑍(𝑔,𝑎)

Proof of Theorem sticksstones18

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sticksstones18.3	. . . . . . . . . . . . . 14 ⊢ 𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}
2	1	eqimssi 4043	. . . . . . . . . . . . 13 ⊢ 𝐴 ⊆ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}
3	2	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)})
4	3	sseld 3981	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑎 ∈ 𝐴 → 𝑎 ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}))
5	4	imp 406	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)})
6		vex 3483	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
7		feq1 6715	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑎 → (𝑔:(1...𝐾)⟶ℕ0 ↔ 𝑎:(1...𝐾)⟶ℕ0))
8		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 = 𝑎 ∧ 𝑖 ∈ (1...𝐾)) → 𝑔 = 𝑎)
9	8	fveq1d 6907	. . . . . . . . . . . . . 14 ⊢ ((𝑔 = 𝑎 ∧ 𝑖 ∈ (1...𝐾)) → (𝑔‘𝑖) = (𝑎‘𝑖))
10	9	sumeq2dv 15739	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑎 → Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = Σ𝑖 ∈ (1...𝐾)(𝑎‘𝑖))
11	10	eqeq1d 2738	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑎 → (Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁 ↔ Σ𝑖 ∈ (1...𝐾)(𝑎‘𝑖) = 𝑁))
12	7, 11	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑎 → ((𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁) ↔ (𝑎:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑎‘𝑖) = 𝑁)))
13	6, 12	elab 3678	. . . . . . . . . 10 ⊢ (𝑎 ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)} ↔ (𝑎:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑎‘𝑖) = 𝑁))
14	5, 13	sylib 218	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑎:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑎‘𝑖) = 𝑁))
15	14	simpld 494	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎:(1...𝐾)⟶ℕ0)
16	15	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝑆) → 𝑎:(1...𝐾)⟶ℕ0)
17		sticksstones18.5	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍:(1...𝐾)–1-1-onto→𝑆)
18		f1ocnv 6859	. . . . . . . . . . 11 ⊢ (𝑍:(1...𝐾)–1-1-onto→𝑆 → ◡𝑍:𝑆–1-1-onto→(1...𝐾))
19	17, 18	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ◡𝑍:𝑆–1-1-onto→(1...𝐾))
20		f1of 6847	. . . . . . . . . 10 ⊢ (◡𝑍:𝑆–1-1-onto→(1...𝐾) → ◡𝑍:𝑆⟶(1...𝐾))
21	19, 20	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ◡𝑍:𝑆⟶(1...𝐾))
22	21	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ◡𝑍:𝑆⟶(1...𝐾))
23	22	ffvelcdmda 7103	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝑆) → (◡𝑍‘𝑥) ∈ (1...𝐾))
24	16, 23	ffvelcdmd 7104	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝑆) → (𝑎‘(◡𝑍‘𝑥)) ∈ ℕ0)
25	24	fmpttd 7134	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))):𝑆⟶ℕ0)
26		eqidd 2737	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) → (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))))
27		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) ∧ 𝑥 = 𝑖) → 𝑥 = 𝑖)
28	27	fveq2d 6909	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) ∧ 𝑥 = 𝑖) → (◡𝑍‘𝑥) = (◡𝑍‘𝑖))
29	28	fveq2d 6909	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) ∧ 𝑥 = 𝑖) → (𝑎‘(◡𝑍‘𝑥)) = (𝑎‘(◡𝑍‘𝑖)))
30		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) → 𝑖 ∈ 𝑆)
31		fvexd 6920	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) → (𝑎‘(◡𝑍‘𝑖)) ∈ V)
32	26, 29, 30, 31	fvmptd 7022	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) → ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖) = (𝑎‘(◡𝑍‘𝑖)))
33	32	sumeq2dv 15739	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑖 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖) = Σ𝑖 ∈ 𝑆 (𝑎‘(◡𝑍‘𝑖)))
34		fveq2 6905	. . . . . . . . 9 ⊢ (𝑛 = (◡𝑍‘𝑖) → (𝑎‘𝑛) = (𝑎‘(◡𝑍‘𝑖)))
35		fzfid 14015	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (1...𝐾) ∈ Fin)
36	17	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑍:(1...𝐾)–1-1-onto→𝑆)
37		f1oenfi 9220	. . . . . . . . . . . 12 ⊢ (((1...𝐾) ∈ Fin ∧ 𝑍:(1...𝐾)–1-1-onto→𝑆) → (1...𝐾) ≈ 𝑆)
38	35, 36, 37	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (1...𝐾) ≈ 𝑆)
39	38	ensymd 9046	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑆 ≈ (1...𝐾))
40		enfii 9227	. . . . . . . . . 10 ⊢ (((1...𝐾) ∈ Fin ∧ 𝑆 ≈ (1...𝐾)) → 𝑆 ∈ Fin)
41	35, 39, 40	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑆 ∈ Fin)
42	19	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ◡𝑍:𝑆–1-1-onto→(1...𝐾))
43		eqidd 2737	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) → (◡𝑍‘𝑖) = (◡𝑍‘𝑖))
44		nn0sscn 12533	. . . . . . . . . . . 12 ⊢ ℕ0 ⊆ ℂ
45	44	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ℕ0 ⊆ ℂ)
46		fss 6751	. . . . . . . . . . 11 ⊢ ((𝑎:(1...𝐾)⟶ℕ0 ∧ ℕ0 ⊆ ℂ) → 𝑎:(1...𝐾)⟶ℂ)
47	15, 45, 46	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎:(1...𝐾)⟶ℂ)
