Theorem distrlem4pru 7698

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

Assertion

Ref	Expression
distrlem4pru	⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑓,𝐴   𝑥,𝐵,𝑦,𝑧,𝑓   𝑥,𝐶,𝑦,𝑧,𝑓

Proof of Theorem distrlem4pru

Dummy variables 𝑤 𝑣 𝑢 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltmnqg 7514	. . . . . . 7 ⊢ ((𝑤 ∈ Q ∧ 𝑣 ∈ Q ∧ 𝑢 ∈ Q) → (𝑤 <Q 𝑣 ↔ (𝑢 ·Q 𝑤) <Q (𝑢 ·Q 𝑣)))
2	1	adantl 277	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) ∧ (𝑤 ∈ Q ∧ 𝑣 ∈ Q ∧ 𝑢 ∈ Q)) → (𝑤 <Q 𝑣 ↔ (𝑢 ·Q 𝑤) <Q (𝑢 ·Q 𝑣)))
3		simp1 1000	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → 𝐴 ∈ P)
4		simpll 527	. . . . . . 7 ⊢ (((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶))) → 𝑥 ∈ (2nd ‘𝐴))
5		prop 7588	. . . . . . . 8 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
6		elprnqu 7595	. . . . . . . 8 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑥 ∈ (2nd ‘𝐴)) → 𝑥 ∈ Q)
7	5, 6	sylan 283	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ (2nd ‘𝐴)) → 𝑥 ∈ Q)
8	3, 4, 7	syl2an 289	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → 𝑥 ∈ Q)
9		simprl 529	. . . . . . 7 ⊢ (((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶))) → 𝑓 ∈ (2nd ‘𝐴))
10		elprnqu 7595	. . . . . . . 8 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) → 𝑓 ∈ Q)
11	5, 10	sylan 283	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) → 𝑓 ∈ Q)
12	3, 9, 11	syl2an 289	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → 𝑓 ∈ Q)
13		simpl3 1005	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → 𝐶 ∈ P)
14		simprrr 540	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → 𝑧 ∈ (2nd ‘𝐶))
15		prop 7588	. . . . . . . 8 ⊢ (𝐶 ∈ P → ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P)
16		elprnqu 7595	. . . . . . . 8 ⊢ ((⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P ∧ 𝑧 ∈ (2nd ‘𝐶)) → 𝑧 ∈ Q)
17	15, 16	sylan 283	. . . . . . 7 ⊢ ((𝐶 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐶)) → 𝑧 ∈ Q)
18	13, 14, 17	syl2anc 411	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → 𝑧 ∈ Q)
19		mulcomnqg 7496	. . . . . . 7 ⊢ ((𝑤 ∈ Q ∧ 𝑣 ∈ Q) → (𝑤 ·Q 𝑣) = (𝑣 ·Q 𝑤))
20	19	adantl 277	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) ∧ (𝑤 ∈ Q ∧ 𝑣 ∈ Q)) → (𝑤 ·Q 𝑣) = (𝑣 ·Q 𝑤))
21	2, 8, 12, 18, 20	caovord2d 6116	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → (𝑥 <Q 𝑓 ↔ (𝑥 ·Q 𝑧) <Q (𝑓 ·Q 𝑧)))
22		mulclnq 7489	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 ·Q 𝑧) ∈ Q)
23	8, 18, 22	syl2anc 411	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → (𝑥 ·Q 𝑧) ∈ Q)
24		mulclnq 7489	. . . . . . 7 ⊢ ((𝑓 ∈ Q ∧ 𝑧 ∈ Q) → (𝑓 ·Q 𝑧) ∈ Q)
25	12, 18, 24	syl2anc 411	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → (𝑓 ·Q 𝑧) ∈ Q)
26		simpl2 1004	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → 𝐵 ∈ P)
27		simprlr 538	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → 𝑦 ∈ (2nd ‘𝐵))
28		prop 7588	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
29		elprnqu 7595	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑦 ∈ (2nd ‘𝐵)) → 𝑦 ∈ Q)
30	28, 29	sylan 283	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝑦 ∈ (2nd ‘𝐵)) → 𝑦 ∈ Q)
31	26, 27, 30	syl2anc 411	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → 𝑦 ∈ Q)
32		mulclnq 7489	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 ·Q 𝑦) ∈ Q)
