Theorem distrlem4pru 6683

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 6499	. . . . . . 7 ⊢ ((𝑤 ∈ Q ∧ 𝑣 ∈ Q ∧ 𝑢 ∈ Q) → (𝑤 <Q 𝑣 ↔ (𝑢 ·Q 𝑤) <Q (𝑢 ·Q 𝑣)))
2	1	adantl 262	. . . . . 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 904	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → 𝐴 ∈ P)
4		simpll 481	. . . . . . 7 ⊢ (((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶))) → 𝑥 ∈ (2nd ‘𝐴))
5		prop 6573	. . . . . . . 8 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
6		elprnqu 6580	. . . . . . . 8 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑥 ∈ (2nd ‘𝐴)) → 𝑥 ∈ Q)
7	5, 6	sylan 267	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ (2nd ‘𝐴)) → 𝑥 ∈ Q)
8	3, 4, 7	syl2an 273	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → 𝑥 ∈ Q)
9		simprl 483	. . . . . . 7 ⊢ (((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶))) → 𝑓 ∈ (2nd ‘𝐴))
10		elprnqu 6580	. . . . . . . 8 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) → 𝑓 ∈ Q)
11	5, 10	sylan 267	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) → 𝑓 ∈ Q)
12	3, 9, 11	syl2an 273	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → 𝑓 ∈ Q)
13		simpl3 909	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → 𝐶 ∈ P)
14		simprrr 492	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → 𝑧 ∈ (2nd ‘𝐶))
15		prop 6573	. . . . . . . 8 ⊢ (𝐶 ∈ P → ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P)
16		elprnqu 6580	. . . . . . . 8 ⊢ ((⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P ∧ 𝑧 ∈ (2nd ‘𝐶)) → 𝑧 ∈ Q)
17	15, 16	sylan 267	. . . . . . 7 ⊢ ((𝐶 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐶)) → 𝑧 ∈ Q)
18	13, 14, 17	syl2anc 391	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → 𝑧 ∈ Q)
19		mulcomnqg 6481	. . . . . . 7 ⊢ ((𝑤 ∈ Q ∧ 𝑣 ∈ Q) → (𝑤 ·Q 𝑣) = (𝑣 ·Q 𝑤))
20	19	adantl 262	. . . . . 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 5670	. . . . 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 6474	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 ·Q 𝑧) ∈ Q)
23	8, 18, 22	syl2anc 391	. . . . . 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 6474	. . . . . . 7 ⊢ ((𝑓 ∈ Q ∧ 𝑧 ∈ Q) → (𝑓 ·Q 𝑧) ∈ Q)
25	12, 18, 24	syl2anc 391	. . . . . 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 908	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → 𝐵 ∈ P)
27		simprlr 490	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → 𝑦 ∈ (2nd ‘𝐵))
28		prop 6573	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
29		elprnqu 6580	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑦 ∈ (2nd ‘𝐵)) → 𝑦 ∈ Q)
30	28, 29	sylan 267	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝑦 ∈ (2nd ‘𝐵)) → 𝑦 ∈ Q)
31	26, 27, 30	syl2anc 391	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → 𝑦 ∈ Q)
32		mulclnq 6474	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 ·Q 𝑦) ∈ Q)
33	8, 31, 32	syl2anc 391	. . . . . 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 6498	. . . . . 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 1135	. . . . 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 177	. . . 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 907	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → 𝐴 ∈ P)
38		addclpr 6635	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 +P 𝐶) ∈ P)
39	38	3adant1 922	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 +P 𝐶) ∈ P)
40	39	adantr 261	. . . . . 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 6670	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ (𝐵 +P 𝐶) ∈ P) → (𝐴 ·P (𝐵 +P 𝐶)) ∈ P)
42	37, 40, 41	syl2anc 391	. . . . 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 6485	. . . . . . 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 1135	. . . . . 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 489	. . . . . . 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 6566	. . . . . . . . . 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 6473	. . . . . . . . . 10 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
48	46, 47	genppreclu 6613	. . . . . . . . 9 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑦 ∈ (2nd ‘𝐵) ∧ 𝑧 ∈ (2nd ‘𝐶)) → (𝑦 +Q 𝑧) ∈ (2nd ‘(𝐵 +P 𝐶))))
