Theorem prplnqu 7704

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 7631	. . . . . . . 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 7681	. . . . . . 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 7662	. . . . . . 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 4060	. . . . 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 7621	. . . . . . . . 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 7559	. . . . . . . . 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 7566	. . . . . . . . 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 7636	. . . . . . 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 7680	. . . . . 6 ⊢ <P Or P
21		ltrelpr 7589	. . . . . 6 ⊢ <P ⊆ (P × P)
22	20, 21	sotri 5066	. . . . 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 7677	. . . . 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 7493	. . 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 7465	. . . . . . . . . 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 2231	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑧 +Q 𝑄) = 𝐴)
36		breq2 4038	. . . . . . . . . 10 ⊢ ((𝑧 +Q 𝑄) = 𝐴 → (𝑙 <Q (𝑧 +Q 𝑄) ↔ 𝑙 <Q 𝐴))
37	36	abbidv 2314	. . . . . . . . 9 ⊢ ((𝑧 +Q 𝑄) = 𝐴 → {𝑙 ∣ 𝑙 <Q (𝑧 +Q 𝑄)} = {𝑙 ∣ 𝑙 <Q 𝐴})
38		breq1 4037	. . . . . . . . . 10 ⊢ ((𝑧 +Q 𝑄) = 𝐴 → ((𝑧 +Q 𝑄) <Q 𝑢 ↔ 𝐴 <Q 𝑢))
39	38	abbidv 2314	. . . . . . . . 9 ⊢ ((𝑧 +Q 𝑄) = 𝐴 → {𝑢 ∣ (𝑧 +Q 𝑄) <Q 𝑢} = {𝑢 ∣ 𝐴 <Q 𝑢})
40	37, 39	opeq12d 3817	. . . . . . . 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 7645	. . . . . . . 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 2231	. . . . . 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 4060	. . . . 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 7703	. . . . . . 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 7631	. . . . . . 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 7662	. . . . . . 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 6097	. . . . 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 7636	. . . . 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 5932	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦 +Q 𝑄) = (𝑧 +Q 𝑄))
60	59	eqeq1d 2205	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝑦 +Q 𝑄) = 𝐴 ↔ (𝑧 +Q 𝑄) = 𝐴))
61	60	rspcev 2868	. . 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 2619	1 ⊢ (𝜑 → ∃𝑦 ∈ (2nd ‘𝑋)(𝑦 +Q 𝑄) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  {cab 2182  ∃wrex 2476  ⟨cop 3626   class class class wbr 4034  ‘cfv 5259  (class class class)co 5925  1^st c1st 6205  2^nd c2nd 6206  Qcnq 7364   +_Q cplq 7366   <_Q cltq 7369  Pcnp 7375   +_P cpp 7377  <_P cltp 7379

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-2o 6484  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437  df-enq0 7508  df-nq0 7509  df-0nq0 7510  df-plq0 7511  df-mq0 7512  df-inp 7550  df-iplp 7552  df-iltp 7554

This theorem is referenced by:  caucvgprprlemexbt  7790