33	8, 31, 32	syl2anc 411	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → (𝑥 ·Q 𝑦) ∈ Q)
34		ltanqg 7513	. . . . . 6 ⊢ (((𝑥 ·Q 𝑧) ∈ Q ∧ (𝑓 ·Q 𝑧) ∈ Q ∧ (𝑥 ·Q 𝑦) ∈ Q) → ((𝑥 ·Q 𝑧) <Q (𝑓 ·Q 𝑧) ↔ ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
35	23, 25, 33, 34	syl3anc 1250	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → ((𝑥 ·Q 𝑧) <Q (𝑓 ·Q 𝑧) ↔ ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
36	21, 35	bitrd 188	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → (𝑥 <Q 𝑓 ↔ ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
37		simpl1 1003	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → 𝐴 ∈ P)
38		addclpr 7650	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 +P 𝐶) ∈ P)
39	38	3adant1 1018	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 +P 𝐶) ∈ P)
40	39	adantr 276	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → (𝐵 +P 𝐶) ∈ P)
41		mulclpr 7685	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ (𝐵 +P 𝐶) ∈ P) → (𝐴 ·P (𝐵 +P 𝐶)) ∈ P)
42	37, 40, 41	syl2anc 411	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → (𝐴 ·P (𝐵 +P 𝐶)) ∈ P)
43		distrnqg 7500	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
44	8, 31, 18, 43	syl3anc 1250	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
45		simprll 537	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → 𝑥 ∈ (2nd ‘𝐴))
46		df-iplp 7581	. . . . . . . . . 10 ⊢ +P = (𝑢 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑤 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (1st ‘𝑢) ∧ ℎ ∈ (1st ‘𝑣) ∧ 𝑤 = (𝑔 +Q ℎ))}, {𝑤 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (2nd ‘𝑢) ∧ ℎ ∈ (2nd ‘𝑣) ∧ 𝑤 = (𝑔 +Q ℎ))}⟩)
47		addclnq 7488	. . . . . . . . . 10 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
48	46, 47	genppreclu 7628	. . . . . . . . 9 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑦 ∈ (2nd ‘𝐵) ∧ 𝑧 ∈ (2nd ‘𝐶)) → (𝑦 +Q 𝑧) ∈ (2nd ‘(𝐵 +P 𝐶))))
49	48	imp 124	. . . . . . . 8 ⊢ (((𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑦 ∈ (2nd ‘𝐵) ∧ 𝑧 ∈ (2nd ‘𝐶))) → (𝑦 +Q 𝑧) ∈ (2nd ‘(𝐵 +P 𝐶)))
50	26, 13, 27, 14, 49	syl22anc 1251	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → (𝑦 +Q 𝑧) ∈ (2nd ‘(𝐵 +P 𝐶)))
51		df-imp 7582	. . . . . . . . 9 ⊢ ·P = (𝑢 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑤 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (1st ‘𝑢) ∧ ℎ ∈ (1st ‘𝑣) ∧ 𝑤 = (𝑔 ·Q ℎ))}, {𝑤 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (2nd ‘𝑢) ∧ ℎ ∈ (2nd ‘𝑣) ∧ 𝑤 = (𝑔 ·Q ℎ))}⟩)
52		mulclnq 7489	. . . . . . . . 9 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 ·Q ℎ) ∈ Q)
53	51, 52	genppreclu 7628	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ (𝐵 +P 𝐶) ∈ P) → ((𝑥 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑧) ∈ (2nd ‘(𝐵 +P 𝐶))) → (𝑥 ·Q (𝑦 +Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
54	53	imp 124	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ (𝐵 +P 𝐶) ∈ P) ∧ (𝑥 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑧) ∈ (2nd ‘(𝐵 +P 𝐶)))) → (𝑥 ·Q (𝑦 +Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
55	37, 40, 45, 50, 54	syl22anc 1251	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → (𝑥 ·Q (𝑦 +Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
56	44, 55	eqeltrrd 2283	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
