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Theorem xmullem 12343
Description: Lemma for rexmul 12350. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xmullem (((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ ¬ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) ∧ ¬ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) → 𝐴 ∈ ℝ)

Proof of Theorem xmullem
StepHypRef Expression
1 ioran 1007 . . . 4 (¬ (𝐴 = 0 ∨ 𝐵 = 0) ↔ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0))
21anbi2i 617 . . 3 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)))
3 ioran 1007 . . . . 5 (¬ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) ↔ (¬ ((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ ¬ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))))
4 ioran 1007 . . . . . 6 (¬ ((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ↔ (¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)))
5 ioran 1007 . . . . . 6 (¬ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) ↔ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞)))
64, 5anbi12i 621 . . . . 5 ((¬ ((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ ¬ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) ↔ ((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))))
73, 6bitri 267 . . . 4 (¬ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) ↔ ((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))))
8 ioran 1007 . . . . 5 (¬ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) ↔ (¬ ((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ ¬ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))))
9 ioran 1007 . . . . . 6 (¬ ((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ↔ (¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)))
10 ioran 1007 . . . . . 6 (¬ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)) ↔ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))
119, 10anbi12i 621 . . . . 5 ((¬ ((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ ¬ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) ↔ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))
128, 11bitri 267 . . . 4 (¬ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) ↔ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))
137, 12anbi12i 621 . . 3 ((¬ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ¬ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) ↔ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))))
14 simplll 792 . . . . 5 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → 𝐴 ∈ ℝ*)
15 elxr 12197 . . . . 5 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
1614, 15sylib 210 . . . 4 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
17 idd 24 . . . . 5 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → (𝐴 ∈ ℝ → 𝐴 ∈ ℝ))
18 simprlr 799 . . . . . . . . 9 ((((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))) → ¬ (𝐵 < 0 ∧ 𝐴 = +∞))
1918adantl 474 . . . . . . . 8 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → ¬ (𝐵 < 0 ∧ 𝐴 = +∞))
2019pm2.21d 119 . . . . . . 7 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → ((𝐵 < 0 ∧ 𝐴 = +∞) → 𝐴 ∈ ℝ))
2120expdimp 445 . . . . . 6 (((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) ∧ 𝐵 < 0) → (𝐴 = +∞ → 𝐴 ∈ ℝ))
22 simplrr 797 . . . . . . . 8 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → ¬ 𝐵 = 0)
2322pm2.21d 119 . . . . . . 7 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → (𝐵 = 0 → (𝐴 = +∞ → 𝐴 ∈ ℝ)))
2423imp 396 . . . . . 6 (((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) ∧ 𝐵 = 0) → (𝐴 = +∞ → 𝐴 ∈ ℝ))
25 simplll 792 . . . . . . . . 9 ((((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))) → ¬ (0 < 𝐵𝐴 = +∞))
2625adantl 474 . . . . . . . 8 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → ¬ (0 < 𝐵𝐴 = +∞))
2726pm2.21d 119 . . . . . . 7 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → ((0 < 𝐵𝐴 = +∞) → 𝐴 ∈ ℝ))
2827expdimp 445 . . . . . 6 (((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) ∧ 0 < 𝐵) → (𝐴 = +∞ → 𝐴 ∈ ℝ))
29 simpllr 794 . . . . . . 7 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → 𝐵 ∈ ℝ*)
30 0xr 10375 . . . . . . 7 0 ∈ ℝ*
31 xrltso 12221 . . . . . . . 8 < Or ℝ*
32 solin 5256 . . . . . . . 8 (( < Or ℝ* ∧ (𝐵 ∈ ℝ* ∧ 0 ∈ ℝ*)) → (𝐵 < 0 ∨ 𝐵 = 0 ∨ 0 < 𝐵))
3331, 32mpan 682 . . . . . . 7 ((𝐵 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝐵 < 0 ∨ 𝐵 = 0 ∨ 0 < 𝐵))
3429, 30, 33sylancl 581 . . . . . 6 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → (𝐵 < 0 ∨ 𝐵 = 0 ∨ 0 < 𝐵))
3521, 24, 28, 34mpjao3dan 1557 . . . . 5 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → (𝐴 = +∞ → 𝐴 ∈ ℝ))
36 simpllr 794 . . . . . . . . 9 ((((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))) → ¬ (𝐵 < 0 ∧ 𝐴 = -∞))
3736adantl 474 . . . . . . . 8 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → ¬ (𝐵 < 0 ∧ 𝐴 = -∞))
3837pm2.21d 119 . . . . . . 7 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → ((𝐵 < 0 ∧ 𝐴 = -∞) → 𝐴 ∈ ℝ))
3938expdimp 445 . . . . . 6 (((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) ∧ 𝐵 < 0) → (𝐴 = -∞ → 𝐴 ∈ ℝ))
4022pm2.21d 119 . . . . . . 7 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → (𝐵 = 0 → (𝐴 = -∞ → 𝐴 ∈ ℝ)))
4140imp 396 . . . . . 6 (((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) ∧ 𝐵 = 0) → (𝐴 = -∞ → 𝐴 ∈ ℝ))
42 simprll 798 . . . . . . . . 9 ((((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))) → ¬ (0 < 𝐵𝐴 = -∞))
4342adantl 474 . . . . . . . 8 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → ¬ (0 < 𝐵𝐴 = -∞))
4443pm2.21d 119 . . . . . . 7 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → ((0 < 𝐵𝐴 = -∞) → 𝐴 ∈ ℝ))
4544expdimp 445 . . . . . 6 (((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) ∧ 0 < 𝐵) → (𝐴 = -∞ → 𝐴 ∈ ℝ))
4639, 41, 45, 34mpjao3dan 1557 . . . . 5 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → (𝐴 = -∞ → 𝐴 ∈ ℝ))
4717, 35, 463jaod 1554 . . . 4 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → 𝐴 ∈ ℝ))
4816, 47mpd 15 . . 3 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → 𝐴 ∈ ℝ)
492, 13, 48syl2anb 592 . 2 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (¬ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ¬ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))))) → 𝐴 ∈ ℝ)
5049anassrs 460 1 (((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ ¬ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) ∧ ¬ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) → 𝐴 ∈ ℝ)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385  wo 874  w3o 1107   = wceq 1653  wcel 2157   class class class wbr 4843   Or wor 5232  cr 10223  0cc0 10224  +∞cpnf 10360  -∞cmnf 10361  *cxr 10362   < clt 10363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-addrcl 10285  ax-rnegex 10295  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-po 5233  df-so 5234  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368
This theorem is referenced by:  xmulcom  12345  xmulneg1  12348  xmulf  12351
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