Theorem distrlem1prl 6558

Description: Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)

Assertion

Ref	Expression
distrlem1prl	⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → (1st ‘(A ·P (B +P 𝐶))) ⊆ (1st ‘((A ·P B) +P (A ·P 𝐶))))

Proof of Theorem distrlem1prl

Dummy variables x y z w v f g ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addclpr 6520	. . . . 5 ⊢ ((B ∈ P ∧ 𝐶 ∈ P) → (B +P 𝐶) ∈ P)
2		df-imp 6452	. . . . . 6 ⊢ ·P = (y ∈ P, z ∈ P ↦ ⟨{f ∈ Q ∣ ∃g ∈ Q ∃ℎ ∈ Q (g ∈ (1st ‘y) ∧ ℎ ∈ (1st ‘z) ∧ f = (g ·Q ℎ))}, {f ∈ Q ∣ ∃g ∈ Q ∃ℎ ∈ Q (g ∈ (2nd ‘y) ∧ ℎ ∈ (2nd ‘z) ∧ f = (g ·Q ℎ))}⟩)
3		mulclnq 6360	. . . . . 6 ⊢ ((g ∈ Q ∧ ℎ ∈ Q) → (g ·Q ℎ) ∈ Q)
4	2, 3	genpelvl 6495	. . . . 5 ⊢ ((A ∈ P ∧ (B +P 𝐶) ∈ P) → (w ∈ (1st ‘(A ·P (B +P 𝐶))) ↔ ∃x ∈ (1st ‘A)∃v ∈ (1st ‘(B +P 𝐶))w = (x ·Q v)))
5	1, 4	sylan2 270	. . . 4 ⊢ ((A ∈ P ∧ (B ∈ P ∧ 𝐶 ∈ P)) → (w ∈ (1st ‘(A ·P (B +P 𝐶))) ↔ ∃x ∈ (1st ‘A)∃v ∈ (1st ‘(B +P 𝐶))w = (x ·Q v)))
6	5	3impb 1099	. . 3 ⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → (w ∈ (1st ‘(A ·P (B +P 𝐶))) ↔ ∃x ∈ (1st ‘A)∃v ∈ (1st ‘(B +P 𝐶))w = (x ·Q v)))
7		df-iplp 6451	. . . . . . . . . . 11 ⊢ +P = (w ∈ P, x ∈ P ↦ ⟨{f ∈ Q ∣ ∃g ∈ Q ∃ℎ ∈ Q (g ∈ (1st ‘w) ∧ ℎ ∈ (1st ‘x) ∧ f = (g +Q ℎ))}, {f ∈ Q ∣ ∃g ∈ Q ∃ℎ ∈ Q (g ∈ (2nd ‘w) ∧ ℎ ∈ (2nd ‘x) ∧ f = (g +Q ℎ))}⟩)
8		addclnq 6359	. . . . . . . . . . 11 ⊢ ((g ∈ Q ∧ ℎ ∈ Q) → (g +Q ℎ) ∈ Q)
9	7, 8	genpelvl 6495	. . . . . . . . . 10 ⊢ ((B ∈ P ∧ 𝐶 ∈ P) → (v ∈ (1st ‘(B +P 𝐶)) ↔ ∃y ∈ (1st ‘B)∃z ∈ (1st ‘𝐶)v = (y +Q z)))
10	9	3adant1 921	. . . . . . . . 9 ⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → (v ∈ (1st ‘(B +P 𝐶)) ↔ ∃y ∈ (1st ‘B)∃z ∈ (1st ‘𝐶)v = (y +Q z)))
11	10	adantr 261	. . . . . . . 8 ⊢ (((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ (x ∈ (1st ‘A) ∧ w = (x ·Q v))) → (v ∈ (1st ‘(B +P 𝐶)) ↔ ∃y ∈ (1st ‘B)∃z ∈ (1st ‘𝐶)v = (y +Q z)))
12		prop 6458	. . . . . . . . . . . . . . . . 17 ⊢ (A ∈ P → ⟨(1st ‘A), (2nd ‘A)⟩ ∈ P)
13		elprnql 6464	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨(1st ‘A), (2nd ‘A)⟩ ∈ P ∧ x ∈ (1st ‘A)) → x ∈ Q)
14	12, 13	sylan 267	. . . . . . . . . . . . . . . 16 ⊢ ((A ∈ P ∧ x ∈ (1st ‘A)) → x ∈ Q)
15	14	3ad2antl1 1065	. . . . . . . . . . . . . . 15 ⊢ (((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ x ∈ (1st ‘A)) → x ∈ Q)
