Theorem mulgt0sr 11030

Description: The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
mulgt0sr	⊢ ((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 ·R 𝐵))

Proof of Theorem mulgt0sr

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

Step	Hyp	Ref	Expression
1		ltrelsr 10993	. . . . 5 ⊢ <R ⊆ (R × R)
2	1	brel 5699	. . . 4 ⊢ (0R <R 𝐴 → (0R ∈ R ∧ 𝐴 ∈ R))
3	2	simprd 495	. . 3 ⊢ (0R <R 𝐴 → 𝐴 ∈ R)
4	1	brel 5699	. . . 4 ⊢ (0R <R 𝐵 → (0R ∈ R ∧ 𝐵 ∈ R))
5	4	simprd 495	. . 3 ⊢ (0R <R 𝐵 → 𝐵 ∈ R)
6	3, 5	anim12i 614	. 2 ⊢ ((0R <R 𝐴 ∧ 0R <R 𝐵) → (𝐴 ∈ R ∧ 𝐵 ∈ R))
7		df-nr 10981	. . 3 ⊢ R = ((P × P) / ~R )
8		breq2 5104	. . . . 5 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (0R <R [⟨𝑥, 𝑦⟩] ~R ↔ 0R <R 𝐴))
9	8	anbi1d 632	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ((0R <R [⟨𝑥, 𝑦⟩] ~R ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R ) ↔ (0R <R 𝐴 ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R )))
10		oveq1 7377	. . . . 5 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) = (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ))
11	10	breq2d 5112	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (0R <R ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) ↔ 0R <R (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R )))
12	9, 11	imbi12d 344	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (((0R <R [⟨𝑥, 𝑦⟩] ~R ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R ) → 0R <R ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R )) ↔ ((0R <R 𝐴 ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R ) → 0R <R (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ))))
13		breq2 5104	. . . . 5 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (0R <R [⟨𝑧, 𝑤⟩] ~R ↔ 0R <R 𝐵))
14	13	anbi2d 631	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ((0R <R 𝐴 ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R ) ↔ (0R <R 𝐴 ∧ 0R <R 𝐵)))
15		oveq2 7378	. . . . 5 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ) = (𝐴 ·R 𝐵))
16	15	breq2d 5112	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (0R <R (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ) ↔ 0R <R (𝐴 ·R 𝐵)))
17	14, 16	imbi12d 344	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (((0R <R 𝐴 ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R ) → 0R <R (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R )) ↔ ((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 ·R 𝐵))))
18		gt0srpr 11003	. . . . 5 ⊢ (0R <R [⟨𝑥, 𝑦⟩] ~R ↔ 𝑦<P 𝑥)
19		gt0srpr 11003	. . . . 5 ⊢ (0R <R [⟨𝑧, 𝑤⟩] ~R ↔ 𝑤<P 𝑧)
20	18, 19	anbi12i 629	. . . 4 ⊢ ((0R <R [⟨𝑥, 𝑦⟩] ~R ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R ) ↔ (𝑦<P 𝑥 ∧ 𝑤<P 𝑧))
21		simprr 773	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → 𝑤 ∈ P)
22		mulclpr 10945	. . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 𝑧 ∈ P) → (𝑥 ·P 𝑧) ∈ P)
23		mulclpr 10945	. . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 𝑤 ∈ P) → (𝑦 ·P 𝑤) ∈ P)
24		addclpr 10943	. . . . . . . 8 ⊢ (((𝑥 ·P 𝑧) ∈ P ∧ (𝑦 ·P 𝑤) ∈ P) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
25	22, 23, 24	syl2an 597	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑦 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
26	25	an4s 661	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
27		ltexpri 10968	. . . . . . . . 9 ⊢ (𝑦<P 𝑥 → ∃𝑣 ∈ P (𝑦 +P 𝑣) = 𝑥)
28		ltexpri 10968	. . . . . . . . 9 ⊢ (𝑤<P 𝑧 → ∃𝑢 ∈ P (𝑤 +P 𝑢) = 𝑧)
29		mulclpr 10945	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ P ∧ 𝑤 ∈ P) → (𝑣 ·P 𝑤) ∈ P)
30		oveq12 7379	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑦 +P 𝑣) = 𝑥 ∧ (𝑤 +P 𝑢) = 𝑧) → ((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) = (𝑥 ·P 𝑧))
31	30	oveq1d 7385	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 +P 𝑣) = 𝑥 ∧ (𝑤 +P 𝑢) = 𝑧) → (((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) +P ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤))) = ((𝑥 ·P 𝑧) +P ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤))))
32		distrpr 10953	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ·P (𝑤 +P 𝑢)) = ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢))
