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Theorem mulgt0sr 7244
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 7205 . . . . 5 <R ⊆ (R × R)
21brel 4451 . . . 4 (0R <R 𝐴 → (0RR𝐴R))
32simprd 112 . . 3 (0R <R 𝐴𝐴R)
41brel 4451 . . . 4 (0R <R 𝐵 → (0RR𝐵R))
54simprd 112 . . 3 (0R <R 𝐵𝐵R)
63, 5anim12i 331 . 2 ((0R <R 𝐴 ∧ 0R <R 𝐵) → (𝐴R𝐵R))
7 df-nr 7194 . . 3 R = ((P × P) / ~R )
8 breq2 3818 . . . . 5 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (0R <R [⟨𝑥, 𝑦⟩] ~R ↔ 0R <R 𝐴))
98anbi1d 453 . . . 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 5601 . . . . 5 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) = (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ))
1110breq2d 3826 . . . 4 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (0R <R ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) ↔ 0R <R (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R )))
129, 11imbi12d 232 . . 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 3818 . . . . 5 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (0R <R [⟨𝑧, 𝑤⟩] ~R ↔ 0R <R 𝐵))
1413anbi2d 452 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ((0R <R 𝐴 ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R ) ↔ (0R <R 𝐴 ∧ 0R <R 𝐵)))
15 oveq2 5602 . . . . 5 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ) = (𝐴 ·R 𝐵))
1615breq2d 3826 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (0R <R (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ) ↔ 0R <R (𝐴 ·R 𝐵)))
1714, 16imbi12d 232 . . 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 7215 . . . . 5 (0R <R [⟨𝑥, 𝑦⟩] ~R𝑦<P 𝑥)
19 gt0srpr 7215 . . . . 5 (0R <R [⟨𝑧, 𝑤⟩] ~R𝑤<P 𝑧)
2018, 19anbi12i 448 . . . 4 ((0R <R [⟨𝑥, 𝑦⟩] ~R ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R ) ↔ (𝑦<P 𝑥𝑤<P 𝑧))
21 ltexpri 7093 . . . . . . 7 (𝑦<P 𝑥 → ∃𝑣P (𝑦 +P 𝑣) = 𝑥)
22 ltexpri 7093 . . . . . . . . 9 (𝑤<P 𝑧 → ∃𝑢P (𝑤 +P 𝑢) = 𝑧)
23 addclpr 7017 . . . . . . . . . . . . . 14 ((𝑓P𝑔P) → (𝑓 +P 𝑔) ∈ P)
2423adantl 271 . . . . . . . . . . . . 13 ((((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) ∈ P)
25 simplrr 503 . . . . . . . . . . . . . . 15 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑦 +P 𝑣) = 𝑥)
26 simplr 497 . . . . . . . . . . . . . . . . 17 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → 𝑦P)
2726ad2antrr 472 . . . . . . . . . . . . . . . 16 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → 𝑦P)
28 simplrl 502 . . . . . . . . . . . . . . . 16 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → 𝑣P)
2924, 27, 28caovcld 5736 . . . . . . . . . . . . . . 15 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑦 +P 𝑣) ∈ P)
3025, 29eqeltrrd 2162 . . . . . . . . . . . . . 14 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → 𝑥P)
31 simplrr 503 . . . . . . . . . . . . . . 15 ((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) → 𝑤P)
3231adantr 270 . . . . . . . . . . . . . 14 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → 𝑤P)
33 mulclpr 7052 . . . . . . . . . . . . . 14 ((𝑥P𝑤P) → (𝑥 ·P 𝑤) ∈ P)
3430, 32, 33syl2anc 403 . . . . . . . . . . . . 13 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑥 ·P 𝑤) ∈ P)
35 simplrl 502 . . . . . . . . . . . . . . 15 ((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) → 𝑧P)
3635adantr 270 . . . . . . . . . . . . . 14 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → 𝑧P)
37 mulclpr 7052 . . . . . . . . . . . . . 14 ((𝑦P𝑧P) → (𝑦 ·P 𝑧) ∈ P)
3827, 36, 37syl2anc 403 . . . . . . . . . . . . 13 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑦 ·P 𝑧) ∈ P)
3924, 34, 38caovcld 5736 . . . . . . . . . . . 12 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
40 simprl 498 . . . . . . . . . . . . 13 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → 𝑢P)
41 mulclpr 7052 . . . . . . . . . . . . 13 ((𝑣P𝑢P) → (𝑣 ·P 𝑢) ∈ P)
4228, 40, 41syl2anc 403 . . . . . . . . . . . 12 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑣 ·P 𝑢) ∈ P)
43 ltaddpr 7077 . . . . . . . . . . . 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 403 . . . . . . . . . . 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 499 . . . . . . . . . . . . . . 15 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑤 +P 𝑢) = 𝑧)