16	15	adantrr 448	. . . . . . . . . . . . . 14 ⊢ (((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ (x ∈ (1st ‘A) ∧ w = (x ·Q v))) → x ∈ Q)
17	16	adantr 261	. . . . . . . . . . . . 13 ⊢ ((((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ (x ∈ (1st ‘A) ∧ w = (x ·Q v))) ∧ ((y ∈ (1st ‘B) ∧ z ∈ (1st ‘𝐶)) ∧ v = (y +Q z))) → x ∈ Q)
18		prop 6458	. . . . . . . . . . . . . . . . . 18 ⊢ (B ∈ P → ⟨(1st ‘B), (2nd ‘B)⟩ ∈ P)
19		elprnql 6464	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨(1st ‘B), (2nd ‘B)⟩ ∈ P ∧ y ∈ (1st ‘B)) → y ∈ Q)
20	18, 19	sylan 267	. . . . . . . . . . . . . . . . 17 ⊢ ((B ∈ P ∧ y ∈ (1st ‘B)) → y ∈ Q)
21		prop 6458	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 ∈ P → ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P)
22		elprnql 6464	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P ∧ z ∈ (1st ‘𝐶)) → z ∈ Q)
23	21, 22	sylan 267	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ P ∧ z ∈ (1st ‘𝐶)) → z ∈ Q)
24	20, 23	anim12i 321	. . . . . . . . . . . . . . . 16 ⊢ (((B ∈ P ∧ y ∈ (1st ‘B)) ∧ (𝐶 ∈ P ∧ z ∈ (1st ‘𝐶))) → (y ∈ Q ∧ z ∈ Q))
25	24	an4s 522	. . . . . . . . . . . . . . 15 ⊢ (((B ∈ P ∧ 𝐶 ∈ P) ∧ (y ∈ (1st ‘B) ∧ z ∈ (1st ‘𝐶))) → (y ∈ Q ∧ z ∈ Q))
26	25	3adantl1 1059	. . . . . . . . . . . . . 14 ⊢ (((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ (y ∈ (1st ‘B) ∧ z ∈ (1st ‘𝐶))) → (y ∈ Q ∧ z ∈ Q))
27	26	ad2ant2r 478	. . . . . . . . . . . . 13 ⊢ ((((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ (x ∈ (1st ‘A) ∧ w = (x ·Q v))) ∧ ((y ∈ (1st ‘B) ∧ z ∈ (1st ‘𝐶)) ∧ v = (y +Q z))) → (y ∈ Q ∧ z ∈ Q))
28		3anass 888	. . . . . . . . . . . . 13 ⊢ ((x ∈ Q ∧ y ∈ Q ∧ z ∈ Q) ↔ (x ∈ Q ∧ (y ∈ Q ∧ z ∈ Q)))
29	17, 27, 28	sylanbrc 394	. . . . . . . . . . . 12 ⊢ ((((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ (x ∈ (1st ‘A) ∧ w = (x ·Q v))) ∧ ((y ∈ (1st ‘B) ∧ z ∈ (1st ‘𝐶)) ∧ v = (y +Q z))) → (x ∈ Q ∧ y ∈ Q ∧ z ∈ Q))
30		simprr 484	. . . . . . . . . . . . 13 ⊢ (((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ (x ∈ (1st ‘A) ∧ w = (x ·Q v))) → w = (x ·Q v))
31		simpr 103	. . . . . . . . . . . . 13 ⊢ (((y ∈ (1st ‘B) ∧ z ∈ (1st ‘𝐶)) ∧ v = (y +Q z)) → v = (y +Q z))
