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Theorem mulgt0sr 7807
Description: The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.)
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 7767 . . . . 5 <R ⊆ (R × R)
21brel 4696 . . . 4 (0R <R 𝐴 → (0RR𝐴R))
32simprd 114 . . 3 (0R <R 𝐴𝐴R)
41brel 4696 . . . 4 (0R <R 𝐵 → (0RR𝐵R))
54simprd 114 . . 3 (0R <R 𝐵𝐵R)
63, 5anim12i 338 . 2 ((0R <R 𝐴 ∧ 0R <R 𝐵) → (𝐴R𝐵R))
7 df-nr 7756 . . 3 R = ((P × P) / ~R )
8 breq2 4022 . . . . 5 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (0R <R [⟨𝑥, 𝑦⟩] ~R ↔ 0R <R 𝐴))
98anbi1d 465 . . . 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 5903 . . . . 5 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) = (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ))
1110breq2d 4030 . . . 4 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (0R <R ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) ↔ 0R <R (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R )))
129, 11imbi12d 234 . . 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 4022 . . . . 5 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (0R <R [⟨𝑧, 𝑤⟩] ~R ↔ 0R <R 𝐵))
1413anbi2d 464 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ((0R <R 𝐴 ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R ) ↔ (0R <R 𝐴 ∧ 0R <R 𝐵)))
15 oveq2 5904 . . . . 5 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ) = (𝐴 ·R 𝐵))
1615breq2d 4030 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (0R <R (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ) ↔ 0R <R (𝐴 ·R 𝐵)))
1714, 16imbi12d 234 . . 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 7777 . . . . 5 (0R <R [⟨𝑥, 𝑦⟩] ~R𝑦<P 𝑥)
19 gt0srpr 7777 . . . . 5 (0R <R [⟨𝑧, 𝑤⟩] ~R𝑤<P 𝑧)
2018, 19anbi12i 460 . . . 4 ((0R <R [⟨𝑥, 𝑦⟩] ~R ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R ) ↔ (𝑦<P 𝑥𝑤<P 𝑧))
21 ltexpri 7642 . . . . . . 7 (𝑦<P 𝑥 → ∃𝑣P (𝑦 +P 𝑣) = 𝑥)
22 ltexpri 7642 . . . . . . . . 9 (𝑤<P 𝑧 → ∃𝑢P (𝑤 +P 𝑢) = 𝑧)
23 addclpr 7566 . . . . . . . . . . . . . 14 ((𝑓P𝑔P) → (𝑓 +P 𝑔) ∈ P)
2423adantl 277 . . . . . . . . . . . . 13 ((((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) ∈ P)
25 simplrr 536 . . . . . . . . . . . . . . 15 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑦 +P 𝑣) = 𝑥)
26 simplr 528 . . . . . . . . . . . . . . . . 17 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → 𝑦P)
2726ad2antrr 488 . . . . . . . . . . . . . . . 16 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → 𝑦P)
28 simplrl 535 . . . . . . . . . . . . . . . 16 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → 𝑣P)
2924, 27, 28caovcld 6050 . . . . . . . . . . . . . . 15 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑦 +P 𝑣) ∈ P)
3025, 29eqeltrrd 2267 . . . . . . . . . . . . . 14 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → 𝑥P)
31 simplrr 536 . . . . . . . . . . . . . . 15 ((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) → 𝑤P)
3231adantr 276 . . . . . . . . . . . . . 14 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → 𝑤P)
33 mulclpr 7601 . . . . . . . . . . . . . 14 ((𝑥P𝑤P) → (𝑥 ·P 𝑤) ∈ P)
3430, 32, 33syl2anc 411 . . . . . . . . . . . . 13 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑥 ·P 𝑤) ∈ P)
35 simplrl 535 . . . . . . . . . . . . . . 15 ((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) → 𝑧P)
3635adantr 276 . . . . . . . . . . . . . 14 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → 𝑧P)
37 mulclpr 7601 . . . . . . . . . . . . . 14 ((𝑦P𝑧P) → (𝑦 ·P 𝑧) ∈ P)
3827, 36, 37syl2anc 411 . . . . . . . . . . . . 13 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑦 ·P 𝑧) ∈ P)
3924, 34, 38caovcld 6050 . . . . . . . . . . . 12 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
40 simprl 529 . . . . . . . . . . . . 13 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → 𝑢P)
41 mulclpr 7601 . . . . . . . . . . . . 13 ((𝑣P𝑢P) → (𝑣 ·P 𝑢) ∈ P)
4228, 40, 41syl2anc 411 . . . . . . . . . . . 12 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑣 ·P 𝑢) ∈ P)
43 ltaddpr 7626 . . . . . . . . . . . 12 ((((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P ∧ (𝑣 ·P 𝑢) ∈ P) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢)))
