Theorem prplnqu 7276

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 7203	. . . . . . . 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 7253	. . . . . . 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 404	. . . . . 6 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩<P (⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩ +P 𝑋))
7		addcomprg 7234	. . . . . . 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 404	. . . . . 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 3891	. . . . 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 7193	. . . . . . . . 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 404	. . . . . . . 8 ⊢ (𝜑 → (𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩) ∈ P)
13		prop 7131	. . . . . . . . 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 7138	. . . . . . . . 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 278	. . . . . . . 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 404	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ Q)
17		nqpru 7208	. . . . . . 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 404	. . . . . 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 146	. . . . 5 ⊢ (𝜑 → (𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)
20		ltsopr 7252	. . . . . 6 ⊢ <P Or P
21		ltrelpr 7161	. . . . . 6 ⊢ <P ⊆ (P × P)
22	20, 21	sotri 4860	. . . . 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 404	. . . 4 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)
24		ltnqpr 7249	. . . . 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 404	. . . 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 166	. . 3 ⊢ (𝜑 → 𝑄 <Q 𝐴)
27		ltexnqi 7065	. . 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 271	. . . . . 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 271	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑄 ∈ Q)
31		simprl 499	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑧 ∈ Q)
32		addcomnqg 7037	. . . . . . . . . 10 ⊢ ((𝑄 ∈ Q ∧ 𝑧 ∈ Q) → (𝑄 +Q 𝑧) = (𝑧 +Q 𝑄))
33	30, 31, 32	syl2anc 404	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑄 +Q 𝑧) = (𝑧 +Q 𝑄))
34		simprr 500	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑄 +Q 𝑧) = 𝐴)
35	33, 34	eqtr3d 2129	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑧 +Q 𝑄) = 𝐴)
36		breq2 3871	. . . . . . . . . 10 ⊢ ((𝑧 +Q 𝑄) = 𝐴 → (𝑙 <Q (𝑧 +Q 𝑄) ↔ 𝑙 <Q 𝐴))
37	36	abbidv 2212	. . . . . . . . 9 ⊢ ((𝑧 +Q 𝑄) = 𝐴 → {𝑙 ∣ 𝑙 <Q (𝑧 +Q 𝑄)} = {𝑙 ∣ 𝑙 <Q 𝐴})
38		breq1 3870	. . . . . . . . . 10 ⊢ ((𝑧 +Q 𝑄) = 𝐴 → ((𝑧 +Q 𝑄) <Q 𝑢 ↔ 𝐴 <Q 𝑢))
39	38	abbidv 2212	. . . . . . . . 9 ⊢ ((𝑧 +Q 𝑄) = 𝐴 → {𝑢 ∣ (𝑧 +Q 𝑄) <Q 𝑢} = {𝑢 ∣ 𝐴 <Q 𝑢})
40	37, 39	opeq12d 3652	. . . . . . . 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 7217	. . . . . . . 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 404	. . . . . . 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 2129	. . . . . 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 3891	. . . . 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 7275	. . . . . . 7 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P) → (𝑓<P 𝑔 ↔ (ℎ +P 𝑓)<P (ℎ +P 𝑔)))
47	46	adantl 272	. . . . . 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 271	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑋 ∈ P)
49		nqprlu 7203	. . . . . . 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 7234	. . . . . . 7 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
53	52	adantl 272	. . . . . 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 5852	. . . . 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 166	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑋<P ⟨{𝑙 ∣ 𝑙 <Q 𝑧}, {𝑢 ∣ 𝑧 <Q 𝑢}⟩)
56		nqpru 7208	. . . . 5 ⊢ ((𝑧 ∈ Q ∧ 𝑋 ∈ P) → (𝑧 ∈ (2nd ‘𝑋) ↔ 𝑋<P ⟨{𝑙 ∣ 𝑙 <Q 𝑧}, {𝑢 ∣ 𝑧 <Q 𝑢}⟩))
57	31, 48, 56	syl2anc 404	. . . 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 166	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑧 ∈ (2nd ‘𝑋))
59		oveq1 5697	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦 +Q 𝑄) = (𝑧 +Q 𝑄))
60	59	eqeq1d 2103	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝑦 +Q 𝑄) = 𝐴 ↔ (𝑧 +Q 𝑄) = 𝐴))
61	60	rspcev 2736	. . 3 ⊢ ((𝑧 ∈ (2nd ‘𝑋) ∧ (𝑧 +Q 𝑄) = 𝐴) → ∃𝑦 ∈ (2nd ‘𝑋)(𝑦 +Q 𝑄) = 𝐴)
62	58, 35, 61	syl2anc 404	. 2 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → ∃𝑦 ∈ (2nd ‘𝑋)(𝑦 +Q 𝑄) = 𝐴)
63	28, 62	rexlimddv 2507	1 ⊢ (𝜑 → ∃𝑦 ∈ (2nd ‘𝑋)(𝑦 +Q 𝑄) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 927   = wceq 1296   ∈ wcel 1445  {cab 2081  ∃wrex 2371  ⟨cop 3469   class class class wbr 3867  ‘cfv 5049  (class class class)co 5690  1^st c1st 5947  2^nd c2nd 5948  Qcnq 6936   +_Q cplq 6938   <_Q cltq 6941  Pcnp 6947   +_P cpp 6949  <_P cltp 6951

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431

This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-eprel 4140  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-irdg 6173  df-1o 6219  df-2o 6220  df-oadd 6223  df-omul 6224  df-er 6332  df-ec 6334  df-qs 6338  df-ni 6960  df-pli 6961  df-mi 6962  df-lti 6963  df-plpq 7000  df-mpq 7001  df-enq 7003  df-nqqs 7004  df-plqqs 7005  df-mqqs 7006  df-1nqqs 7007  df-rq 7008  df-ltnqqs 7009  df-enq0 7080  df-nq0 7081  df-0nq0 7082  df-plq0 7083  df-mq0 7084  df-inp 7122  df-iplp 7124  df-iltp 7126

This theorem is referenced by:  caucvgprprlemexbt  7362