32	30, 31	anim12i 321	. . . . . . . . . . . 12 ⊢ ((((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ (x ∈ (1st ‘A) ∧ w = (x ·Q v))) ∧ ((y ∈ (1st ‘B) ∧ z ∈ (1st ‘𝐶)) ∧ v = (y +Q z))) → (w = (x ·Q v) ∧ v = (y +Q z)))
33		oveq2 5463	. . . . . . . . . . . . . . 15 ⊢ (v = (y +Q z) → (x ·Q v) = (x ·Q (y +Q z)))
34	33	eqeq2d 2048	. . . . . . . . . . . . . 14 ⊢ (v = (y +Q z) → (w = (x ·Q v) ↔ w = (x ·Q (y +Q z))))
35	34	biimpac 282	. . . . . . . . . . . . 13 ⊢ ((w = (x ·Q v) ∧ v = (y +Q z)) → w = (x ·Q (y +Q z)))
36		distrnqg 6371	. . . . . . . . . . . . . 14 ⊢ ((x ∈ Q ∧ y ∈ Q ∧ z ∈ Q) → (x ·Q (y +Q z)) = ((x ·Q y) +Q (x ·Q z)))
37	36	eqeq2d 2048	. . . . . . . . . . . . 13 ⊢ ((x ∈ Q ∧ y ∈ Q ∧ z ∈ Q) → (w = (x ·Q (y +Q z)) ↔ w = ((x ·Q y) +Q (x ·Q z))))
38	35, 37	syl5ib 143	. . . . . . . . . . . 12 ⊢ ((x ∈ Q ∧ y ∈ Q ∧ z ∈ Q) → ((w = (x ·Q v) ∧ v = (y +Q z)) → w = ((x ·Q y) +Q (x ·Q z))))
39	29, 32, 38	sylc 56	. . . . . . . . . . 11 ⊢ ((((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ (x ∈ (1st ‘A) ∧ w = (x ·Q v))) ∧ ((y ∈ (1st ‘B) ∧ z ∈ (1st ‘𝐶)) ∧ v = (y +Q z))) → w = ((x ·Q y) +Q (x ·Q z)))
40		mulclpr 6553	. . . . . . . . . . . . . 14 ⊢ ((A ∈ P ∧ B ∈ P) → (A ·P B) ∈ P)
41	40	3adant3 923	. . . . . . . . . . . . 13 ⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → (A ·P B) ∈ P)
42	41	ad2antrr 457	. . . . . . . . . . . 12 ⊢ ((((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ (x ∈ (1st ‘A) ∧ w = (x ·Q v))) ∧ ((y ∈ (1st ‘B) ∧ z ∈ (1st ‘𝐶)) ∧ v = (y +Q z))) → (A ·P B) ∈ P)
43		mulclpr 6553	. . . . . . . . . . . . . 14 ⊢ ((A ∈ P ∧ 𝐶 ∈ P) → (A ·P 𝐶) ∈ P)
44	43	3adant2 922	. . . . . . . . . . . . 13 ⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → (A ·P 𝐶) ∈ P)
45	44	ad2antrr 457	. . . . . . . . . . . 12 ⊢ ((((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ (x ∈ (1st ‘A) ∧ w = (x ·Q v))) ∧ ((y ∈ (1st ‘B) ∧ z ∈ (1st ‘𝐶)) ∧ v = (y +Q z))) → (A ·P 𝐶) ∈ P)
46		simpll 481	. . . . . . . . . . . . 13 ⊢ (((y ∈ (1st ‘B) ∧ z ∈ (1st ‘𝐶)) ∧ v = (y +Q z)) → y ∈ (1st ‘B))
47	2, 3	genpprecll 6497	. . . . . . . . . . . . . . . 16 ⊢ ((A ∈ P ∧ B ∈ P) → ((x ∈ (1st ‘A) ∧ y ∈ (1st ‘B)) → (x ·Q y) ∈ (1st ‘(A ·P B))))
