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Theorem xmullem 12650
Description: Lemma for rexmul 12657. (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 979 . . . 4 (¬ (𝐴 = 0 ∨ 𝐵 = 0) ↔ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0))
21anbi2i 622 . . 3 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)))
3 ioran 979 . . . . 5 (¬ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) ↔ (¬ ((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ ¬ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))))
4 ioran 979 . . . . . 6 (¬ ((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ↔ (¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)))
5 ioran 979 . . . . . 6 (¬ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) ↔ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞)))
64, 5anbi12i 626 . . . . 5 ((¬ ((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ ¬ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) ↔ ((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))))
73, 6bitri 276 . . . 4 (¬ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) ↔ ((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))))
8 ioran 979 . . . . 5 (¬ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) ↔ (¬ ((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ ¬ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))))
9 ioran 979 . . . . . 6 (¬ ((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ↔ (¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)))
10 ioran 979 . . . . . 6 (¬ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)) ↔ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))
119, 10anbi12i 626 . . . . 5 ((¬ ((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ ¬ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) ↔ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))
128, 11bitri 276 . . . 4 (¬ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) ↔ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))
137, 12anbi12i 626 . . 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 771 . . . . 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 12504 . . . . 5 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
1614, 15sylib 219 . . . 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 776 . . . . . . . . 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 482 . . . . . . . 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 121 . . . . . . 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 453 . . . . . 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 774 . . . . . . . 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 121 . . . . . . 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 407 . . . . . 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 771 . . . . . . . . 9 ((((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))) → ¬ (0 < 𝐵𝐴 = +∞))
2625adantl 482 . . . . . . . 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 121 . . . . . . 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 453 . . . . . 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 772 . . . . . . 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 10680 . . . . . . 7 0 ∈ ℝ*
31 xrltso 12527 . . . . . . . 8 < Or ℝ*
32 solin 5496 . . . . . . . 8 (( < Or ℝ* ∧ (𝐵 ∈ ℝ* ∧ 0 ∈ ℝ*)) → (𝐵 < 0 ∨ 𝐵 = 0 ∨ 0 < 𝐵))
3331, 32mpan 686 . . . . . . 7 ((𝐵 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝐵 < 0 ∨ 𝐵 = 0 ∨ 0 < 𝐵))
3429, 30, 33sylancl 586 . . . . . 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 1425 . . . . 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 772 . . . . . . . . 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 482 . . . . . . . 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 121 . . . . . . 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 453 . . . . . 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 121 . . . . . . 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 407 . . . . . 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 775 . . . . . . . . 9 ((((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))) → ¬ (0 < 𝐵𝐴 = -∞))
4342adantl 482 . . . . . . . 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 121 . . . . . . 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 453 . . . . . 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 1425 . . . . 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 1422 . . . 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 597 . 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 468 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 396  wo 843  w3o 1080   = wceq 1530  wcel 2107   class class class wbr 5062   Or wor 5471  cr 10528  0cc0 10529  +∞cpnf 10664  -∞cmnf 10665  *cxr 10666   < clt 10667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-addrcl 10590  ax-rnegex 10600  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-po 5472  df-so 5473  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672
This theorem is referenced by:  xmulcom  12652  xmulneg1  12655  xmulf  12658
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