Theorem mulgt0sr 10516

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 10479	. . . . 5 ⊢ <R ⊆ (R × R)
2	1	brel 5581	. . . 4 ⊢ (0R <R 𝐴 → (0R ∈ R ∧ 𝐴 ∈ R))
3	2	simprd 499	. . 3 ⊢ (0R <R 𝐴 → 𝐴 ∈ R)
4	1	brel 5581	. . . 4 ⊢ (0R <R 𝐵 → (0R ∈ R ∧ 𝐵 ∈ R))
5	4	simprd 499	. . 3 ⊢ (0R <R 𝐵 → 𝐵 ∈ R)
6	3, 5	anim12i 615	. 2 ⊢ ((0R <R 𝐴 ∧ 0R <R 𝐵) → (𝐴 ∈ R ∧ 𝐵 ∈ R))
7		df-nr 10467	. . 3 ⊢ R = ((P × P) / ~R )
8		breq2 5034	. . . . 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 7142	. . . . 5 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) = (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ))
11	10	breq2d 5042	. . . 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 348	. . 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 5034	. . . . 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 7143	. . . . 5 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ) = (𝐴 ·R 𝐵))
16	15	breq2d 5042	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (0R <R (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ) ↔ 0R <R (𝐴 ·R 𝐵)))
17	14, 16	imbi12d 348	. . 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 10489	. . . . 5 ⊢ (0R <R [⟨𝑥, 𝑦⟩] ~R ↔ 𝑦<P 𝑥)
19		gt0srpr 10489	. . . . 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 772	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → 𝑤 ∈ P)
22		mulclpr 10431	. . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 𝑧 ∈ P) → (𝑥 ·P 𝑧) ∈ P)
23		mulclpr 10431	. . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 𝑤 ∈ P) → (𝑦 ·P 𝑤) ∈ P)
24		addclpr 10429	. . . . . . . 8 ⊢ (((𝑥 ·P 𝑧) ∈ P ∧ (𝑦 ·P 𝑤) ∈ P) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
25	22, 23, 24	syl2an 598	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑦 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
26	25	an4s 659	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
27		ltexpri 10454	. . . . . . . . 9 ⊢ (𝑦<P 𝑥 → ∃𝑣 ∈ P (𝑦 +P 𝑣) = 𝑥)
28		ltexpri 10454	. . . . . . . . 9 ⊢ (𝑤<P 𝑧 → ∃𝑢 ∈ P (𝑤 +P 𝑢) = 𝑧)
29		mulclpr 10431	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ P ∧ 𝑤 ∈ P) → (𝑣 ·P 𝑤) ∈ P)
30		oveq12 7144	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑦 +P 𝑣) = 𝑥 ∧ (𝑤 +P 𝑢) = 𝑧) → ((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) = (𝑥 ·P 𝑧))
31	30	oveq1d 7150	. . . . . . . . . . . . . . . . . . . . 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 10439	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ·P (𝑤 +P 𝑢)) = ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢))
33		oveq2 7143	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑤 +P 𝑢) = 𝑧 → (𝑦 ·P (𝑤 +P 𝑢)) = (𝑦 ·P 𝑧))
34	32, 33	syl5eqr 2847	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑤 +P 𝑢) = 𝑧 → ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢)) = (𝑦 ·P 𝑧))
35	34	oveq1d 7150	. . . . . . . . . . . . . . . . . . . . . . 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 3444	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 𝑦 ∈ V
37		vex 3444	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 𝑣 ∈ V
38		vex 3444	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 𝑤 ∈ V
39		mulcompr 10434	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 ·P 𝑔) = (𝑔 ·P 𝑓)
40		distrpr 10439	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 ·P (𝑔 +P ℎ)) = ((𝑓 ·P 𝑔) +P (𝑓 ·P ℎ))
41	36, 37, 38, 39, 40	caovdir 7362	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 +P 𝑣) ·P 𝑤) = ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤))
42		vex 3444	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 𝑢 ∈ V
43	36, 37, 42, 39, 40	caovdir 7362	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 +P 𝑣) ·P 𝑢) = ((𝑦 ·P 𝑢) +P (𝑣 ·P 𝑢))
44	41, 43	oveq12i 7147	. . . . . . . . . . . . . . . . . . . . . . . 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 10439	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) = (((𝑦 +P 𝑣) ·P 𝑤) +P ((𝑦 +P 𝑣) ·P 𝑢))
46		ovex 7168	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ·P 𝑤) ∈ V
47		ovex 7168	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ·P 𝑢) ∈ V
48		ovex 7168	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑣 ·P 𝑤) ∈ V
49		addcompr 10432	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 +P 𝑔) = (𝑔 +P 𝑓)
50		addasspr 10433	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 +P 𝑔) +P ℎ) = (𝑓 +P (𝑔 +P ℎ))
