Theorem prplnqu 7621

Description: Membership in the upper cut of a sum of a positive real and a fraction. (Contributed by Jim Kingdon, 16-Jun-2021.)

Hypotheses

Ref	Expression
prplnqu.x	⊢ (𝜑 → 𝑋 ∈ P)
prplnqu.q	⊢ (𝜑 → 𝑄 ∈ Q)
prplnqu.sum	⊢ (𝜑 → 𝐴 ∈ (2nd ‘(𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)))

Assertion

Ref	Expression
prplnqu	⊢ (𝜑 → ∃𝑦 ∈ (2nd ‘𝑋)(𝑦 +Q 𝑄) = 𝐴)

Distinct variable groups:   𝐴,𝑙,𝑢   𝑦,𝐴   𝑄,𝑙,𝑢   𝑦,𝑄   𝑦,𝑋

Allowed substitution hints:   𝜑(𝑦,𝑢,𝑙)   𝑋(𝑢,𝑙)

Proof of Theorem prplnqu

Dummy variables 𝑓 𝑔 ℎ 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prplnqu.q	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ Q)
2		nqprlu 7548	. . . . . . . 8 ⊢ (𝑄 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩ ∈ P)
3	1, 2	syl 14	. . . . . . 7 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩ ∈ P)
4		prplnqu.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ P)
5		ltaddpr 7598	. . . . . . 7 ⊢ ((⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩ ∈ P ∧ 𝑋 ∈ P) → ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩<P (⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩ +P 𝑋))
6	3, 4, 5	syl2anc 411	. . . . . 6 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩<P (⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩ +P 𝑋))
7		addcomprg 7579	. . . . . . 7 ⊢ ((⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩ ∈ P ∧ 𝑋 ∈ P) → (⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩ +P 𝑋) = (𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩))
8	3, 4, 7	syl2anc 411	. . . . . 6 ⊢ (𝜑 → (⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩ +P 𝑋) = (𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩))
9	6, 8	breqtrd 4031	. . . . 5 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩<P (𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩))
10		prplnqu.sum	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (2nd ‘(𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)))
11		addclpr 7538	. . . . . . . . 9 ⊢ ((𝑋 ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩ ∈ P) → (𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩) ∈ P)
12	4, 3, 11	syl2anc 411	. . . . . . . 8 ⊢ (𝜑 → (𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩) ∈ P)
13		prop 7476	. . . . . . . . 9 ⊢ ((𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩) ∈ P → ⟨(1st ‘(𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)), (2nd ‘(𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩))⟩ ∈ P)
14		elprnqu 7483	. . . . . . . . 9 ⊢ ((⟨(1st ‘(𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)), (2nd ‘(𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩))⟩ ∈ P ∧ 𝐴 ∈ (2nd ‘(𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩))) → 𝐴 ∈ Q)
15	13, 14	sylan 283	. . . . . . . 8 ⊢ (((𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩) ∈ P ∧ 𝐴 ∈ (2nd ‘(𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩))) → 𝐴 ∈ Q)
16	12, 10, 15	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ Q)
17		nqpru 7553	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ (𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩) ∈ P) → (𝐴 ∈ (2nd ‘(𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)) ↔ (𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))
18	16, 12, 17	syl2anc 411	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ (2nd ‘(𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)) ↔ (𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))
19	10, 18	mpbid 147	. . . . 5 ⊢ (𝜑 → (𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)
20		ltsopr 7597	. . . . . 6 ⊢ <P Or P
21		ltrelpr 7506	. . . . . 6 ⊢ <P ⊆ (P × P)
22	20, 21	sotri 5026	. . . . 5 ⊢ ((⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩<P (𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩) ∧ (𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) → ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)
23	9, 19, 22	syl2anc 411	. . . 4 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)
24		ltnqpr 7594	. . . . 5 ⊢ ((𝑄 ∈ Q ∧ 𝐴 ∈ Q) → (𝑄 <Q 𝐴 ↔ ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))
25	1, 16, 24	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝑄 <Q 𝐴 ↔ ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))
26	23, 25	mpbird 167	. . 3 ⊢ (𝜑 → 𝑄 <Q 𝐴)
27		ltexnqi 7410	. . 3 ⊢ (𝑄 <Q 𝐴 → ∃𝑧 ∈ Q (𝑄 +Q 𝑧) = 𝐴)
28	26, 27	syl 14	. 2 ⊢ (𝜑 → ∃𝑧 ∈ Q (𝑄 +Q 𝑧) = 𝐴)