4439, 42, 43syl2anc 411 . . . . . . . . . . 11 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢)))
45 simprr 531 . . . . . . . . . . . . . . 15 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑤 +P 𝑢) = 𝑧)
46 oveq12 5905 . . . . . . . . . . . . . . . 16 (((𝑦 +P 𝑣) = 𝑥 ∧ (𝑤 +P 𝑢) = 𝑧) → ((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) = (𝑥 ·P 𝑧))
4746oveq1d 5911 . . . . . . . . . . . . . . 15 (((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))))
4825, 45, 47syl2anc 411 . . . . . . . . . . . . . 14 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (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))))
49 distrprg 7617 . . . . . . . . . . . . . . . . . . 19 ((𝑦P𝑤P𝑢P) → (𝑦 ·P (𝑤 +P 𝑢)) = ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢)))
5027, 32, 40, 49syl3anc 1249 . . . . . . . . . . . . . . . . . 18 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑦 ·P (𝑤 +P 𝑢)) = ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢)))
51 oveq2 5904 . . . . . . . . . . . . . . . . . . . 20 ((𝑤 +P 𝑢) = 𝑧 → (𝑦 ·P (𝑤 +P 𝑢)) = (𝑦 ·P 𝑧))
5251adantl 277 . . . . . . . . . . . . . . . . . . 19 ((𝑢P ∧ (𝑤 +P 𝑢) = 𝑧) → (𝑦 ·P (𝑤 +P 𝑢)) = (𝑦 ·P 𝑧))
5352adantl 277 . . . . . . . . . . . . . . . . . 18 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑦 ·P (𝑤 +P 𝑢)) = (𝑦 ·P 𝑧))
5450, 53eqtr3d 2224 . . . . . . . . . . . . . . . . 17 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢)) = (𝑦 ·P 𝑧))
5554oveq1d 5911 . . . . . . . . . . . . . . . 16 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (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))))
56 distrprg 7617 . . . . . . . . . . . . . . . . . . . 20 ((𝑓P𝑔PP) → (𝑓 ·P (𝑔 +P )) = ((𝑓 ·P 𝑔) +P (𝑓 ·P )))
5756adantl 277 . . . . . . . . . . . . . . . . . . 19 ((((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) ∧ (𝑓P𝑔PP)) → (𝑓 ·P (𝑔 +P )) = ((𝑓 ·P 𝑔) +P (𝑓 ·P )))
58 mulcomprg 7609 . . . . . . . . . . . . . . . . . . . 20 ((𝑓P𝑔P) → (𝑓 ·P 𝑔) = (𝑔 ·P 𝑓))
5958adantl 277 . . . . . . . . . . . . . . . . . . 19 ((((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) ∧ (𝑓P𝑔P)) → (𝑓 ·P 𝑔) = (𝑔 ·P 𝑓))
6057, 27, 28, 32, 24, 59caovdir2d 6073 . . . . . . . . . . . . . . . . . 18 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑦 +P 𝑣) ·P 𝑤) = ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤)))
6157, 27, 28, 40, 24, 59caovdir2d 6073 . . . . . . . . . . . . . . . . . 18 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑦 +P 𝑣) ·P 𝑢) = ((𝑦 ·P 𝑢) +P (𝑣 ·P 𝑢)))
6260, 61oveq12d 5914 . . . . . . . . . . . . . . . . 17 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (((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))))
63 distrprg 7617 . . . . . . . . . . . . . . . . . 18 (((𝑦 +P 𝑣) ∈ P𝑤P𝑢P) → ((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) = (((𝑦 +P 𝑣) ·P 𝑤) +P ((𝑦 +P 𝑣) ·P 𝑢)))
6429, 32, 40, 63syl3anc 1249 . . . . . . . . . . . . . . . . 17 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) = (((𝑦 +P 𝑣) ·P 𝑤) +P ((𝑦 +P 𝑣) ·P 𝑢)))
65 mulclpr 7601 . . . . . . . . . . . . . . . . . . 19 ((𝑦P𝑤P) → (𝑦 ·P 𝑤) ∈ P)
6627, 32, 65syl2anc 411 . . . . . . . . . . . . . . . . . 18 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑦 ·P 𝑤) ∈ P)
67 mulclpr 7601 . . . . . . . . . . . . . . . . . . 19 ((𝑦P𝑢P) → (𝑦 ·P 𝑢) ∈ P)
6827, 40, 67syl2anc 411 . . . . . . . . . . . . . . . . . 18 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑦 ·P 𝑢) ∈ P)
69 mulclpr 7601 . . . . . . . . . . . . . . . . . . 19 ((𝑣P𝑤P) → (𝑣 ·P 𝑤) ∈ P)
7028, 32, 69syl2anc 411 . . . . . . . . . . . . . . . . . 18 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑣 ·P 𝑤) ∈ P)
71 addcomprg 7607 . . . . . . . . . . . . . . . . . . 19 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
7271adantl 277 . . . . . . . . . . . . . . . . . 18 ((((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
73 addassprg 7608 . . . . . . . . . . . . . . . . . . 19 ((𝑓P𝑔PP) → ((𝑓 +P 𝑔) +P ) = (𝑓 +P (𝑔 +P )))
7473adantl 277 . . . . . . . . . . . . . . . . . 18 ((((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) ∧ (𝑓P𝑔PP)) → ((𝑓 +P 𝑔) +P ) = (𝑓 +P (𝑔 +P )))