48	47	ffvelcdmda 7103	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑛 ∈ (1...𝐾)) → (𝑎‘𝑛) ∈ ℂ)
49	34, 41, 42, 43, 48	fsumf1o 15760	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑛 ∈ (1...𝐾)(𝑎‘𝑛) = Σ𝑖 ∈ 𝑆 (𝑎‘(◡𝑍‘𝑖)))
50	49	eqcomd 2742	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑖 ∈ 𝑆 (𝑎‘(◡𝑍‘𝑖)) = Σ𝑛 ∈ (1...𝐾)(𝑎‘𝑛))
51		fveq2 6905	. . . . . . . . . 10 ⊢ (𝑛 = 𝑖 → (𝑎‘𝑛) = (𝑎‘𝑖))
52	51	cbvsumv 15733	. . . . . . . . 9 ⊢ Σ𝑛 ∈ (1...𝐾)(𝑎‘𝑛) = Σ𝑖 ∈ (1...𝐾)(𝑎‘𝑖)
53	52	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑛 ∈ (1...𝐾)(𝑎‘𝑛) = Σ𝑖 ∈ (1...𝐾)(𝑎‘𝑖))
54	14	simprd 495	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑖 ∈ (1...𝐾)(𝑎‘𝑖) = 𝑁)
55	53, 54	eqtrd 2776	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑛 ∈ (1...𝐾)(𝑎‘𝑛) = 𝑁)
56	50, 55	eqtrd 2776	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑖 ∈ 𝑆 (𝑎‘(◡𝑍‘𝑖)) = 𝑁)
57	33, 56	eqtrd 2776	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑖 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖) = 𝑁)
58	25, 57	jca 511	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))):𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖) = 𝑁))
59		fzfid 14015	. . . . . . . 8 ⊢ (𝜑 → (1...𝐾) ∈ Fin)
60	59	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (1...𝐾) ∈ Fin)
61	59, 17, 37	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (1...𝐾) ≈ 𝑆)
62	61	ensymd 9046	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ≈ (1...𝐾))
63	62	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑆 ≈ (1...𝐾))
64	60, 63, 40	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑆 ∈ Fin)
65	64	mptexd 7245	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) ∈ V)
66		feq1 6715	. . . . . . 7 ⊢ (ℎ = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) → (ℎ:𝑆⟶ℕ0 ↔ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))):𝑆⟶ℕ0))
67		simpl 482	. . . . . . . . . 10 ⊢ ((ℎ = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) ∧ 𝑖 ∈ 𝑆) → ℎ = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))))
68	67	fveq1d 6907	. . . . . . . . 9 ⊢ ((ℎ = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) ∧ 𝑖 ∈ 𝑆) → (ℎ‘𝑖) = ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖))
69	68	sumeq2dv 15739	. . . . . . . 8 ⊢ (ℎ = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) → Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = Σ𝑖 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖))
70	69	eqeq1d 2738	. . . . . . 7 ⊢ (ℎ = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) → (Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁 ↔ Σ𝑖 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖) = 𝑁))
71	66, 70	anbi12d 632	. . . . . 6 ⊢ (ℎ = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) → ((ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁) ↔ ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))):𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖) = 𝑁)))
72	71	elabg 3675	. . . . 5 ⊢ ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) ∈ V → ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) ∈ {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)} ↔ ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))):𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖) = 𝑁)))
73	65, 72	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) ∈ {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)} ↔ ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))):𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖) = 𝑁)))
74	58, 73	mpbird 257	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) ∈ {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)})
75		sticksstones18.4	. . . 4 ⊢ 𝐵 = {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)}
76	75	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐵 = {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)})
77	74, 76	eleqtrrd 2843	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) ∈ 𝐵)
78		sticksstones18.6	. 2 ⊢ 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))))
79	77, 78	fmptd 7133	1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  {cab 2713  Vcvv 3479   ⊆ wss 3950   class class class wbr 5142   ↦ cmpt 5224  ^◡ccnv 5683  ⟶wf 6556  –1-1-onto→wf1o 6559  ‘cfv 6560  (class class class)co 7432   ≈ cen 8983  Fincfn 8986  ℂcc 11154  1c1 11157  ℕ₀cn0 12528  ...cfz 13548  Σcsu 15723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-oi 9551  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-n0 12529  df-z 12616  df-uz 12880  df-rp 13036  df-fz 13549  df-fzo 13696  df-seq 14044  df-exp 14104  df-hash 14371  df-cj 15139  df-re 15140  df-im 15141  df-sqrt 15275  df-abs 15276  df-clim 15525  df-sum 15724

This theorem is referenced by:  sticksstones19  42167