MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulgt0sr Structured version   Visualization version   GIF version

Theorem mulgt0sr 11041
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 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelsr 11004 . . . . 5 <R ⊆ (R × R)
21brel 5697 . . . 4 (0R <R 𝐴 → (0RR𝐴R))
32simprd 496 . . 3 (0R <R 𝐴𝐴R)
41brel 5697 . . . 4 (0R <R 𝐵 → (0RR𝐵R))
54simprd 496 . . 3 (0R <R 𝐵𝐵R)
63, 5anim12i 613 . 2 ((0R <R 𝐴 ∧ 0R <R 𝐵) → (𝐴R𝐵R))
7 df-nr 10992 . . 3 R = ((P × P) / ~R )
8 breq2 5109 . . . . 5 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (0R <R [⟨𝑥, 𝑦⟩] ~R ↔ 0R <R 𝐴))
98anbi1d 630 . . . 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 7364 . . . . 5 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) = (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ))
1110breq2d 5117 . . . 4 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (0R <R ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) ↔ 0R <R (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R )))
129, 11imbi12d 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 5109 . . . . 5 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (0R <R [⟨𝑧, 𝑤⟩] ~R ↔ 0R <R 𝐵))
1413anbi2d 629 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ((0R <R 𝐴 ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R ) ↔ (0R <R 𝐴 ∧ 0R <R 𝐵)))
15 oveq2 7365 . . . . 5 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ) = (𝐴 ·R 𝐵))
1615breq2d 5117 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (0R <R (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ) ↔ 0R <R (𝐴 ·R 𝐵)))
1714, 16imbi12d 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 11014 . . . . 5 (0R <R [⟨𝑥, 𝑦⟩] ~R𝑦<P 𝑥)
19 gt0srpr 11014 . . . . 5 (0R <R [⟨𝑧, 𝑤⟩] ~R𝑤<P 𝑧)
2018, 19anbi12i 627 . . . 4 ((0R <R [⟨𝑥, 𝑦⟩] ~R ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R ) ↔ (𝑦<P 𝑥𝑤<P 𝑧))
21 simprr 771 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → 𝑤P)
22 mulclpr 10956 . . . . . . . 8 ((𝑥P𝑧P) → (𝑥 ·P 𝑧) ∈ P)
23 mulclpr 10956 . . . . . . . 8 ((𝑦P𝑤P) → (𝑦 ·P 𝑤) ∈ P)
24 addclpr 10954 . . . . . . . 8 (((𝑥 ·P 𝑧) ∈ P ∧ (𝑦 ·P 𝑤) ∈ P) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
2522, 23, 24syl2an 596 . . . . . . 7 (((𝑥P𝑧P) ∧ (𝑦P𝑤P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
2625an4s 658 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
27 ltexpri 10979 . . . . . . . . 9 (𝑦<P 𝑥 → ∃𝑣P (𝑦 +P 𝑣) = 𝑥)
28 ltexpri 10979 . . . . . . . . 9 (𝑤<P 𝑧 → ∃𝑢P (𝑤 +P 𝑢) = 𝑧)
29 mulclpr 10956 . . . . . . . . . . . . . . . . 17 ((𝑣P𝑤P) → (𝑣 ·P 𝑤) ∈ P)
30 oveq12 7366 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦 +P 𝑣) = 𝑥 ∧ (𝑤 +P 𝑢) = 𝑧) → ((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) = (𝑥 ·P 𝑧))
3130oveq1d 7372 . . . . . . . . . . . . . . . . . . . . 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 10964 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ·P (𝑤 +P 𝑢)) = ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢))
33 oveq2 7365 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑤 +P 𝑢) = 𝑧 → (𝑦 ·P (𝑤 +P 𝑢)) = (𝑦 ·P 𝑧))
3432, 33eqtr3id 2790 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑤 +P 𝑢) = 𝑧 → ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢)) = (𝑦 ·P 𝑧))
3534oveq1d 7372 . . . . . . . . . . . . . . . . . . . . . . 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 3449 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑦 ∈ V
37 vex 3449 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑣 ∈ V
38 vex 3449 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑤 ∈ V
39 mulcompr 10959 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 ·P 𝑔) = (𝑔 ·P 𝑓)
40 distrpr 10964 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 ·P (𝑔 +P )) = ((𝑓 ·P 𝑔) +P (𝑓 ·P ))
4136, 37, 38, 39, 40caovdir 7588 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 +P 𝑣) ·P 𝑤) = ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤))
