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Theorem mulgt0sr 9805
 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 9768 . . . . 5 <R ⊆ (R × R)
21brel 5090 . . . 4 (0R <R 𝐴 → (0RR𝐴R))
32simprd 478 . . 3 (0R <R 𝐴𝐴R)
41brel 5090 . . . 4 (0R <R 𝐵 → (0RR𝐵R))
54simprd 478 . . 3 (0R <R 𝐵𝐵R)
63, 5anim12i 588 . 2 ((0R <R 𝐴 ∧ 0R <R 𝐵) → (𝐴R𝐵R))
7 df-nr 9757 . . 3 R = ((P × P) / ~R )
8 breq2 4587 . . . . 5 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (0R <R [⟨𝑥, 𝑦⟩] ~R ↔ 0R <R 𝐴))
98anbi1d 737 . . . 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 6556 . . . . 5 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) = (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ))
1110breq2d 4595 . . . 4 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (0R <R ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) ↔ 0R <R (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R )))
129, 11imbi12d 333 . . 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 4587 . . . . 5 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (0R <R [⟨𝑧, 𝑤⟩] ~R ↔ 0R <R 𝐵))
1413anbi2d 736 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ((0R <R 𝐴 ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R ) ↔ (0R <R 𝐴 ∧ 0R <R 𝐵)))
15 oveq2 6557 . . . . 5 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ) = (𝐴 ·R 𝐵))
1615breq2d 4595 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (0R <R (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ) ↔ 0R <R (𝐴 ·R 𝐵)))
1714, 16imbi12d 333 . . 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 9778 . . . . 5 (0R <R [⟨𝑥, 𝑦⟩] ~R𝑦<P 𝑥)
19 gt0srpr 9778 . . . . 5 (0R <R [⟨𝑧, 𝑤⟩] ~R𝑤<P 𝑧)
2018, 19anbi12i 729 . . . 4 ((0R <R [⟨𝑥, 𝑦⟩] ~R ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R ) ↔ (𝑦<P 𝑥𝑤<P 𝑧))
21 simprr 792 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → 𝑤P)
22 mulclpr 9721 . . . . . . . 8 ((𝑥P𝑧P) → (𝑥 ·P 𝑧) ∈ P)
23 mulclpr 9721 . . . . . . . 8 ((𝑦P𝑤P) → (𝑦 ·P 𝑤) ∈ P)
24 addclpr 9719 . . . . . . . 8 (((𝑥 ·P 𝑧) ∈ P ∧ (𝑦 ·P 𝑤) ∈ P) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
2522, 23, 24syl2an 493 . . . . . . 7 (((𝑥P𝑧P) ∧ (𝑦P𝑤P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
2625an4s 865 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
27 ltexpri 9744 . . . . . . . . 9 (𝑦<P 𝑥 → ∃𝑣P (𝑦 +P 𝑣) = 𝑥)
28 ltexpri 9744 . . . . . . . . 9 (𝑤<P 𝑧 → ∃𝑢P (𝑤 +P 𝑢) = 𝑧)
29 mulclpr 9721 . . . . . . . . . . . . . . . . 17 ((𝑣P𝑤P) → (𝑣 ·P 𝑤) ∈ P)
30 oveq12 6558 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦 +P 𝑣) = 𝑥 ∧ (𝑤 +P 𝑢) = 𝑧) → ((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) = (𝑥 ·P 𝑧))
3130oveq1d 6564 . . . . . . . . . . . . . . . . . . . . 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 9729 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ·P (𝑤 +P 𝑢)) = ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢))
33 oveq2 6557 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑤 +P 𝑢) = 𝑧 → (𝑦 ·P (𝑤 +P 𝑢)) = (𝑦 ·P 𝑧))
3432, 33syl5eqr 2658 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑤 +P 𝑢) = 𝑧 → ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢)) = (𝑦 ·P 𝑧))
3534oveq1d 6564 . . . . . . . . . . . . . . . . . . . . . . 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 3176 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑦 ∈ V
37 vex 3176 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑣 ∈ V
38 vex 3176 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑤 ∈ V
39 mulcompr 9724 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 ·P 𝑔) = (𝑔 ·P 𝑓)
40 distrpr 9729 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 ·P (𝑔 +P )) = ((𝑓 ·P 𝑔) +P (𝑓 ·P ))
4136, 37, 38, 39, 40caovdir 6766 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 +P 𝑣) ·P 𝑤) = ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤))
42 vex 3176 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑢 ∈ V
4336, 37, 42, 39, 40caovdir 6766 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 +P 𝑣) ·P 𝑢) = ((𝑦 ·P 𝑢) +P (𝑣 ·P 𝑢))
4441, 43oveq12i 6561 . . . . . . . . . . . . . . . . . . . . . . . 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 9729 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) = (((𝑦 +P 𝑣) ·P 𝑤) +P ((𝑦 +P 𝑣) ·P 𝑢))
46 ovex 6577 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ·P 𝑤) ∈ V
47 ovex 6577 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ·P 𝑢) ∈ V
48 ovex 6577 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑣 ·P 𝑤) ∈ V
49 addcompr 9722 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 +P 𝑔) = (𝑔 +P 𝑓)