7566, 68, 70, 72, 74, 42, 24caov4d 6081 . . . . . . . . . . . . . . . . 17 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (((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))))
7662, 64, 753eqtr4d 2232 . . . . . . . . . . . . . . . 16 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) = (((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢)) +P ((𝑣 ·P 𝑤) +P (𝑣 ·P 𝑢))))
7770, 38, 42, 72, 74caov12d 6078 . . . . . . . . . . . . . . . 16 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑣 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢))) = ((𝑦 ·P 𝑧) +P ((𝑣 ·P 𝑤) +P (𝑣 ·P 𝑢))))
7855, 76, 773eqtr4d 2232 . . . . . . . . . . . . . . 15 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) = ((𝑣 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢))))
79 oveq1 5903 . . . . . . . . . . . . . . . . . 18 ((𝑦 +P 𝑣) = 𝑥 → ((𝑦 +P 𝑣) ·P 𝑤) = (𝑥 ·P 𝑤))
8079adantl 277 . . . . . . . . . . . . . . . . 17 ((𝑣P ∧ (𝑦 +P 𝑣) = 𝑥) → ((𝑦 +P 𝑣) ·P 𝑤) = (𝑥 ·P 𝑤))
8180ad2antlr 489 . . . . . . . . . . . . . . . 16 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑦 +P 𝑣) ·P 𝑤) = (𝑥 ·P 𝑤))
8260, 81eqtr3d 2224 . . . . . . . . . . . . . . 15 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤)) = (𝑥 ·P 𝑤))
8378, 82oveq12d 5914 . . . . . . . . . . . . . 14 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (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)))
8448, 83eqtr3d 2224 . . . . . . . . . . . . 13 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (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)))
85 mulclpr 7601 . . . . . . . . . . . . . . . 16 ((𝑥P𝑧P) → (𝑥 ·P 𝑧) ∈ P)
8630, 36, 85syl2anc 411 . . . . . . . . . . . . . . 15 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑥 ·P 𝑧) ∈ P)
87 addassprg 7608 . . . . . . . . . . . . . . 15 (((𝑥 ·P 𝑧) ∈ P ∧ (𝑦 ·P 𝑤) ∈ P ∧ (𝑣 ·P 𝑤) ∈ P) → (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) +P (𝑣 ·P 𝑤)) = ((𝑥 ·P 𝑧) +P ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤))))
8886, 66, 70, 87syl3anc 1249 . . . . . . . . . . . . . 14 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) +P (𝑣 ·P 𝑤)) = ((𝑥 ·P 𝑧) +P ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤))))
89 addclpr 7566 . . . . . . . . . . . . . . . 16 (((𝑥 ·P 𝑧) ∈ P ∧ (𝑦 ·P 𝑤) ∈ P) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
9086, 66, 89syl2anc 411 . . . . . . . . . . . . . . 15 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
91 addcomprg 7607 . . . . . . . . . . . . . . 15 ((((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P ∧ (𝑣 ·P 𝑤) ∈ P) → (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) +P (𝑣 ·P 𝑤)) = ((𝑣 ·P 𝑤) +P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))
9290, 70, 91syl2anc 411 . . . . . . . . . . . . . 14 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) +P (𝑣 ·P 𝑤)) = ((𝑣 ·P 𝑤) +P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))
9388, 92eqtr3d 2224 . . . . . . . . . . . . 13 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑥 ·P 𝑧) +P ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤))) = ((𝑣 ·P 𝑤) +P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))
9424, 38, 42caovcld 6050 . . . . . . . . . . . . . . 15 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢)) ∈ P)
95 addassprg 7608 . . . . . . . . . . . . . . 15 (((𝑣 ·P 𝑤) ∈ P ∧ (𝑥 ·P 𝑤) ∈ P ∧ ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢)) ∈ P) → (((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)))))
9670, 34, 94, 95syl3anc 1249 . . . . . . . . . . . . . 14 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (((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)))))
9770, 94, 34, 72, 74caov32d 6077 . . . . . . . . . . . . . 14 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (((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))))
98 addassprg 7608 . . . . . . . . . . . . . . . 16 (((𝑥 ·P 𝑤) ∈ P ∧ (𝑦 ·P 𝑧) ∈ P ∧ (𝑣 ·P 𝑢) ∈ P) → (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢)) = ((𝑥 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢))))
