Theorem sticksstones18 43405

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)⟶1-1onto𝑆)
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 3973	. . . . . . . . . . . . 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 3912	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑎 ∈ 𝐴 → 𝑎 ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}))
5	4	imp 412	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)})
6		vex 3458	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
7		feq1 6706	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑎 → (𝑔:(1...𝐾)⟶ℕ0 ↔ 𝑎:(1...𝐾)⟶ℕ0))
8		simpl 488	. . . . . . . . . . . . . . 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 15895	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑎 → Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = Σ𝑖 ∈ (1...𝐾)(𝑎‘𝑖))
11	10	eqeq1d 2765	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑎 → (Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁 ↔ Σ𝑖 ∈ (1...𝐾)(𝑎‘𝑖) = 𝑁))
12	7, 11	anbi12d 644	. . . . . . . . . . 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 3636	. . . . . . . . . 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 221	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑎:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑎‘𝑖) = 𝑁))
15	14	simpld 500	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎:(1...𝐾)⟶ℕ0)
16	15	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝑆) → 𝑎:(1...𝐾)⟶ℕ0)
17		sticksstones18.5	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍:(1...𝐾)⟶1-1onto𝑆)
18		f1ocnv 6856	. . . . . . . . . . 11 ⊢ (𝑍:(1...𝐾)⟶1-1onto𝑆 → ◡𝑍:𝑆⟶1-1onto(1...𝐾))
19	17, 18	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ◡𝑍:𝑆⟶1-1onto(1...𝐾))
20		f1of 6843	. . . . . . . . . 10 ⊢ (◡𝑍:𝑆⟶1-1onto(1...𝐾) → ◡𝑍:𝑆⟶(1...𝐾))
21	19, 20	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ◡𝑍:𝑆⟶(1...𝐾))
22	21	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ◡𝑍:𝑆⟶(1...𝐾))
23	22	ffvelcdmda 7105	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝑆) → (◡𝑍‘𝑥) ∈ (1...𝐾))
24	16, 23	ffvelcdmd 7106	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝑆) → (𝑎‘(◡𝑍‘𝑥)) ∈ ℕ0)
25	24	fmpttd 7136	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))):𝑆⟶ℕ0)
26		eqidd 2764	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) → (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))))
27		simpr 490	. . . . . . . . . 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 490	. . . . . . . 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 7021	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) → ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖) = (𝑎‘(◡𝑍‘𝑖)))
33	32	sumeq2dv 15895	. . . . . 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 14140	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (1...𝐾) ∈ Fin)
36	17	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑍:(1...𝐾)⟶1-1onto𝑆)
37		f1oenfi 9219	. . . . . . . . . . . 12 ⊢ (((1...𝐾) ∈ Fin ∧ 𝑍:(1...𝐾)⟶1-1onto𝑆) → (1...𝐾) ≈ 𝑆)
38	35, 36, 37	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (1...𝐾) ≈ 𝑆)
39	38	ensymd 9057	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑆 ≈ (1...𝐾))
40		enfii 9226	. . . . . . . . . 10 ⊢ (((1...𝐾) ∈ Fin ∧ 𝑆 ≈ (1...𝐾)) → 𝑆 ∈ Fin)
41	35, 39, 40	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑆 ∈ Fin)
42	19	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ◡𝑍:𝑆⟶1-1onto(1...𝐾))
43		eqidd 2764	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) → (◡𝑍‘𝑖) = (◡𝑍‘𝑖))
44		nn0sscn 12635	. . . . . . . . . . . 12 ⊢ ℕ0 ⊆ ℂ
45	44	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ℕ0 ⊆ ℂ)
46		fss 6745	. . . . . . . . . . 11 ⊢ ((𝑎:(1...𝐾)⟶ℕ0 ∧ ℕ0 ⊆ ℂ) → 𝑎:(1...𝐾)⟶ℂ)
47	15, 45, 46	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎:(1...𝐾)⟶ℂ)
48	47	ffvelcdmda 7105	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑛 ∈ (1...𝐾)) → (𝑎‘𝑛) ∈ ℂ)
49	34, 41, 42, 43, 48	fsumf1o 15915	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑛 ∈ (1...𝐾)(𝑎‘𝑛) = Σ𝑖 ∈ 𝑆 (𝑎‘(◡𝑍‘𝑖)))
50	49	eqcomd 2769	. . . . . . 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 15889	. . . . . . . . 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 501	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑖 ∈ (1...𝐾)(𝑎‘𝑖) = 𝑁)
55	53, 54	eqtrd 2798	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑛 ∈ (1...𝐾)(𝑎‘𝑛) = 𝑁)
56	50, 55	eqtrd 2798	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑖 ∈ 𝑆 (𝑎‘(◡𝑍‘𝑖)) = 𝑁)
57	33, 56	eqtrd 2798	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑖 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖) = 𝑁)
58	25, 57	jca 521	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))):𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖) = 𝑁))
59		fzfid 14140	. . . . . . . 8 ⊢ (𝜑 → (1...𝐾) ∈ Fin)
60	59	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (1...𝐾) ∈ Fin)
61	59, 17, 37	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (1...𝐾) ≈ 𝑆)
62	61	ensymd 9057	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ≈ (1...𝐾))
63	62	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑆 ≈ (1...𝐾))
64	60, 63, 40	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑆 ∈ Fin)
65	64	mptexd 7250	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) ∈ V)
66		feq1 6706	. . . . . . 7 ⊢ (ℎ = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) → (ℎ:𝑆⟶ℕ0 ↔ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))):𝑆⟶ℕ0))
67		simpl 488	. . . . . . . . . 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 15895	. . . . . . . 8 ⊢ (ℎ = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) → Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = Σ𝑖 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖))
70	69	eqeq1d 2765	. . . . . . 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 644	. . . . . 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 3633	. . . . 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 18	. . . 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 260	. . 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 2867	. 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 7135	1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  Vcvv 3454   ⊆ wss 3900   class class class wbr 5106   ↦ cmpt 5189  ^◡ccnv 5654  ⟶wf 6551  ⟶1-1ontowf1o 6554  ‘cfv 6555  (class class class)co 7440   ≈ cen 8995  Fincfn 8998  ℂcc 11222  1c1 11225  ℕ₀cn0 12630  ...cfz 13663  Σcsu 15879

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-inf2 9666  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301  ax-pre-sup 11302

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-isom 6564  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-er 8742  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-sup 9458  df-oi 9528  df-card 10044  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-div 11998  df-nn 12360  df-2 12429  df-3 12430  df-n0 12631  df-z 12718  df-uz 12990  df-rp 13145  df-fz 13664  df-fzo 13813  df-seq 14169  df-exp 14229  df-hash 14499  df-cj 15292  df-re 15293  df-im 15294  df-sqrt 15428  df-abs 15429  df-clim 15681  df-sum 15880

This theorem is used by:  sticksstones19  43406