49	48	imp 115	. . . . . . . 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 1136	. . . . . . 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 6567	. . . . . . . . 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 6474	. . . . . . . . 9 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 ·Q ℎ) ∈ Q)
53	51, 52	genppreclu 6613	. . . . . . . 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 115	. . . . . . 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 1136	. . . . . 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 2115	. . . . 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 6573	. . . . . 6 ⊢ ((𝐴 ·P (𝐵 +P 𝐶)) ∈ P → ⟨(1st ‘(𝐴 ·P (𝐵 +P 𝐶))), (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))⟩ ∈ P)
58		prcunqu 6583	. . . . . 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 267	. . . . 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 391	. . . 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 139	. . 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 5670	. . . . 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 6498	. . . . . . 7 ⊢ ((𝑤 ∈ Q ∧ 𝑣 ∈ Q ∧ 𝑢 ∈ Q) → (𝑤 <Q 𝑣 ↔ (𝑢 +Q 𝑤) <Q (𝑢 +Q 𝑣)))
64	63	adantl 262	. . . . . 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 6474	. . . . . . 7 ⊢ ((𝑓 ∈ Q ∧ 𝑦 ∈ Q) → (𝑓 ·Q 𝑦) ∈ Q)
66	12, 31, 65	syl2anc 391	. . . . . 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 6479	. . . . . . 7 ⊢ ((𝑤 ∈ Q ∧ 𝑣 ∈ Q) → (𝑤 +Q 𝑣) = (𝑣 +Q 𝑤))
68	67	adantl 262	. . . . . 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 5670	. . . . 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 177	. . . 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 6485	. . . . . . 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 1135	. . . . . 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 491	. . . . . . 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 6613	. . . . . . . 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 115	. . . . . . 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 1136	. . . . . 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 2115	. . . . 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 6583	. . . . . 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 267	. . . . 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 391	. . . 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 139	. . 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 637	. 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 6496	. . . . 5 ⊢ <Q Or Q
84		nqtri3or 6494	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 <Q 𝑓 ∨ 𝑥 = 𝑓 ∨ 𝑓 <Q 𝑥))
85	83, 84	sotritrieq 4062	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 = 𝑓 ↔ ¬ (𝑥 <Q 𝑓 ∨ 𝑓 <Q 𝑥)))
86	8, 12, 85	syl2anc 391	. . 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 5519	. . . . . . 7 ⊢ (𝑥 = 𝑓 → (𝑥 ·Q 𝑧) = (𝑓 ·Q 𝑧))
88	87	oveq2d 5528	. . . . . 6 ⊢ (𝑥 = 𝑓 → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
89	44, 88	sylan9eq 2092	. . . . 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 261	. . . . 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 2115	. . . 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 108	. . 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 159	. 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 6495	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → DECID 𝑥 <Q 𝑓)
95		ltdcnq 6495	. . . . . 6 ⊢ ((𝑓 ∈ Q ∧ 𝑥 ∈ Q) → DECID 𝑓 <Q 𝑥)
96	95	ancoms 255	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → DECID 𝑓 <Q 𝑥)
97		dcor 843	. . . . 5 ⊢ (DECID 𝑥 <Q 𝑓 → (DECID 𝑓 <Q 𝑥 → DECID (𝑥 <Q 𝑓 ∨ 𝑓 <Q 𝑥)))
98	94, 96, 97	sylc 56	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → DECID (𝑥 <Q 𝑓 ∨ 𝑓 <Q 𝑥))
99	8, 12, 98	syl2anc 391	. . 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 743	. . 3 ⊢ (DECID (𝑥 <Q 𝑓 ∨ 𝑓 <Q 𝑥) ↔ ((𝑥 <Q 𝑓 ∨ 𝑓 <Q 𝑥) ∨ ¬ (𝑥 <Q 𝑓 ∨ 𝑓 <Q 𝑥)))
101	99, 100	sylib 127	. 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 638	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 97   ↔ wb 98   ∨ wo 629  DECID wdc 742   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ⟨cop 3378   class class class wbr 3764  ‘cfv 4902  (class class class)co 5512  1^st c1st 5765  2^nd c2nd 5766  Qcnq 6378   +_Q cplq 6380   ·_Q cmq 6381   <_Q cltq 6383  Pcnp 6389   +_P cpp 6391   ·_P cmp 6392

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-iplp 6566  df-imp 6567

This theorem is referenced by:  distrlem5pru  6685