Theorem prplnqu 7575

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 7502	. . . . . . . 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 7552	. . . . . . 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 409	. . . . . 6 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩<P (⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩ +P 𝑋))
7		addcomprg 7533	. . . . . . 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 409	. . . . . 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 4013	. . . . 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 7492	. . . . . . . . 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 409	. . . . . . . 8 ⊢ (𝜑 → (𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩) ∈ P)
13		prop 7430	. . . . . . . . 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 7437	. . . . . . . . 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 281	. . . . . . . 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 409	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ Q)
17		nqpru 7507	. . . . . . 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 409	. . . . . 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 7551	. . . . . 6 ⊢ <P Or P
21		ltrelpr 7460	. . . . . 6 ⊢ <P ⊆ (P × P)
22	20, 21	sotri 5004	. . . . 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 409	. . . 4 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)
24		ltnqpr 7548	. . . . 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 409	. . . 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 7364	. . 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 274	. . . . . 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 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑄 ∈ Q)
31		simprl 526	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑧 ∈ Q)
32		addcomnqg 7336	. . . . . . . . . 10 ⊢ ((𝑄 ∈ Q ∧ 𝑧 ∈ Q) → (𝑄 +Q 𝑧) = (𝑧 +Q 𝑄))
33	30, 31, 32	syl2anc 409	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑄 +Q 𝑧) = (𝑧 +Q 𝑄))
34		simprr 527	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑄 +Q 𝑧) = 𝐴)
35	33, 34	eqtr3d 2205	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → (𝑧 +Q 𝑄) = 𝐴)
36		breq2 3991	. . . . . . . . . 10 ⊢ ((𝑧 +Q 𝑄) = 𝐴 → (𝑙 <Q (𝑧 +Q 𝑄) ↔ 𝑙 <Q 𝐴))
37	36	abbidv 2288	. . . . . . . . 9 ⊢ ((𝑧 +Q 𝑄) = 𝐴 → {𝑙 ∣ 𝑙 <Q (𝑧 +Q 𝑄)} = {𝑙 ∣ 𝑙 <Q 𝐴})
38		breq1 3990	. . . . . . . . . 10 ⊢ ((𝑧 +Q 𝑄) = 𝐴 → ((𝑧 +Q 𝑄) <Q 𝑢 ↔ 𝐴 <Q 𝑢))
39	38	abbidv 2288	. . . . . . . . 9 ⊢ ((𝑧 +Q 𝑄) = 𝐴 → {𝑢 ∣ (𝑧 +Q 𝑄) <Q 𝑢} = {𝑢 ∣ 𝐴 <Q 𝑢})
40	37, 39	opeq12d 3771	. . . . . . . 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 7516	. . . . . . . 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 409	. . . . . . 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 2205	. . . . . 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 4013	. . . . 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 7574	. . . . . . 7 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P) → (𝑓<P 𝑔 ↔ (ℎ +P 𝑓)<P (ℎ +P 𝑔)))
47	46	adantl 275	. . . . . 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 274	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → 𝑋 ∈ P)
49		nqprlu 7502	. . . . . . 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 7533	. . . . . . 7 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
53	52	adantl 275	. . . . . 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 6020	. . . . 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 7507	. . . . 5 ⊢ ((𝑧 ∈ Q ∧ 𝑋 ∈ P) → (𝑧 ∈ (2nd ‘𝑋) ↔ 𝑋<P ⟨{𝑙 ∣ 𝑙 <Q 𝑧}, {𝑢 ∣ 𝑧 <Q 𝑢}⟩))
57	31, 48, 56	syl2anc 409	. . . 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 5858	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦 +Q 𝑄) = (𝑧 +Q 𝑄))
60	59	eqeq1d 2179	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝑦 +Q 𝑄) = 𝐴 ↔ (𝑧 +Q 𝑄) = 𝐴))
61	60	rspcev 2834	. . 3 ⊢ ((𝑧 ∈ (2nd ‘𝑋) ∧ (𝑧 +Q 𝑄) = 𝐴) → ∃𝑦 ∈ (2nd ‘𝑋)(𝑦 +Q 𝑄) = 𝐴)
62	58, 35, 61	syl2anc 409	. 2 ⊢ ((𝜑 ∧ (𝑧 ∈ Q ∧ (𝑄 +Q 𝑧) = 𝐴)) → ∃𝑦 ∈ (2nd ‘𝑋)(𝑦 +Q 𝑄) = 𝐴)
63	28, 62	rexlimddv 2592	1 ⊢ (𝜑 → ∃𝑦 ∈ (2nd ‘𝑋)(𝑦 +Q 𝑄) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 973   = wceq 1348   ∈ wcel 2141  {cab 2156  ∃wrex 2449  ⟨cop 3584   class class class wbr 3987  ‘cfv 5196  (class class class)co 5851  1^st c1st 6115  2^nd c2nd 6116  Qcnq 7235   +_Q cplq 7237   <_Q cltq 7240  Pcnp 7246   +_P cpp 7248  <_P cltp 7250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-eprel 4272  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-irdg 6347  df-1o 6393  df-2o 6394  df-oadd 6397  df-omul 6398  df-er 6511  df-ec 6513  df-qs 6517  df-ni 7259  df-pli 7260  df-mi 7261  df-lti 7262  df-plpq 7299  df-mpq 7300  df-enq 7302  df-nqqs 7303  df-plqqs 7304  df-mqqs 7305  df-1nqqs 7306  df-rq 7307  df-ltnqqs 7308  df-enq0 7379  df-nq0 7380  df-0nq0 7381  df-plq0 7382  df-mq0 7383  df-inp 7421  df-iplp 7423  df-iltp 7425

This theorem is referenced by:  caucvgprprlemexbt  7661