42 vex 3449 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑢 ∈ V
4336, 37, 42, 39, 40caovdir 7588 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 +P 𝑣) ·P 𝑢) = ((𝑦 ·P 𝑢) +P (𝑣 ·P 𝑢))
4441, 43oveq12i 7369 . . . . . . . . . . . . . . . . . . . . . . . 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 10964 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) = (((𝑦 +P 𝑣) ·P 𝑤) +P ((𝑦 +P 𝑣) ·P 𝑢))
46 ovex 7390 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ·P 𝑤) ∈ V
47 ovex 7390 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ·P 𝑢) ∈ V
48 ovex 7390 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑣 ·P 𝑤) ∈ V
49 addcompr 10957 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 +P 𝑔) = (𝑔 +P 𝑓)
50 addasspr 10958 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓 +P 𝑔) +P ) = (𝑓 +P (𝑔 +P ))
51 ovex 7390 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑣 ·P 𝑢) ∈ V
5246, 47, 48, 49, 50, 51caov4 7585 . . . . . . . . . . . . . . . . . . . . . . . 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)))
5344, 45, 523eqtr4i 2774 . . . . . . . . . . . . . . . . . . . . . . 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 7390 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ·P 𝑧) ∈ V
5548, 54, 51, 49, 50caov12 7582 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑣 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢))) = ((𝑦 ·P 𝑧) +P ((𝑣 ·P 𝑤) +P (𝑣 ·P 𝑢)))
5635, 53, 553eqtr4g 2801 . . . . . . . . . . . . . . . . . . . . . 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 7364 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 +P 𝑣) = 𝑥 → ((𝑦 +P 𝑣) ·P 𝑤) = (𝑥 ·P 𝑤))
5841, 57eqtr3id 2790 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 +P 𝑣) = 𝑥 → ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤)) = (𝑥 ·P 𝑤))
5956, 58oveqan12rd 7377 . . . . . . . . . . . . . . . . . . . . 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)))
6031, 59eqtr3d 2778 . . . . . . . . . . . . . . . . . . . 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 10958 . . . . . . . . . . . . . . . . . . . . 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 10957 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) +P (𝑣 ·P 𝑤)) = ((𝑣 ·P 𝑤) +P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)))
6361, 62eqtr3i 2766 . . . . . . . . . . . . . . . . . . . 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 10958 . . . . . . . . . . . . . . . . . . . . 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 7390 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢)) ∈ V
66 ovex 7390 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ·P 𝑤) ∈ V
6748, 65, 66, 49, 50caov32 7581 . . . . . . . . . . . . . . . . . . . . 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 10958 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢)) = ((𝑥 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢)))
6968oveq2i 7368 . . . . . . . . . . . . . . . . . . . . 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))))
7064, 67, 693eqtr4i 2774 . . . . . . . . . . . . . . . . . . . 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)))
7160, 63, 703eqtr3g 2799 . . . . . . . . . . . . . . . . . . 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 10982 . . . . . . . . . . . . . . . . . . 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))))
7371, 72syl5 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 2743 . . . . . . . . . . . . . . . . . . . 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 10971 . . . . . . . . . . . . . . . . . . . 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))))
7674, 75biimtrid 241 . . . . . . . . . . . . . . . . . . 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))))
7776adantl 482 . . . . . . . . . . . . . . . . . 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))))
7873, 77syld 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))))