48	47	3adant3 923	. . . . . . . . . . . . . . 15 ⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → ((x ∈ (1st ‘A) ∧ y ∈ (1st ‘B)) → (x ·Q y) ∈ (1st ‘(A ·P B))))
49	48	impl 362	. . . . . . . . . . . . . 14 ⊢ ((((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ x ∈ (1st ‘A)) ∧ y ∈ (1st ‘B)) → (x ·Q y) ∈ (1st ‘(A ·P B)))
50	49	adantlrr 452	. . . . . . . . . . . . 13 ⊢ ((((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ (x ∈ (1st ‘A) ∧ w = (x ·Q v))) ∧ y ∈ (1st ‘B)) → (x ·Q y) ∈ (1st ‘(A ·P B)))
51	46, 50	sylan2 270	. . . . . . . . . . . 12 ⊢ ((((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ (x ∈ (1st ‘A) ∧ w = (x ·Q v))) ∧ ((y ∈ (1st ‘B) ∧ z ∈ (1st ‘𝐶)) ∧ v = (y +Q z))) → (x ·Q y) ∈ (1st ‘(A ·P B)))
52		simplr 482	. . . . . . . . . . . . 13 ⊢ (((y ∈ (1st ‘B) ∧ z ∈ (1st ‘𝐶)) ∧ v = (y +Q z)) → z ∈ (1st ‘𝐶))
53	2, 3	genpprecll 6497	. . . . . . . . . . . . . . . 16 ⊢ ((A ∈ P ∧ 𝐶 ∈ P) → ((x ∈ (1st ‘A) ∧ z ∈ (1st ‘𝐶)) → (x ·Q z) ∈ (1st ‘(A ·P 𝐶))))
54	53	3adant2 922	. . . . . . . . . . . . . . 15 ⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → ((x ∈ (1st ‘A) ∧ z ∈ (1st ‘𝐶)) → (x ·Q z) ∈ (1st ‘(A ·P 𝐶))))
55	54	impl 362	. . . . . . . . . . . . . 14 ⊢ ((((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ x ∈ (1st ‘A)) ∧ z ∈ (1st ‘𝐶)) → (x ·Q z) ∈ (1st ‘(A ·P 𝐶)))
56	55	adantlrr 452	. . . . . . . . . . . . 13 ⊢ ((((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ (x ∈ (1st ‘A) ∧ w = (x ·Q v))) ∧ z ∈ (1st ‘𝐶)) → (x ·Q z) ∈ (1st ‘(A ·P 𝐶)))
57	52, 56	sylan2 270	. . . . . . . . . . . 12 ⊢ ((((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ (x ∈ (1st ‘A) ∧ w = (x ·Q v))) ∧ ((y ∈ (1st ‘B) ∧ z ∈ (1st ‘𝐶)) ∧ v = (y +Q z))) → (x ·Q z) ∈ (1st ‘(A ·P 𝐶)))
58	7, 8	genpprecll 6497	. . . . . . . . . . . . 13 ⊢ (((A ·P B) ∈ P ∧ (A ·P 𝐶) ∈ P) → (((x ·Q y) ∈ (1st ‘(A ·P B)) ∧ (x ·Q z) ∈ (1st ‘(A ·P 𝐶))) → ((x ·Q y) +Q (x ·Q z)) ∈ (1st ‘((A ·P B) +P (A ·P 𝐶)))))
59	58	imp 115	. . . . . . . . . . . 12 ⊢ ((((A ·P B) ∈ P ∧ (A ·P 𝐶) ∈ P) ∧ ((x ·Q y) ∈ (1st ‘(A ·P B)) ∧ (x ·Q z) ∈ (1st ‘(A ·P 𝐶)))) → ((x ·Q y) +Q (x ·Q z)) ∈ (1st ‘((A ·P B) +P (A ·P 𝐶))))
60	42, 45, 51, 57, 59	syl22anc 1135	. . . . . . . . . . 11 ⊢ ((((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ (x ∈ (1st ‘A) ∧ w = (x ·Q v))) ∧ ((y ∈ (1st ‘B) ∧ z ∈ (1st ‘𝐶)) ∧ v = (y +Q z))) → ((x ·Q y) +Q (x ·Q z)) ∈ (1st ‘((A ·P B) +P (A ·P 𝐶))))