51		ovex 7168	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑣 ·P 𝑢) ∈ V
52	46, 47, 48, 49, 50, 51	caov4 7359	. . . . . . . . . . . . . . . . . . . . . . . 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 2831	. . . . . . . . . . . . . . . . . . . . . . 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 7168	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ·P 𝑧) ∈ V
55	48, 54, 51, 49, 50	caov12 7356	. . . . . . . . . . . . . . . . . . . . . . 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 2858	. . . . . . . . . . . . . . . . . . . . . 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 7142	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 +P 𝑣) = 𝑥 → ((𝑦 +P 𝑣) ·P 𝑤) = (𝑥 ·P 𝑤))
58	41, 57	syl5eqr 2847	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 +P 𝑣) = 𝑥 → ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤)) = (𝑥 ·P 𝑤))
59	56, 58	oveqan12rd 7155	. . . . . . . . . . . . . . . . . . . . 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 2835	. . . . . . . . . . . . . . . . . . . 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 10433	. . . . . . . . . . . . . . . . . . . . 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 10432	. . . . . . . . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . . . . . . . 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 10433	. . . . . . . . . . . . . . . . . . . . 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 7168	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢)) ∈ V
66		ovex 7168	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ·P 𝑤) ∈ V
67	48, 65, 66, 49, 50	caov32 7355	. . . . . . . . . . . . . . . . . . . . 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 10433	. . . . . . . . . . . . . . . . . . . . . 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 7146	. . . . . . . . . . . . . . . . . . . . 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 2831	. . . . . . . . . . . . . . . . . . . 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 2856	. . . . . . . . . . . . . . . . . . 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 10457	. . . . . . . . . . . . . . . . . . 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 2805	. . . . . . . . . . . . . . . . . . . 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 10446	. . . . . . . . . . . . . . . . . . . 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	syl5bi 245	. . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . 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 435	. . . . . . . . . . . . . 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 3242	. . . . . . . . . . . 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 461	. . . . . . . . . . 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 3239	. . . . . . . . 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 410	. . . . . . 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 587	. . . . 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 10487	. . . . . . 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 5042	. . . . . 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 10489	. . . . . 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	syl6bb 290	. . . . 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 262	. . . 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	syl5bi 245	. . 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 8371	. 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 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  ⟨cop 4531   class class class wbr 5030  (class class class)co 7135  [cec 8270  Pcnp 10270   +_P cpp 10272   ·_P cmp 10273  <_P cltp 10274   ~_R cer 10275  Rcnr 10276  0_Rc0r 10277   ·_R cmr 10281   <_R cltr 10282

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-omul 8090  df-er 8272  df-ec 8274  df-qs 8278  df-ni 10283  df-pli 10284  df-mi 10285  df-lti 10286  df-plpq 10319  df-mpq 10320  df-ltpq 10321  df-enq 10322  df-nq 10323  df-erq 10324  df-plq 10325  df-mq 10326  df-1nq 10327  df-rq 10328  df-ltnq 10329  df-np 10392  df-1p 10393  df-plp 10394  df-mp 10395  df-ltp 10396  df-enr 10466  df-nr 10467  df-mr 10469  df-ltr 10470  df-0r 10471

This theorem is referenced by:  sqgt0sr  10517  axpre-mulgt0  10579