29	19	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)
30	1	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑄 ∈ Q)
31		simprl 529	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑧 ∈ Q)
32		addcomnqg 7382	. . . . . . . . . 10 ⊢ ((𝑄 ∈ Q ∧ 𝑧 ∈ Q) → (𝑄 +Q 𝑧) = (𝑧 +Q 𝑄))
33	30, 31, 32	syl2anc 411	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑄 +Q 𝑧) = (𝑧 +Q 𝑄))
34		simprr 531	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑄 +Q 𝑧) = 𝐴)
35	33, 34	eqtr3d 2212	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑧 +Q 𝑄) = 𝐴)
36		breq2 4009	. . . . . . . . . 10 ⊢ ((𝑧 +Q 𝑄) = 𝐴 → (𝑙 <Q (𝑧 +Q 𝑄) ↔ 𝑙 <Q 𝐴))
37	36	abbidv 2295	. . . . . . . . 9 ⊢ ((𝑧 +Q 𝑄) = 𝐴 → {𝑙 ∣ 𝑙 <Q (𝑧 +Q 𝑄)} = {𝑙 ∣ 𝑙 <Q 𝐴})
38		breq1 4008	. . . . . . . . . 10 ⊢ ((𝑧 +Q 𝑄) = 𝐴 → ((𝑧 +Q 𝑄) <Q 𝑢 ↔ 𝐴 <Q 𝑢))
39	38	abbidv 2295	. . . . . . . . 9 ⊢ ((𝑧 +Q 𝑄) = 𝐴 → {𝑢 ∣ (𝑧 +Q 𝑄) <Q 𝑢} = {𝑢 ∣ 𝐴 <Q 𝑢})
40	37, 39	opeq12d 3788	. . . . . . . 8 ⊢ ((𝑧 +Q 𝑄) = 𝐴 → ⟨{𝑙 ∣ 𝑙 <Q (𝑧 +Q 𝑄)}, {𝑢 ∣ (𝑧 +Q 𝑄) <Q 𝑢}⟩ = ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)
41	35, 40	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → ⟨{𝑙 ∣ 𝑙 <Q (𝑧 +Q 𝑄)}, {𝑢 ∣ (𝑧 +Q 𝑄) <Q 𝑢}⟩ = ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)
42		addnqpr 7562	. . . . . . . 8 ⊢ ((𝑧 ∈ Q ∧ 𝑄 ∈ Q) → ⟨{𝑙 ∣ 𝑙 <Q (𝑧 +Q 𝑄)}, {𝑢 ∣ (𝑧 +Q 𝑄) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q 𝑧}, {𝑢 ∣ 𝑧 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩))
43	31, 30, 42	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → ⟨{𝑙 ∣ 𝑙 <Q (𝑧 +Q 𝑄)}, {𝑢 ∣ (𝑧 +Q 𝑄) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q 𝑧}, {𝑢 ∣ 𝑧 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩))
44	41, 43	eqtr3d 2212	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q 𝑧}, {𝑢 ∣ 𝑧 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩))
45	29, 44	breqtrd 4031	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)<P (⟨{𝑙 ∣ 𝑙 <Q 𝑧}, {𝑢 ∣ 𝑧 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩))
46		ltaprg 7620	. . . . . . 7 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P) → (𝑓<P 𝑔 ↔ (ℎ +P 𝑓)<P (ℎ +P 𝑔)))
47	46	adantl 277	. . . . . 6 ⊢ (((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P)) → (𝑓<P 𝑔 ↔ (ℎ +P 𝑓)<P (ℎ +P 𝑔)))
48	4	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑋 ∈ P)
49		nqprlu 7548	. . . . . . 7 ⊢ (𝑧 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝑧}, {𝑢 ∣ 𝑧 <Q 𝑢}⟩ ∈ P)
50	31, 49	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → ⟨{𝑙 ∣ 𝑙 <Q 𝑧}, {𝑢 ∣ 𝑧 <Q 𝑢}⟩ ∈ P)
51	30, 2	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩ ∈ P)
52		addcomprg 7579	. . . . . . 7 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
53	52	adantl 277	. . . . . 6 ⊢ (((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
54	47, 48, 50, 51, 53	caovord2d 6046	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑋<P ⟨{𝑙 ∣ 𝑙 <Q 𝑧}, {𝑢 ∣ 𝑧 <Q 𝑢}⟩ ↔ (𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)<P (⟨{𝑙 ∣ 𝑙 <Q 𝑧}, {𝑢 ∣ 𝑧 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)))
55	45, 54	mpbird 167	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑋<P ⟨{𝑙 ∣ 𝑙 <Q 𝑧}, {𝑢 ∣ 𝑧 <Q 𝑢}⟩)
56		nqpru 7553	. . . . 5 ⊢ ((𝑧 ∈ Q ∧ 𝑋 ∈ P) → (𝑧 ∈ (2nd ‘𝑋) ↔ 𝑋<P ⟨{𝑙 ∣ 𝑙 <Q 𝑧}, {𝑢 ∣ 𝑧 <Q 𝑢}⟩))
57	31, 48, 56	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑧 ∈ (2nd ‘𝑋) ↔ 𝑋<P ⟨{𝑙 ∣ 𝑙 <Q 𝑧}, {𝑢 ∣ 𝑧 <Q 𝑢}⟩))
58	55, 57	mpbird 167	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑧 ∈ (2nd ‘𝑋))
59		oveq1 5884	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦 +Q 𝑄) = (𝑧 +Q 𝑄))
60	59	eqeq1d 2186	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝑦 +Q 𝑄) = 𝐴 ↔ (𝑧 +Q 𝑄) = 𝐴))
61	60	rspcev 2843	. . 3 ⊢ ((𝑧 ∈ (2nd ‘𝑋) ∧ (𝑧 +Q 𝑄) = 𝐴) → ∃𝑦 ∈ (2nd ‘𝑋)(𝑦 +Q 𝑄) = 𝐴)
62	58, 35, 61	syl2anc 411	. 2 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → ∃𝑦 ∈ (2nd ‘𝑋)(𝑦 +Q 𝑄) = 𝐴)
63	28, 62	rexlimddv 2599	1 ⊢ (𝜑 → ∃𝑦 ∈ (2nd ‘𝑋)(𝑦 +Q 𝑄) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  {cab 2163  ∃wrex 2456  ⟨cop 3597   class class class wbr 4005  ‘cfv 5218  (class class class)co 5877  1^st c1st 6141  2^nd c2nd 6142  Qcnq 7281   +_Q cplq 7283   <_Q cltq 7286  Pcnp 7292   +_P cpp 7294  <_P cltp 7296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-2o 6420  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-enq0 7425  df-nq0 7426  df-0nq0 7427  df-plq0 7428  df-mq0 7429  df-inp 7467  df-iplp 7469  df-iltp 7471

This theorem is referenced by:  caucvgprprlemexbt  7707