57		prop 7588	. . . . . 6 ⊢ ((𝐴 ·P (𝐵 +P 𝐶)) ∈ P → ⟨(1st ‘(𝐴 ·P (𝐵 +P 𝐶))), (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))⟩ ∈ P)
58		prcunqu 7598	. . . . . 6 ⊢ ((⟨(1st ‘(𝐴 ·P (𝐵 +P 𝐶))), (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))⟩ ∈ P ∧ ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))) → (((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
59	57, 58	sylan 283	. . . . 5 ⊢ (((𝐴 ·P (𝐵 +P 𝐶)) ∈ P ∧ ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))) → (((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
60	42, 56, 59	syl2anc 411	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → (((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
61	36, 60	sylbid 150	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → (𝑥 <Q 𝑓 → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
62	2, 12, 8, 31, 20	caovord2d 6116	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → (𝑓 <Q 𝑥 ↔ (𝑓 ·Q 𝑦) <Q (𝑥 ·Q 𝑦)))
63		ltanqg 7513	. . . . . . 7 ⊢ ((𝑤 ∈ Q ∧ 𝑣 ∈ Q ∧ 𝑢 ∈ Q) → (𝑤 <Q 𝑣 ↔ (𝑢 +Q 𝑤) <Q (𝑢 +Q 𝑣)))
64	63	adantl 277	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) ∧ (𝑤 ∈ Q ∧ 𝑣 ∈ Q ∧ 𝑢 ∈ Q)) → (𝑤 <Q 𝑣 ↔ (𝑢 +Q 𝑤) <Q (𝑢 +Q 𝑣)))
65		mulclnq 7489	. . . . . . 7 ⊢ ((𝑓 ∈ Q ∧ 𝑦 ∈ Q) → (𝑓 ·Q 𝑦) ∈ Q)
66	12, 31, 65	syl2anc 411	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → (𝑓 ·Q 𝑦) ∈ Q)
67		addcomnqg 7494	. . . . . . 7 ⊢ ((𝑤 ∈ Q ∧ 𝑣 ∈ Q) → (𝑤 +Q 𝑣) = (𝑣 +Q 𝑤))
68	67	adantl 277	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) ∧ (𝑤 ∈ Q ∧ 𝑣 ∈ Q)) → (𝑤 +Q 𝑣) = (𝑣 +Q 𝑤))
69	64, 66, 33, 25, 68	caovord2d 6116	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → ((𝑓 ·Q 𝑦) <Q (𝑥 ·Q 𝑦) ↔ ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
70	62, 69	bitrd 188	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → (𝑓 <Q 𝑥 ↔ ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
71		distrnqg 7500	. . . . . . 7 ⊢ ((𝑓 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑓 ·Q (𝑦 +Q 𝑧)) = ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
72	12, 31, 18, 71	syl3anc 1250	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → (𝑓 ·Q (𝑦 +Q 𝑧)) = ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
73		simprrl 539	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → 𝑓 ∈ (2nd ‘𝐴))
74	51, 52	genppreclu 7628	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ (𝐵 +P 𝐶) ∈ P) → ((𝑓 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑧) ∈ (2nd ‘(𝐵 +P 𝐶))) → (𝑓 ·Q (𝑦 +Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
75	74	imp 124	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ (𝐵 +P 𝐶) ∈ P) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑧) ∈ (2nd ‘(𝐵 +P 𝐶)))) → (𝑓 ·Q (𝑦 +Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
76	37, 40, 73, 50, 75	syl22anc 1251	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → (𝑓 ·Q (𝑦 +Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
77	72, 76	eqeltrrd 2283	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
78		prcunqu 7598	. . . . . 6 ⊢ ((⟨(1st ‘(𝐴 ·P (𝐵 +P 𝐶))), (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))⟩ ∈ P ∧ ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))) → (((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
79	57, 78	sylan 283	. . . . 5 ⊢ (((𝐴 ·P (𝐵 +P 𝐶)) ∈ P ∧ ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))) → (((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