9934, 38, 42, 98syl3anc 1249 . . . . . . . . . . . . . . 15 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢)) = ((𝑥 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢))))
10099oveq2d 5912 . . . . . . . . . . . . . 14 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((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)))))
10196, 97, 1003eqtr4d 2232 . . . . . . . . . . . . 13 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (((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))))
10284, 93, 1013eqtr3d 2230 . . . . . . . . . . . 12 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (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))))
10324, 39, 42caovcld 6050 . . . . . . . . . . . . 13 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢)) ∈ P)
104 addcanprg 7645 . . . . . . . . . . . . 13 (((𝑣 ·P 𝑤) ∈ P ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P ∧ (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢)) ∈ 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))))
10570, 90, 103, 104syl3anc 1249 . . . . . . . . . . . 12 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (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))) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) = (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢))))
106102, 105mpd 13 . . . . . . . . . . 11 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) = (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢)))
10744, 106breqtrrd 4046 . . . . . . . . . 10 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)))
108107rexlimdvaa 2608 . . . . . . . . 9 ((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) → (∃𝑢P (𝑤 +P 𝑢) = 𝑧 → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))
10922, 108syl5 32 . . . . . . . 8 ((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) → (𝑤<P 𝑧 → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))
110109rexlimdvaa 2608 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (∃𝑣P (𝑦 +P 𝑣) = 𝑥 → (𝑤<P 𝑧 → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)))))
11121, 110syl5 32 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (𝑦<P 𝑥 → (𝑤<P 𝑧 → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)))))
112111impd 254 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑦<P 𝑥𝑤<P 𝑧) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))
113 mulsrpr 7775 . . . . . . 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 )
114113breq2d 4030 . . . . . 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 ))
115 gt0srpr 7777 . . . . . 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)))
116114, 115bitrdi 196 . . . . 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))))
117112, 116sylibrd 169 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑦<P 𝑥𝑤<P 𝑧) → 0R <R ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R )))
11820, 117biimtrid 152 . . 3 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((0R <R [⟨𝑥, 𝑦⟩] ~R ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R ) → 0R <R ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R )))
1197, 12, 17, 1182ecoptocl 6649 . 2 ((𝐴R𝐵R) → ((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 ·R 𝐵)))
1206, 119mpcom 36 1 ((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 ·R 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 980   = wceq 1364  wcel 2160  wrex 2469  cop 3610   class class class wbr 4018  (class class class)co 5896  [cec 6557  Pcnp 7320   +P cpp 7322   ·P cmp 7323  <P cltp 7324   ~R cer 7325  Rcnr 7326  0Rc0r 7327   ·R cmr 7331   <R cltr 7332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-irdg 6395  df-1o 6441  df-2o 6442  df-oadd 6445  df-omul 6446  df-er 6559  df-ec 6561  df-qs 6565  df-ni 7333  df-pli 7334  df-mi 7335  df-lti 7336  df-plpq 7373  df-mpq 7374  df-enq 7376  df-nqqs 7377  df-plqqs 7378  df-mqqs 7379  df-1nqqs 7380  df-rq 7381  df-ltnqqs 7382  df-enq0 7453  df-nq0 7454  df-0nq0 7455  df-plq0 7456  df-mq0 7457  df-inp 7495  df-i1p 7496  df-iplp 7497  df-imp 7498  df-iltp 7499  df-enr 7755  df-nr 7756  df-mr 7758  df-ltr 7759  df-0r 7760
This theorem is referenced by:  axpre-mulgt0  7916
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