33		oveq2 7378	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑤 +P 𝑢) = 𝑧 → (𝑦 ·P (𝑤 +P 𝑢)) = (𝑦 ·P 𝑧))
34	32, 33	eqtr3id 2786	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑤 +P 𝑢) = 𝑧 → ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢)) = (𝑦 ·P 𝑧))
35	34	oveq1d 7385	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑤 +P 𝑢) = 𝑧 → (((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢)) +P ((𝑣 ·P 𝑤) +P (𝑣 ·P 𝑢))) = ((𝑦 ·P 𝑧) +P ((𝑣 ·P 𝑤) +P (𝑣 ·P 𝑢))))
36		vex 3446	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 𝑦 ∈ V
37		vex 3446	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 𝑣 ∈ V
38		vex 3446	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 𝑤 ∈ V
39		mulcompr 10948	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 ·P 𝑔) = (𝑔 ·P 𝑓)
40		distrpr 10953	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 ·P (𝑔 +P ℎ)) = ((𝑓 ·P 𝑔) +P (𝑓 ·P ℎ))
41	36, 37, 38, 39, 40	caovdir 7604	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 +P 𝑣) ·P 𝑤) = ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤))
42		vex 3446	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 𝑢 ∈ V
43	36, 37, 42, 39, 40	caovdir 7604	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 +P 𝑣) ·P 𝑢) = ((𝑦 ·P 𝑢) +P (𝑣 ·P 𝑢))
44	41, 43	oveq12i 7382	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑦 +P 𝑣) ·P 𝑤) +P ((𝑦 +P 𝑣) ·P 𝑢)) = (((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤)) +P ((𝑦 ·P 𝑢) +P (𝑣 ·P 𝑢)))
45		distrpr 10953	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) = (((𝑦 +P 𝑣) ·P 𝑤) +P ((𝑦 +P 𝑣) ·P 𝑢))
46		ovex 7403	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ·P 𝑤) ∈ V
47		ovex 7403	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ·P 𝑢) ∈ V
48		ovex 7403	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑣 ·P 𝑤) ∈ V
49		addcompr 10946	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 +P 𝑔) = (𝑔 +P 𝑓)
50		addasspr 10947	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 +P 𝑔) +P ℎ) = (𝑓 +P (𝑔 +P ℎ))
51		ovex 7403	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑣 ·P 𝑢) ∈ V
52	46, 47, 48, 49, 50, 51	caov4 7601	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢)) +P ((𝑣 ·P 𝑤) +P (𝑣 ·P 𝑢))) = (((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤)) +P ((𝑦 ·P 𝑢) +P (𝑣 ·P 𝑢)))
53	44, 45, 52	3eqtr4i 2770	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) = (((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢)) +P ((𝑣 ·P 𝑤) +P (𝑣 ·P 𝑢)))
54		ovex 7403	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ·P 𝑧) ∈ V
55	48, 54, 51, 49, 50	caov12 7598	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑣 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢))) = ((𝑦 ·P 𝑧) +P ((𝑣 ·P 𝑤) +P (𝑣 ·P 𝑢)))
56	35, 53, 55	3eqtr4g 2797	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑤 +P 𝑢) = 𝑧 → ((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) = ((𝑣 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢))))
57		oveq1 7377	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 +P 𝑣) = 𝑥 → ((𝑦 +P 𝑣) ·P 𝑤) = (𝑥 ·P 𝑤))
58	41, 57	eqtr3id 2786	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 +P 𝑣) = 𝑥 → ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤)) = (𝑥 ·P 𝑤))
59	56, 58	oveqan12rd 7390	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 +P 𝑣) = 𝑥 ∧ (𝑤 +P 𝑢) = 𝑧) → (((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) +P ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤))) = (((𝑣 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢))) +P (𝑥 ·P 𝑤)))
60	31, 59	eqtr3d 2774	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 +P 𝑣) = 𝑥 ∧ (𝑤 +P 𝑢) = 𝑧) → ((𝑥 ·P 𝑧) +P ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤))) = (((𝑣 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢))) +P (𝑥 ·P 𝑤)))
61		addasspr 10947	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) +P (𝑣 ·P 𝑤)) = ((𝑥 ·P 𝑧) +P ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤)))
62		addcompr 10946	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) +P (𝑣 ·P 𝑤)) = ((𝑣 ·P 𝑤) +P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)))