80	42, 77, 79	syl2anc 411	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → (((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
81	70, 80	sylbid 150	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → (𝑓 <Q 𝑥 → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
82	61, 81	jaod 719	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → ((𝑥 <Q 𝑓 ∨ 𝑓 <Q 𝑥) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
83		ltsonq 7511	. . . . 5 ⊢ <Q Or Q
84		nqtri3or 7509	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 <Q 𝑓 ∨ 𝑥 = 𝑓 ∨ 𝑓 <Q 𝑥))
85	83, 84	sotritrieq 4372	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 = 𝑓 ↔ ¬ (𝑥 <Q 𝑓 ∨ 𝑓 <Q 𝑥)))
86	8, 12, 85	syl2anc 411	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → (𝑥 = 𝑓 ↔ ¬ (𝑥 <Q 𝑓 ∨ 𝑓 <Q 𝑥)))
87		oveq1 5951	. . . . . . 7 ⊢ (𝑥 = 𝑓 → (𝑥 ·Q 𝑧) = (𝑓 ·Q 𝑧))
88	87	oveq2d 5960	. . . . . 6 ⊢ (𝑥 = 𝑓 → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
89	44, 88	sylan9eq 2258	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) ∧ 𝑥 = 𝑓) → (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
90	55	adantr 276	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) ∧ 𝑥 = 𝑓) → (𝑥 ·Q (𝑦 +Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
91	89, 90	eqeltrrd 2283	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) ∧ 𝑥 = 𝑓) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
92	91	ex 115	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → (𝑥 = 𝑓 → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
93	86, 92	sylbird 170	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → (¬ (𝑥 <Q 𝑓 ∨ 𝑓 <Q 𝑥) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
94		ltdcnq 7510	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → DECID 𝑥 <Q 𝑓)
95		ltdcnq 7510	. . . . . 6 ⊢ ((𝑓 ∈ Q ∧ 𝑥 ∈ Q) → DECID 𝑓 <Q 𝑥)
96	95	ancoms 268	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → DECID 𝑓 <Q 𝑥)
97		dcor 938	. . . . 5 ⊢ (DECID 𝑥 <Q 𝑓 → (DECID 𝑓 <Q 𝑥 → DECID (𝑥 <Q 𝑓 ∨ 𝑓 <Q 𝑥)))
98	94, 96, 97	sylc 62	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → DECID (𝑥 <Q 𝑓 ∨ 𝑓 <Q 𝑥))
99	8, 12, 98	syl2anc 411	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → DECID (𝑥 <Q 𝑓 ∨ 𝑓 <Q 𝑥))
100		df-dc 837	. . 3 ⊢ (DECID (𝑥 <Q 𝑓 ∨ 𝑓 <Q 𝑥) ↔ ((𝑥 <Q 𝑓 ∨ 𝑓 <Q 𝑥) ∨ ¬ (𝑥 <Q 𝑓 ∨ 𝑓 <Q 𝑥)))
101	99, 100	sylib 122	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → ((𝑥 <Q 𝑓 ∨ 𝑓 <Q 𝑥) ∨ ¬ (𝑥 <Q 𝑓 ∨ 𝑓 <Q 𝑥)))
102	82, 93, 101	mpjaod 720	1 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 710  DECID wdc 836   ∧ w3a 981   = wceq 1373   ∈ wcel 2176  ⟨cop 3636   class class class wbr 4044  ‘cfv 5271  (class class class)co 5944  1^st c1st 6224  2^nd c2nd 6225  Qcnq 7393   +_Q cplq 7395   ·_Q cmq 7396   <_Q cltq 7398  Pcnp 7404   +_P cpp 7406   ·_P cmp 7407

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-eprel 4336  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-irdg 6456  df-1o 6502  df-2o 6503  df-oadd 6506  df-omul 6507  df-er 6620  df-ec 6622  df-qs 6626  df-ni 7417  df-pli 7418  df-mi 7419  df-lti 7420  df-plpq 7457  df-mpq 7458  df-enq 7460  df-nqqs 7461  df-plqqs 7462  df-mqqs 7463  df-1nqqs 7464  df-rq 7465  df-ltnqqs 7466  df-enq0 7537  df-nq0 7538  df-0nq0 7539  df-plq0 7540  df-mq0 7541  df-inp 7579  df-iplp 7581  df-imp 7582

This theorem is referenced by:  distrlem5pru  7700