46 oveq12 5603 . . . . . . . . . . . . . . . 16 (((𝑦 +P 𝑣) = 𝑥 ∧ (𝑤 +P 𝑢) = 𝑧) → ((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) = (𝑥 ·P 𝑧))
4746oveq1d 5609 . . . . . . . . . . . . . . 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 403 . . . . . . . . . . . . . 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 7068 . . . . . . . . . . . . . . . . . . 19 ((𝑦P𝑤P𝑢P) → (𝑦 ·P (𝑤 +P 𝑢)) = ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢)))
5027, 32, 40, 49syl3anc 1172 . . . . . . . . . . . . . . . . . 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 5602 . . . . . . . . . . . . . . . . . . . 20 ((𝑤 +P 𝑢) = 𝑧 → (𝑦 ·P (𝑤 +P 𝑢)) = (𝑦 ·P 𝑧))
5251adantl 271 . . . . . . . . . . . . . . . . . . 19 ((𝑢P ∧ (𝑤 +P 𝑢) = 𝑧) → (𝑦 ·P (𝑤 +P 𝑢)) = (𝑦 ·P 𝑧))
5352adantl 271 . . . . . . . . . . . . . . . . . 18 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑦 ·P (𝑤 +P 𝑢)) = (𝑦 ·P 𝑧))
5450, 53eqtr3d 2119 . . . . . . . . . . . . . . . . 17 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢)) = (𝑦 ·P 𝑧))
5554oveq1d 5609 . . . . . . . . . . . . . . . 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 7068 . . . . . . . . . . . . . . . . . . . 20 ((𝑓P𝑔PP) → (𝑓 ·P (𝑔 +P )) = ((𝑓 ·P 𝑔) +P (𝑓 ·P )))
5756adantl 271 . . . . . . . . . . . . . . . . . . 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 7060 . . . . . . . . . . . . . . . . . . . 20 ((𝑓P𝑔P) → (𝑓 ·P 𝑔) = (𝑔 ·P 𝑓))
5958adantl 271 . . . . . . . . . . . . . . . . . . 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 5759 . . . . . . . . . . . . . . . . . 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 5759 . . . . . . . . . . . . . . . . . 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 5612 . . . . . . . . . . . . . . . . 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 7068 . . . . . . . . . . . . . . . . . 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 1172 . . . . . . . . . . . . . . . . 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 7052 . . . . . . . . . . . . . . . . . . 19 ((𝑦P𝑤P) → (𝑦 ·P 𝑤) ∈ P)
6627, 32, 65syl2anc 403 . . . . . . . . . . . . . . . . . 18 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑦 ·P 𝑤) ∈ P)
67 mulclpr 7052 . . . . . . . . . . . . . . . . . . 19 ((𝑦P𝑢P) → (𝑦 ·P 𝑢) ∈ P)
6827, 40, 67syl2anc 403 . . . . . . . . . . . . . . . . . 18 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑦 ·P 𝑢) ∈ P)
69 mulclpr 7052 . . . . . . . . . . . . . . . . . . 19 ((𝑣P𝑤P) → (𝑣 ·P 𝑤) ∈ P)
7028, 32, 69syl2anc 403 . . . . . . . . . . . . . . . . . 18 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑣 ·P 𝑤) ∈ P)
71 addcomprg 7058 . . . . . . . . . . . . . . . . . . 19 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
7271adantl 271 . . . . . . . . . . . . . . . . . 18 ((((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
73 addassprg 7059 . . . . . . . . . . . . . . . . . . 19 ((𝑓P𝑔PP) → ((𝑓 +P 𝑔) +P ) = (𝑓 +P (𝑔 +P )))
7473adantl 271 . . . . . . . . . . . . . . . . . 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 5767 . . . . . . . . . . . . . . . . 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 2127 . . . . . . . . . . . . . . . 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 5764 . . . . . . . . . . . . . . . 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 2127 . . . . . . . . . . . . . . 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 5601 . . . . . . . . . . . . . . . . . 18 ((𝑦 +P 𝑣) = 𝑥 → ((𝑦 +P 𝑣) ·P 𝑤) = (𝑥 ·P 𝑤))
8079adantl 271 . . . . . . . . . . . . . . . . 17 ((𝑣P ∧ (𝑦 +P 𝑣) = 𝑥) → ((𝑦 +P 𝑣) ·P 𝑤) = (𝑥 ·P 𝑤))
8180ad2antlr 473 . . . . . . . . . . . . . . . 16 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑦 +P 𝑣) ·P 𝑤) = (𝑥 ·P 𝑤))
8260, 81eqtr3d 2119 . . . . . . . . . . . . . . 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 5612 . . . . . . . . . . . . . 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 2119 . . . . . . . . . . . . 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 7052 . . . . . . . . . . . . . . . 16 ((𝑥P𝑧P) → (𝑥 ·P 𝑧) ∈ P)