63	61, 62	eqtr3i 2762	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ·P 𝑧) +P ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤))) = ((𝑣 ·P 𝑤) +P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)))
64		addasspr 10947	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑣 ·P 𝑤) +P (𝑥 ·P 𝑤)) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢))) = ((𝑣 ·P 𝑤) +P ((𝑥 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢))))
65		ovex 7403	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢)) ∈ V
66		ovex 7403	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ·P 𝑤) ∈ V
67	48, 65, 66, 49, 50	caov32 7597	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑣 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢))) +P (𝑥 ·P 𝑤)) = (((𝑣 ·P 𝑤) +P (𝑥 ·P 𝑤)) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢)))
68		addasspr 10947	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢)) = ((𝑥 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢)))
69	68	oveq2i 7381	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 ·P 𝑤) +P (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢))) = ((𝑣 ·P 𝑤) +P ((𝑥 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢))))
70	64, 67, 69	3eqtr4i 2770	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑣 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢))) +P (𝑥 ·P 𝑤)) = ((𝑣 ·P 𝑤) +P (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢)))
71	60, 63, 70	3eqtr3g 2795	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑦 +P 𝑣) = 𝑥 ∧ (𝑤 +P 𝑢) = 𝑧) → ((𝑣 ·P 𝑤) +P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))) = ((𝑣 ·P 𝑤) +P (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢))))
72		addcanpr 10971	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑣 ·P 𝑤) ∈ P ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) → (((𝑣 ·P 𝑤) +P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))) = ((𝑣 ·P 𝑤) +P (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢))) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) = (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢))))
73	71, 72	syl5 34	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑣 ·P 𝑤) ∈ P ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) → (((𝑦 +P 𝑣) = 𝑥 ∧ (𝑤 +P 𝑢) = 𝑧) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) = (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢))))
74		eqcom 2744	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) = (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢)) ↔ (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢)) = ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)))
75		ltaddpr2 10960	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P → ((((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢)) = ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))
76	74, 75	biimtrid 242	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P → (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) = (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))
77	76	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑣 ·P 𝑤) ∈ P ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) → (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) = (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))
78	73, 77	syld 47	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑣 ·P 𝑤) ∈ P ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) → (((𝑦 +P 𝑣) = 𝑥 ∧ (𝑤 +P 𝑢) = 𝑧) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))
79	29, 78	sylan 581	. . . . . . . . . . . . . . . 16 ⊢ (((𝑣 ∈ P ∧ 𝑤 ∈ P) ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) → (((𝑦 +P 𝑣) = 𝑥 ∧ (𝑤 +P 𝑢) = 𝑧) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))
80	79	a1d 25	. . . . . . . . . . . . . . 15 ⊢ (((𝑣 ∈ P ∧ 𝑤 ∈ P) ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) → (𝑢 ∈ P → (((𝑦 +P 𝑣) = 𝑥 ∧ (𝑤 +P 𝑢) = 𝑧) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)))))
