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Theorem xmullem 12645

Description: Lemma for rexmul 12652. (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 981	. . . 4 ⊢ (¬ (𝐴 = 0 ∨ 𝐵 = 0) ↔ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0))
2	1	anbi2i 625	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ↔ ((𝐴 ∈ ℝ^ ∧ 𝐵 ∈ ℝ^*) ∧ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0)))
3		ioran 981	. . . . 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 981	. . . . . 6 ⊢ (¬ ((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ↔ (¬ (0 < 𝐵 ∧ 𝐴 = +∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = -∞)))
5		ioran 981	. . . . . 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 278	. . . 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 981	. . . . 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 981	. . . . . 6 ⊢ (¬ ((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ↔ (¬ (0 < 𝐵 ∧ 𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)))
10		ioran 981	. . . . . 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 278	. . . 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 774	. . . . 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 12499	. . . . 5 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
16	14, 15	sylib 221	. . . 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 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 = +∞)))) → ¬ (𝐵 < 0 ∧ 𝐴 = +∞))
19	18	adantl 485	. . . . . . . 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 456	. . . . . 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 777	. . . . . . . 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 410	. . . . . 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 774	. . . . . . . . 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 485	. . . . . . . 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 456	. . . . . 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 775	. . . . . . 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 10677	. . . . . . 7 ⊢ 0 ∈ ℝ^*
31		xrltso 12522	. . . . . . . 8 ⊢ < Or ℝ^*
32		solin 5462	. . . . . . . 8 ⊢ (( < Or ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ∈ ℝ^*)) → (𝐵 < 0 ∨ 𝐵 = 0 ∨ 0 < 𝐵))
33	31, 32	mpan 689	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (𝐵 < 0 ∨ 𝐵 = 0 ∨ 0 < 𝐵))
34	29, 30, 33	sylancl 589	. . . . . 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 1428	. . . . 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 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 = +∞)))) → ¬ (𝐵 < 0 ∧ 𝐴 = -∞))
37	36	adantl 485	. . . . . . . 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 456	. . . . . 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 410	. . . . . 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 778	. . . . . . . . 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 485	. . . . . . . 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 456	. . . . . 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 1428	. . . . 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 1425	. . . 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 600	. 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 471	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 399 ∨ wo 844 ∨ w3o 1083 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 Or wor 5437 ℝcr 10525 0cc0 10526 +∞cpnf 10661 -∞cmnf 10662 ℝ^*cxr 10663 < clt 10664

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-addrcl 10587 ax-rnegex 10597 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669

This theorem is referenced by: xmulcom 12647 xmulneg1 12650 xmulf 12653

W3C validator