Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  xmullem Structured version   Visualization version   GIF version

Theorem xmullem 11966
 Description: Lemma for rexmul 11973. (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 510 . . . 4 (¬ (𝐴 = 0 ∨ 𝐵 = 0) ↔ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0))
21anbi2i 726 . . 3 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)))
3 ioran 510 . . . . 5 (¬ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) ↔ (¬ ((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ ¬ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))))
4 ioran 510 . . . . . 6 (¬ ((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ↔ (¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)))
5 ioran 510 . . . . . 6 (¬ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) ↔ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞)))
64, 5anbi12i 729 . . . . 5 ((¬ ((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ ¬ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) ↔ ((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))))
73, 6bitri 263 . . . 4 (¬ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) ↔ ((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))))
8 ioran 510 . . . . 5 (¬ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) ↔ (¬ ((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ ¬ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))))
9 ioran 510 . . . . . 6 (¬ ((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ↔ (¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)))
10 ioran 510 . . . . . 6 (¬ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)) ↔ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))
119, 10anbi12i 729 . . . . 5 ((¬ ((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ ¬ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) ↔ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))
128, 11bitri 263 . . . 4 (¬ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) ↔ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))
137, 12anbi12i 729 . . 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 794 . . . . 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 11826 . . . . 5 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
1614, 15sylib 207 . . . 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 481 . . . . . . . 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 117 . . . . . . 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 452 . . . . . 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 117 . . . . . . 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 444 . . . . . 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 794 . . . . . . . . 9 ((((¬ (0 < 𝐵𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))) → ¬ (0 < 𝐵𝐴 = +∞))
2625adantl 481 . . . . . . . 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 117 . . . . . . 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 452 . . . . . 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 795 . . . . . . 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 9965 . . . . . . 7 0 ∈ ℝ*
31 xrltso 11850 . . . . . . . 8 < Or ℝ*
32 solin 4982 . . . . . . . 8 (( < Or ℝ* ∧ (𝐵 ∈ ℝ* ∧ 0 ∈ ℝ*)) → (𝐵 < 0 ∨ 𝐵 = 0 ∨ 0 < 𝐵))
3331, 32mpan 702 . . . . . . 7 ((𝐵 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝐵 < 0 ∨ 𝐵 = 0 ∨ 0 < 𝐵))
3429, 30, 33sylancl 693 . . . . . 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 1387 . . . . 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 795 . . . . . . . . 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 481 . . . . . . . 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 117 . . . . . . 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 452 . . . . . 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 117 . . . . . . 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 444 . . . . . 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 481 . . . . . . . 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 117 . . . . . . 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 452 . . . . . 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 1387 . . . . 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 1384 . . . 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 495 . 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 678 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   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   = wceq 1475   ∈ wcel 1977   class class class wbr 4583   Or wor 4958  ℝcr 9814  0cc0 9815  +∞cpnf 9950  -∞cmnf 9951  ℝ*cxr 9952   < clt 9953 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-i2m1 9883  ax-1ne0 9884  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958 This theorem is referenced by:  xmulcom  11968  xmulneg1  11971  xmulf  11974
 Copyright terms: Public domain W3C validator