81	80	exp4a 431	. . . . . . . . . . . . . 14 ⊢ (((𝑣 ∈ P ∧ 𝑤 ∈ P) ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) → (𝑢 ∈ P → ((𝑦 +P 𝑣) = 𝑥 → ((𝑤 +P 𝑢) = 𝑧 → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))))
82	81	com34 91	. . . . . . . . . . . . 13 ⊢ (((𝑣 ∈ P ∧ 𝑤 ∈ P) ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) → (𝑢 ∈ P → ((𝑤 +P 𝑢) = 𝑧 → ((𝑦 +P 𝑣) = 𝑥 → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))))
83	82	rexlimdv 3137	. . . . . . . . . . . 12 ⊢ (((𝑣 ∈ P ∧ 𝑤 ∈ P) ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) → (∃𝑢 ∈ P (𝑤 +P 𝑢) = 𝑧 → ((𝑦 +P 𝑣) = 𝑥 → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)))))
84	83	expl 457	. . . . . . . . . . 11 ⊢ (𝑣 ∈ P → ((𝑤 ∈ P ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) → (∃𝑢 ∈ P (𝑤 +P 𝑢) = 𝑧 → ((𝑦 +P 𝑣) = 𝑥 → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))))
85	84	com24 95	. . . . . . . . . 10 ⊢ (𝑣 ∈ P → ((𝑦 +P 𝑣) = 𝑥 → (∃𝑢 ∈ P (𝑤 +P 𝑢) = 𝑧 → ((𝑤 ∈ P ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))))
86	85	rexlimiv 3132	. . . . . . . . 9 ⊢ (∃𝑣 ∈ P (𝑦 +P 𝑣) = 𝑥 → (∃𝑢 ∈ P (𝑤 +P 𝑢) = 𝑧 → ((𝑤 ∈ P ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)))))
87	27, 28, 86	syl2im 40	. . . . . . . 8 ⊢ (𝑦<P 𝑥 → (𝑤<P 𝑧 → ((𝑤 ∈ P ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)))))
88	87	imp 406	. . . . . . 7 ⊢ ((𝑦<P 𝑥 ∧ 𝑤<P 𝑧) → ((𝑤 ∈ P ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))
89	88	com12 32	. . . . . 6 ⊢ ((𝑤 ∈ P ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) → ((𝑦<P 𝑥 ∧ 𝑤<P 𝑧) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))
90	21, 26, 89	syl2anc 585	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑦<P 𝑥 ∧ 𝑤<P 𝑧) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))
91		mulsrpr 11001	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) = [⟨((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)), ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))⟩] ~R )
92	91	breq2d 5112	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (0R <R ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) ↔ 0R <R [⟨((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)), ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))⟩] ~R ))
93		gt0srpr 11003	. . . . . 6 ⊢ (0R <R [⟨((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)), ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))⟩] ~R ↔ ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)))
94	92, 93	bitrdi 287	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (0R <R ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) ↔ ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))
95	90, 94	sylibrd 259	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑦<P 𝑥 ∧ 𝑤<P 𝑧) → 0R <R ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R )))
96	20, 95	biimtrid 242	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((0R <R [⟨𝑥, 𝑦⟩] ~R ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R ) → 0R <R ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R )))
97	7, 12, 17, 96	2ecoptocl 8759	. 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → ((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 ·R 𝐵)))
98	6, 97	mpcom 38	1 ⊢ ((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 ·R 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  ⟨cop 4588   class class class wbr 5100  (class class class)co 7370  [cec 8645  Pcnp 10784   +_P cpp 10786   ·_P cmp 10787  <_P cltp 10788   ~_R cer 10789  Rcnr 10790  0_Rc0r 10791   ·_R cmr 10795   <_R cltr 10796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-inf2 9564

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-1st 7945  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-oadd 8413  df-omul 8414  df-er 8647  df-ec 8649  df-qs 8653  df-ni 10797  df-pli 10798  df-mi 10799  df-lti 10800  df-plpq 10833  df-mpq 10834  df-ltpq 10835  df-enq 10836  df-nq 10837  df-erq 10838  df-plq 10839  df-mq 10840  df-1nq 10841  df-rq 10842  df-ltnq 10843  df-np 10906  df-1p 10907  df-plp 10908  df-mp 10909  df-ltp 10910  df-enr 10980  df-nr 10981  df-mr 10983  df-ltr 10984  df-0r 10985

This theorem is referenced by:  sqgt0sr  11031  axpre-mulgt0  11093