50 addasspr 9723 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓 +P 𝑔) +P ) = (𝑓 +P (𝑔 +P ))
51 ovex 6577 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑣 ·P 𝑢) ∈ V
5246, 47, 48, 49, 50, 51caov4 6763 . . . . . . . . . . . . . . . . . . . . . . . 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 2642 . . . . . . . . . . . . . . . . . . . . . . 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 6577 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ·P 𝑧) ∈ V
5548, 54, 51, 49, 50caov12 6760 . . . . . . . . . . . . . . . . . . . . . . 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 2669 . . . . . . . . . . . . . . . . . . . . . 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 6556 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 +P 𝑣) = 𝑥 → ((𝑦 +P 𝑣) ·P 𝑤) = (𝑥 ·P 𝑤))
5841, 57syl5eqr 2658 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 +P 𝑣) = 𝑥 → ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤)) = (𝑥 ·P 𝑤))
5956, 58oveqan12rd 6569 . . . . . . . . . . . . . . . . . . . . 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 2646 . . . . . . . . . . . . . . . . . . . 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 9723 . . . . . . . . . . . . . . . . . . . . 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 9722 . . . . . . . . . . . . . . . . . . . . 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 2634 . . . . . . . . . . . . . . . . . . . 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 9723 . . . . . . . . . . . . . . . . . . . . 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 6577 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢)) ∈ V
66 ovex 6577 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ·P 𝑤) ∈ V
6748, 65, 66, 49, 50caov32 6759 . . . . . . . . . . . . . . . . . . . . 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 9723 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢)) = ((𝑥 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢)))
6968oveq2i 6560 . . . . . . . . . . . . . . . . . . . . 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 2642 . . . . . . . . . . . . . . . . . . . 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 2667 . . . . . . . . . . . . . . . . . . 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 9747 . . . . . . . . . . . . . . . . . . 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 33 . . . . . . . . . . . . . . . . . 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 2617 . . . . . . . . . . . . . . . . . . . 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 9736 . . . . . . . . . . . . . . . . . . . 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, 75syl5bi 231 . . . . . . . . . . . . . . . . . . 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 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))))
7873, 77syld 46 . . . . . . . . . . . . . . . . 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 487 . . . . . . . . . . . . . . . 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 631 . . . . . . . . . . . . . 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 89 . . . . . . . . . . . . 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 3012 . . . . . . . . . . . 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 646 . . . . . . . . . . 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 93 . . . . . . . . . 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 3009 . . . . . . . . 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 39 . . . . . . . 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 444 . . . . . . 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 691 . . . . 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 9776 . . . . . . 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 4595 . . . . . 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 9778 . . . . . 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, 93syl6bb 275 . . . . 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 248 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑦<P 𝑥𝑤<P 𝑧) → 0R <R ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R )))
9620, 95syl5bi 231 . . 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 7725 . 2 ((𝐴R𝐵R) → ((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 ·R 𝐵)))
986, 97mpcom 37 1 ((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 ·R 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  ⟨cop 4131   class class class wbr 4583  (class class class)co 6549  [cec 7627  Pcnp 9560   +P cpp 9562   ·P cmp 9563
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