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Theorem xmullem 13191

Description: Lemma for rexmul 13198. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xmullem	⊢ (((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ ¬ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) ∧ ¬ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) → 𝐴 ∈ ℝ)

**Proof of Theorem xmullem**
Step	Hyp	Ref	Expression
1		ioran 986	. . . 4 ⊢ (¬ (𝐴 = 0 ∨ 𝐵 = 0) ↔ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0))
2	1	anbi2i 624	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ↔ ((𝐴 ∈ ℝ^ ∧ 𝐵 ∈ ℝ^*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)))
3		ioran 986	. . . . 5 ⊢ (¬ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) ↔ (¬ ((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ ¬ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))))
4		ioran 986	. . . . . 6 ⊢ (¬ ((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ↔ (¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)))
5		ioran 986	. . . . . 6 ⊢ (¬ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) ↔ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞)))
6	4, 5	anbi12i 629	. . . . 5 ⊢ ((¬ ((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ ¬ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) ↔ ((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))))
7	3, 6	bitri 275	. . . 4 ⊢ (¬ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) ↔ ((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))))
8		ioran 986	. . . . 5 ⊢ (¬ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) ↔ (¬ ((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ ¬ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))))
9		ioran 986	. . . . . 6 ⊢ (¬ ((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ↔ (¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)))
10		ioran 986	. . . . . 6 ⊢ (¬ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)) ↔ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))
11	9, 10	anbi12i 629	. . . . 5 ⊢ ((¬ ((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ ¬ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) ↔ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))
12	8, 11	bitri 275	. . . 4 ⊢ (¬ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) ↔ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))
13	7, 12	anbi12i 629	. . 3 ⊢ ((¬ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ¬ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) ↔ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))))
14		simplll 775	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → 𝐴 ∈ ℝ^)
15		elxr 13042	. . . . 5 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
16	14, 15	sylib 218	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
17		idd 24	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → (𝐴 ∈ ℝ → 𝐴 ∈ ℝ))
18		simprlr 780	. . . . . . . . 9 ⊢ ((((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))) → ¬ (𝐵 < 0 ∧ 𝐴 = +∞))
19	18	adantl 481	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → ¬ (𝐵 < 0 ∧ 𝐴 = +∞))
20	19	pm2.21d 121	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → ((𝐵 < 0 ∧ 𝐴 = +∞) → 𝐴 ∈ ℝ))
21	20	expdimp 452	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) ∧ 𝐵 < 0) → (𝐴 = +∞ → 𝐴 ∈ ℝ))
22		simplrr 778	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → ¬ 𝐵 = 0)
23	22	pm2.21d 121	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → (𝐵 = 0 → (𝐴 = +∞ → 𝐴 ∈ ℝ)))
24	23	imp 406	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) ∧ 𝐵 = 0) → (𝐴 = +∞ → 𝐴 ∈ ℝ))
25		simplll 775	. . . . . . . . 9 ⊢ ((((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))) → ¬ (0 < 𝐵 ∧ 𝐴 = +∞))
26	25	adantl 481	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → ¬ (0 < 𝐵 ∧ 𝐴 = +∞))
27	26	pm2.21d 121	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → ((0 < 𝐵 ∧ 𝐴 = +∞) → 𝐴 ∈ ℝ))
28	27	expdimp 452	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) ∧ 0 < 𝐵) → (𝐴 = +∞ → 𝐴 ∈ ℝ))
29		simpllr 776	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → 𝐵 ∈ ℝ^)
30		0xr 11191	. . . . . . 7 ⊢ 0 ∈ ℝ^*
31		xrltso 13067	. . . . . . . 8 ⊢ < Or ℝ^*
32		solin 5567	. . . . . . . 8 ⊢ (( < Or ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ∈ ℝ^*)) → (𝐵 < 0 ∨ 𝐵 = 0 ∨ 0 < 𝐵))
33	31, 32	mpan 691	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (𝐵 < 0 ∨ 𝐵 = 0 ∨ 0 < 𝐵))
34	29, 30, 33	sylancl 587	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → (𝐵 < 0 ∨ 𝐵 = 0 ∨ 0 < 𝐵))
35	21, 24, 28, 34	mpjao3dan 1435	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → (𝐴 = +∞ → 𝐴 ∈ ℝ))
36		simpllr 776	. . . . . . . . 9 ⊢ ((((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))) → ¬ (𝐵 < 0 ∧ 𝐴 = -∞))
37	36	adantl 481	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → ¬ (𝐵 < 0 ∧ 𝐴 = -∞))
38	37	pm2.21d 121	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → ((𝐵 < 0 ∧ 𝐴 = -∞) → 𝐴 ∈ ℝ))
39	38	expdimp 452	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) ∧ 𝐵 < 0) → (𝐴 = -∞ → 𝐴 ∈ ℝ))
40	22	pm2.21d 121	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → (𝐵 = 0 → (𝐴 = -∞ → 𝐴 ∈ ℝ)))
41	40	imp 406	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) ∧ 𝐵 = 0) → (𝐴 = -∞ → 𝐴 ∈ ℝ))
42		simprll 779	. . . . . . . . 9 ⊢ ((((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))) → ¬ (0 < 𝐵 ∧ 𝐴 = -∞))
43	42	adantl 481	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → ¬ (0 < 𝐵 ∧ 𝐴 = -∞))
44	43	pm2.21d 121	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → ((0 < 𝐵 ∧ 𝐴 = -∞) → 𝐴 ∈ ℝ))
45	44	expdimp 452	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) ∧ 0 < 𝐵) → (𝐴 = -∞ → 𝐴 ∈ ℝ))
46	39, 41, 45, 34	mpjao3dan 1435	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → (𝐴 = -∞ → 𝐴 ∈ ℝ))
47	17, 35, 46	3jaod 1432	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → 𝐴 ∈ ℝ))
48	16, 47	mpd 15	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)) ∧ (((¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = +∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ((¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ (¬ (0 < 𝐴 ∧ 𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))) → 𝐴 ∈ ℝ)
49	2, 13, 48	syl2anb 599	. 2 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (¬ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) ∧ ¬ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))))) → 𝐴 ∈ ℝ)
50	49	anassrs 467	1 ⊢ (((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ ¬ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) ∧ ¬ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) → 𝐴 ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 848 ∨ w3o 1086 = wceq 1542 ∈ wcel 2114 class class class wbr 5100 Or wor 5539 ℝcr 11037 0cc0 11038 +∞cpnf 11175 -∞cmnf 11176 ℝ^*cxr 11177 < clt 11178

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-addrcl 11099 ax-rnegex 11109 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-po 5540 df-so 5541 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183

This theorem is referenced by: xmulcom 13193 xmulneg1 13196 xmulf 13199

W3C validator