61	39, 60	eqeltrd 2111	. . . . . . . . . 10 ⊢ ((((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ (x ∈ (1st ‘A) ∧ w = (x ·Q v))) ∧ ((y ∈ (1st ‘B) ∧ z ∈ (1st ‘𝐶)) ∧ v = (y +Q z))) → w ∈ (1st ‘((A ·P B) +P (A ·P 𝐶))))
62	61	exp32 347	. . . . . . . . 9 ⊢ (((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ (x ∈ (1st ‘A) ∧ w = (x ·Q v))) → ((y ∈ (1st ‘B) ∧ z ∈ (1st ‘𝐶)) → (v = (y +Q z) → w ∈ (1st ‘((A ·P B) +P (A ·P 𝐶))))))
63	62	rexlimdvv 2433	. . . . . . . 8 ⊢ (((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ (x ∈ (1st ‘A) ∧ w = (x ·Q v))) → (∃y ∈ (1st ‘B)∃z ∈ (1st ‘𝐶)v = (y +Q z) → w ∈ (1st ‘((A ·P B) +P (A ·P 𝐶)))))
64	11, 63	sylbid 139	. . . . . . 7 ⊢ (((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) ∧ (x ∈ (1st ‘A) ∧ w = (x ·Q v))) → (v ∈ (1st ‘(B +P 𝐶)) → w ∈ (1st ‘((A ·P B) +P (A ·P 𝐶)))))
65	64	exp32 347	. . . . . 6 ⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → (x ∈ (1st ‘A) → (w = (x ·Q v) → (v ∈ (1st ‘(B +P 𝐶)) → w ∈ (1st ‘((A ·P B) +P (A ·P 𝐶)))))))
66	65	com34 77	. . . . 5 ⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → (x ∈ (1st ‘A) → (v ∈ (1st ‘(B +P 𝐶)) → (w = (x ·Q v) → w ∈ (1st ‘((A ·P B) +P (A ·P 𝐶)))))))
67	66	impd 242	. . . 4 ⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → ((x ∈ (1st ‘A) ∧ v ∈ (1st ‘(B +P 𝐶))) → (w = (x ·Q v) → w ∈ (1st ‘((A ·P B) +P (A ·P 𝐶))))))
68	67	rexlimdvv 2433	. . 3 ⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → (∃x ∈ (1st ‘A)∃v ∈ (1st ‘(B +P 𝐶))w = (x ·Q v) → w ∈ (1st ‘((A ·P B) +P (A ·P 𝐶)))))
69	6, 68	sylbid 139	. 2 ⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → (w ∈ (1st ‘(A ·P (B +P 𝐶))) → w ∈ (1st ‘((A ·P B) +P (A ·P 𝐶)))))
70	69	ssrdv 2945	1 ⊢ ((A ∈ P ∧ B ∈ P ∧ 𝐶 ∈ P) → (1st ‘(A ·P (B +P 𝐶))) ⊆ (1st ‘((A ·P B) +P (A ·P 𝐶))))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  ∃wrex 2301   ⊆ wss 2911  ⟨cop 3370  ‘cfv 4845  (class class class)co 5455  1^st c1st 5707  2^nd c2nd 5708  Qcnq 6264   +_Q cplq 6266   ·_Q cmq 6267  Pcnp 6275   +_P cpp 6277   ·_P cmp 6278

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-iplp 6451  df-imp 6452

This theorem is referenced by:  distrprg  6564