8630, 36, 85syl2anc 403 . . . . . . . . . . . . . . 15 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑥 ·P 𝑧) ∈ P)
87 addassprg 7059 . . . . . . . . . . . . . . 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 1172 . . . . . . . . . . . . . 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 7017 . . . . . . . . . . . . . . . 16 (((𝑥 ·P 𝑧) ∈ P ∧ (𝑦 ·P 𝑤) ∈ P) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
9086, 66, 89syl2anc 403 . . . . . . . . . . . . . . 15 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
91 addcomprg 7058 . . . . . . . . . . . . . . 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 403 . . . . . . . . . . . . . 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 2119 . . . . . . . . . . . . 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 5736 . . . . . . . . . . . . . . 15 (((((𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑣P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢)) ∈ P)
95 addassprg 7059 . . . . . . . . . . . . . . 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 1172 . . . . . . . . . . . . . 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 5763 . . . . . . . . . . . . . 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 7059 . . . . . . . . . . . . . . . 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 1172 . . . . . . . . . . . . . . 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 5610 . . . . . . . . . . . . . 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 2127 . . . . . . . . . . . . 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 2125 . . . . . . . . . . . 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 5736 . . . . . . . . . . . . 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 7096 . . . . . . . . . . . . 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 1172 . . . . . . . . . . . 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 3840 . . . . . . . . . 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 2486 . . . . . . . . 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 2486 . . . . . . 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 251 . . . . 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 7213 . . . . . . 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 3826 . . . . . 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 7215 . . . . . 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, 115syl6bb 194 . . . . 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 167 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑦<P 𝑥𝑤<P 𝑧) → 0R <R ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R )))
11820, 117syl5bi 150 . . 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 6313 . 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 102  w3a 922   = wceq 1287  wcel 1436  wrex 2356  cop 3428   class class class wbr 3814  (class class class)co 5594  [cec 6223  Pcnp 6771   +P cpp 6773   ·P cmp 6774  <P cltp 6775   ~R cer 6776  Rcnr 6777  0Rc0r 6778   ·R cmr 6782   <R cltr 6783
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-nul 3933  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-iinf 4369
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-tr 3905  df-eprel 4083  df-id 4087  df-po 4090  df-iso 4091  df-iord 4160  df-on 4162  df-suc 4165  df-iom 4372  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-1st 5849  df-2nd 5850  df-recs 6005  df-irdg 6070  df-1o 6116  df-2o 6117  df-oadd 6120  df-omul 6121  df-er 6225  df-ec 6227  df-qs 6231  df-ni 6784  df-pli 6785  df-mi 6786  df-lti 6787  df-plpq 6824  df-mpq 6825  df-enq 6827  df-nqqs 6828  df-plqqs 6829  df-mqqs 6830  df-1nqqs 6831  df-rq 6832  df-ltnqqs 6833  df-enq0 6904  df-nq0 6905  df-0nq0 6906  df-plq0 6907  df-mq0 6908  df-inp 6946  df-i1p 6947  df-iplp 6948  df-imp 6949  df-iltp 6950  df-enr 7193  df-nr 7194  df-mr 7196  df-ltr 7197  df-0r 7198
This theorem is referenced by:  axpre-mulgt0  7343
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