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Theorem mulgt0sr 7752
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 7712 . . . . 5 <R ⊆ (R × R)
21brel 4672 . . . 4 (0R <R 𝐴 → (0RR𝐴R))
32simprd 114 . . 3 (0R <R 𝐴𝐴R)
41brel 4672 . . . 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 7701 . . 3 R = ((P × P) / ~R )
8 breq2 4002 . . . . 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 5872 . . . . 5 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) = (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ))
1110breq2d 4010 . . . 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 4002 . . . . 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 5873 . . . . 5 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ) = (𝐴 ·R 𝐵))
1615breq2d 4010 . . . 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 7722 . . . . 5 (0R <R [⟨𝑥, 𝑦⟩] ~R𝑦<P 𝑥)
19 gt0srpr 7722 . . . . 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 7587 . . . . . . 7 (𝑦<P 𝑥 → ∃𝑣P (𝑦 +P 𝑣) = 𝑥)
22 ltexpri 7587 . . . . . . . . 9 (𝑤<P 𝑧 → ∃𝑢P (𝑤 +P 𝑢) = 𝑧)
23 addclpr 7511 . . . . . . . . . . . . . 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 6018 . . . . . . . . . . . . . . 15 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑦 +P 𝑣) ∈ P)
3025, 29eqeltrrd 2253 . . . . . . . . . . . . . 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 7546 . . . . . . . . . . . . . 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 7546 . . . . . . . . . . . . . 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 6018 . . . . . . . . . . . 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 7546 . . . . . . . . . . . . 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 7571 . . . . . . . . . . . 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 5874 . . . . . . . . . . . . . . . 16 (((𝑦 +P 𝑣) = 𝑥 ∧ (𝑤 +P 𝑢) = 𝑧) → ((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) = (𝑥 ·P 𝑧))
4746oveq1d 5880 . . . . . . . . . . . . . . 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 7562 . . . . . . . . . . . . . . . . . . 19 ((𝑦P𝑤P𝑢P) → (𝑦 ·P (𝑤 +P 𝑢)) = ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢)))
5027, 32, 40, 49syl3anc 1238 . . . . . . . . . . . . . . . . . 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 5873 . . . . . . . . . . . . . . . . . . . 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 2210 . . . . . . . . . . . . . . . . 17 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢)) = (𝑦 ·P 𝑧))
5554oveq1d 5880 . . . . . . . . . . . . . . . 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 7562 . . . . . . . . . . . . . . . . . . . 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 7554 . . . . . . . . . . . . . . . . . . . 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 6041 . . . . . . . . . . . . . . . . . 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 6041 . . . . . . . . . . . . . . . . . 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 5883 . . . . . . . . . . . . . . . . 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 7562 . . . . . . . . . . . . . . . . . 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 1238 . . . . . . . . . . . . . . . . 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 7546 . . . . . . . . . . . . . . . . . . 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 7546 . . . . . . . . . . . . . . . . . . 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 7546 . . . . . . . . . . . . . . . . . . 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 7552 . . . . . . . . . . . . . . . . . . 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 7553 . . . . . . . . . . . . . . . . . . 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 6049 . . . . . . . . . . . . . . . . 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 2218 . . . . . . . . . . . . . . . 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 6046 . . . . . . . . . . . . . . . 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 2218 . . . . . . . . . . . . . . 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 5872 . . . . . . . . . . . . . . . . . 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 2210 . . . . . . . . . . . . . . 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 5883 . . . . . . . . . . . . . 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 2210 . . . . . . . . . . . . 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 7546 . . . . . . . . . . . . . . . 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 7553 . . . . . . . . . . . . . . 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 1238 . . . . . . . . . . . . . 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 7511 . . . . . . . . . . . . . . . 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 7552 . . . . . . . . . . . . . . 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 2210 . . . . . . . . . . . . 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 6018 . . . . . . . . . . . . . . 15 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢)) ∈ P)
95 addassprg 7553 . . . . . . . . . . . . . . 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 1238 . . . . . . . . . . . . . 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 6045 . . . . . . . . . . . . . 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 7553 . . . . . . . . . . . . . . . 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 1238 . . . . . . . . . . . . . . 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 5881 . . . . . . . . . . . . . 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 2218 . . . . . . . . . . . . 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 2216 . . . . . . . . . . . 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 6018 . . . . . . . . . . . . 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 7590 . . . . . . . . . . . . 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 1238 . . . . . . . . . . . 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 4026 . . . . . . . . . 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 2593 . . . . . . . . 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 2593 . . . . . . 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 7720 . . . . . . 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 4010 . . . . . 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 7722 . . . . . 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 6613 . 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 978   = wceq 1353  wcel 2146  wrex 2454  cop 3592   class class class wbr 3998  (class class class)co 5865  [cec 6523  Pcnp 7265   +P cpp 7267   ·P cmp 7268  <P cltp 7269   ~R cer 7270  Rcnr 7271  0Rc0r 7272   ·R cmr 7276   <R cltr 7277
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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-eprel 4283  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-1o 6407  df-2o 6408  df-oadd 6411  df-omul 6412  df-er 6525  df-ec 6527  df-qs 6531  df-ni 7278  df-pli 7279  df-mi 7280  df-lti 7281  df-plpq 7318  df-mpq 7319  df-enq 7321  df-nqqs 7322  df-plqqs 7323  df-mqqs 7324  df-1nqqs 7325  df-rq 7326  df-ltnqqs 7327  df-enq0 7398  df-nq0 7399  df-0nq0 7400  df-plq0 7401  df-mq0 7402  df-inp 7440  df-i1p 7441  df-iplp 7442  df-imp 7443  df-iltp 7444  df-enr 7700  df-nr 7701  df-mr 7703  df-ltr 7704  df-0r 7705
This theorem is referenced by:  axpre-mulgt0  7861
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