Theorem sticksstones18 42357

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 3992	. . . . . . . . . . . . 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 3930	. . . . . . . . . . 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 3442	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
7		feq1 6638	. . . . . . . . . . . 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 6834	. . . . . . . . . . . . . 14 ⊢ ((𝑔 = 𝑎 ∧ 𝑖 ∈ (1...𝐾)) → (𝑔‘𝑖) = (𝑎‘𝑖))
10	9	sumeq2dv 15623	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑎 → Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = Σ𝑖 ∈ (1...𝐾)(𝑎‘𝑖))
11	10	eqeq1d 2736	. . . . . . . . . . . 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 3632	. . . . . . . . . 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 6784	. . . . . . . . . . 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 6772	. . . . . . . . . 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 7027	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝑆) → (◡𝑍‘𝑥) ∈ (1...𝐾))
24	16, 23	ffvelcdmd 7028	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝑆) → (𝑎‘(◡𝑍‘𝑥)) ∈ ℕ0)
25	24	fmpttd 7058	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))):𝑆⟶ℕ0)
26		eqidd 2735	. . . . . . . 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 6836	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) ∧ 𝑥 = 𝑖) → (◡𝑍‘𝑥) = (◡𝑍‘𝑖))
29	28	fveq2d 6836	. . . . . . . 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 6847	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) → (𝑎‘(◡𝑍‘𝑖)) ∈ V)
32	26, 29, 30, 31	fvmptd 6946	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) → ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖) = (𝑎‘(◡𝑍‘𝑖)))
33	32	sumeq2dv 15623	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑖 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖) = Σ𝑖 ∈ 𝑆 (𝑎‘(◡𝑍‘𝑖)))
34		fveq2 6832	. . . . . . . . 9 ⊢ (𝑛 = (◡𝑍‘𝑖) → (𝑎‘𝑛) = (𝑎‘(◡𝑍‘𝑖)))
35		fzfid 13894	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (1...𝐾) ∈ Fin)
36	17	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑍:(1...𝐾)–1-1-onto→𝑆)
37		f1oenfi 9101	. . . . . . . . . . . 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 8940	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑆 ≈ (1...𝐾))
40		enfii 9108	. . . . . . . . . 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 2735	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) → (◡𝑍‘𝑖) = (◡𝑍‘𝑖))
44		nn0sscn 12404	. . . . . . . . . . . 12 ⊢ ℕ0 ⊆ ℂ
45	44	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ℕ0 ⊆ ℂ)
46		fss 6676	. . . . . . . . . . 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 7027	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑛 ∈ (1...𝐾)) → (𝑎‘𝑛) ∈ ℂ)
49	34, 41, 42, 43, 48	fsumf1o 15644	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑛 ∈ (1...𝐾)(𝑎‘𝑛) = Σ𝑖 ∈ 𝑆 (𝑎‘(◡𝑍‘𝑖)))
50	49	eqcomd 2740	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑖 ∈ 𝑆 (𝑎‘(◡𝑍‘𝑖)) = Σ𝑛 ∈ (1...𝐾)(𝑎‘𝑛))
51		fveq2 6832	. . . . . . . . . 10 ⊢ (𝑛 = 𝑖 → (𝑎‘𝑛) = (𝑎‘𝑖))
52	51	cbvsumv 15617	. . . . . . . . 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 2769	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑛 ∈ (1...𝐾)(𝑎‘𝑛) = 𝑁)
56	50, 55	eqtrd 2769	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑖 ∈ 𝑆 (𝑎‘(◡𝑍‘𝑖)) = 𝑁)
57	33, 56	eqtrd 2769	. . . . 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 13894	. . . . . . . 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 8940	. . . . . . . 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 7168	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) ∈ V)
66		feq1 6638	. . . . . . 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 6834	. . . . . . . . 9 ⊢ ((ℎ = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) ∧ 𝑖 ∈ 𝑆) → (ℎ‘𝑖) = ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖))
69	68	sumeq2dv 15623	. . . . . . . 8 ⊢ (ℎ = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) → Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = Σ𝑖 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖))
70	69	eqeq1d 2736	. . . . . . 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 3629	. . . . 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 2837	. 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 7057	1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2712  Vcvv 3438   ⊆ wss 3899   class class class wbr 5096   ↦ cmpt 5177  ^◡ccnv 5621  ⟶wf 6486  –1-1-onto→wf1o 6489  ‘cfv 6490  (class class class)co 7356   ≈ cen 8878  Fincfn 8881  ℂcc 11022  1c1 11025  ℕ₀cn0 12399  ...cfz 13421  Σcsu 15607

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-inf2 9548  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101  ax-pre-sup 11102

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-sup 9343  df-oi 9413  df-card 9849  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-div 11793  df-nn 12144  df-2 12206  df-3 12207  df-n0 12400  df-z 12487  df-uz 12750  df-rp 12904  df-fz 13422  df-fzo 13569  df-seq 13923  df-exp 13983  df-hash 14252  df-cj 15020  df-re 15021  df-im 15022  df-sqrt 15156  df-abs 15157  df-clim 15409  df-sum 15608

This theorem is referenced by:  sticksstones19  42358