7929, 78sylan 580 . . . . . . . . . . . . . . . 16 (((𝑣P𝑤P) ∧ ((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))))
8079a1d 25 . . . . . . . . . . . . . . 15 (((𝑣P𝑤P) ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) → (𝑢P → (((𝑦 +P 𝑣) = 𝑥 ∧ (𝑤 +P 𝑢) = 𝑧) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)))))
8180exp4a 432 . . . . . . . . . . . . . 14 (((𝑣P𝑤P) ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) → (𝑢P → ((𝑦 +P 𝑣) = 𝑥 → ((𝑤 +P 𝑢) = 𝑧 → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))))
8281com34 91 . . . . . . . . . . . . 13 (((𝑣P𝑤P) ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) → (𝑢P → ((𝑤 +P 𝑢) = 𝑧 → ((𝑦 +P 𝑣) = 𝑥 → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))))
8382rexlimdv 3150 . . . . . . . . . . . 12 (((𝑣P𝑤P) ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) → (∃𝑢P (𝑤 +P 𝑢) = 𝑧 → ((𝑦 +P 𝑣) = 𝑥 → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)))))
8483expl 458 . . . . . . . . . . 11 (𝑣P → ((𝑤P ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) → (∃𝑢P (𝑤 +P 𝑢) = 𝑧 → ((𝑦 +P 𝑣) = 𝑥 → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))))
8584com24 95 . . . . . . . . . 10 (𝑣P → ((𝑦 +P 𝑣) = 𝑥 → (∃𝑢P (𝑤 +P 𝑢) = 𝑧 → ((𝑤P ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))))
8685rexlimiv 3145 . . . . . . . . 9 (∃𝑣P (𝑦 +P 𝑣) = 𝑥 → (∃𝑢P (𝑤 +P 𝑢) = 𝑧 → ((𝑤P ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)))))
8727, 28, 86syl2im 40 . . . . . . . 8 (𝑦<P 𝑥 → (𝑤<P 𝑧 → ((𝑤P ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)))))
8887imp 407 . . . . . . 7 ((𝑦<P 𝑥𝑤<P 𝑧) → ((𝑤P ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))
8988com12 32 . . . . . 6 ((𝑤P ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) → ((𝑦<P 𝑥𝑤<P 𝑧) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))
9021, 26, 89syl2anc 584 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑦<P 𝑥𝑤<P 𝑧) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))
91 mulsrpr 11012 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) = [⟨((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)), ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))⟩] ~R )
9291breq2d 5117 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (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 11014 . . . . . 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)))
9492, 93bitrdi 286 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (0R <R ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) ↔ ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))
9590, 94sylibrd 258 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑦<P 𝑥𝑤<P 𝑧) → 0R <R ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R )))
9620, 95biimtrid 241 . . 3 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((0R <R [⟨𝑥, 𝑦⟩] ~R ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R ) → 0R <R ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R )))
977, 12, 17, 962ecoptocl 8747 . 2 ((𝐴R𝐵R) → ((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 ·R 𝐵)))
986, 97mpcom 38 1 ((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 ·R 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wrex 3073  cop 4592   class class class wbr 5105  (class class class)co 7357  [cec 8646  Pcnp 10795   +P cpp 10797   ·P cmp 10798  <P cltp 10799   ~R cer 10800  Rcnr 10801  0Rc0r 10802   ·R cmr 10806   <R cltr 10807
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-omul 8417  df-er 8648  df-ec 8650  df-qs 8654  df-ni 10808  df-pli 10809  df-mi 10810  df-lti 10811  df-plpq 10844  df-mpq 10845  df-ltpq 10846  df-enq 10847  df-nq 10848  df-erq 10849  df-plq 10850  df-mq 10851  df-1nq 10852  df-rq 10853  df-ltnq 10854  df-np 10917  df-1p 10918  df-plp 10919  df-mp 10920  df-ltp 10921  df-enr 10991  df-nr 10992  df-mr 10994  df-ltr 10995  df-0r 10996
This theorem is referenced by:  sqgt0sr  11042  axpre-mulgt0  11104
  Copyright terms: Public domain W3C validator