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Theorem xmulneg1 13290

Description: Extended real version of mulneg1 11678. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xmulneg1	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 ·_e 𝐵) = -_𝑒(𝐴 ·_e 𝐵))

**Proof of Theorem xmulneg1**
Step	Hyp	Ref	Expression
1		xneg0 13233	. . . . . . . . 9 ⊢ -_𝑒0 = 0
2	1	eqeq2i 2749	. . . . . . . 8 ⊢ (-_𝑒𝐴 = -_𝑒0 ↔ -_𝑒𝐴 = 0)
3		0xr 11287	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
4		xneg11 13236	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (-_𝑒𝐴 = -_𝑒0 ↔ 𝐴 = 0))
5	3, 4	mpan2 691	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ^* → (-_𝑒𝐴 = -_𝑒0 ↔ 𝐴 = 0))
6	2, 5	bitr3id 285	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → (-_𝑒𝐴 = 0 ↔ 𝐴 = 0))
7	6	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 = 0 ↔ 𝐴 = 0))
8	7	orbi1d 916	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((-_𝑒𝐴 = 0 ∨ 𝐵 = 0) ↔ (𝐴 = 0 ∨ 𝐵 = 0)))
9	8	ifbid 4529	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → if((-_𝑒𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ -_𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = -∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = +∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -_𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = +∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-_𝑒𝐴 · 𝐵)))) = if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ -_𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = -∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = +∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -_𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = +∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-_𝑒𝐴 · 𝐵)))))
10		xnegpnf 13230	. . . . . . . . . . . . . 14 ⊢ -_𝑒+∞ = -∞
11	10	eqeq2i 2749	. . . . . . . . . . . . 13 ⊢ (-_𝑒𝐴 = -_𝑒+∞ ↔ -_𝑒𝐴 = -∞)
12		simpll 766	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → 𝐴 ∈ ℝ^)
13		pnfxr 11294	. . . . . . . . . . . . . 14 ⊢ +∞ ∈ ℝ^*
14		xneg11 13236	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (-_𝑒𝐴 = -_𝑒+∞ ↔ 𝐴 = +∞))
15	12, 13, 14	sylancl 586	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (-_𝑒𝐴 = -_𝑒+∞ ↔ 𝐴 = +∞))
16	11, 15	bitr3id 285	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (-_𝑒𝐴 = -∞ ↔ 𝐴 = +∞))
17	16	anbi2d 630	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → ((0 < 𝐵 ∧ -_𝑒𝐴 = -∞) ↔ (0 < 𝐵 ∧ 𝐴 = +∞)))
18		xnegmnf 13231	. . . . . . . . . . . . . 14 ⊢ -_𝑒-∞ = +∞
19	18	eqeq2i 2749	. . . . . . . . . . . . 13 ⊢ (-_𝑒𝐴 = -_𝑒-∞ ↔ -_𝑒𝐴 = +∞)
20		mnfxr 11297	. . . . . . . . . . . . . 14 ⊢ -∞ ∈ ℝ^*
21		xneg11 13236	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ^* ∧ -∞ ∈ ℝ^*) → (-_𝑒𝐴 = -_𝑒-∞ ↔ 𝐴 = -∞))
22	12, 20, 21	sylancl 586	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (-_𝑒𝐴 = -_𝑒-∞ ↔ 𝐴 = -∞))
23	19, 22	bitr3id 285	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (-_𝑒𝐴 = +∞ ↔ 𝐴 = -∞))
24	23	anbi2d 630	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → ((𝐵 < 0 ∧ -_𝑒𝐴 = +∞) ↔ (𝐵 < 0 ∧ 𝐴 = -∞)))
25	17, 24	orbi12d 918	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (((0 < 𝐵 ∧ -_𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = +∞)) ↔ ((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞))))
26		xlt0neg1 13240	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ^* → (𝐴 < 0 ↔ 0 < -_𝑒𝐴))
27	26	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (𝐴 < 0 ↔ 0 < -_𝑒𝐴))
28	27	bicomd 223	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (0 < -_𝑒𝐴 ↔ 𝐴 < 0))
29	28	anbi1d 631	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → ((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ↔ (𝐴 < 0 ∧ 𝐵 = -∞)))
30		xlt0neg2 13241	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ^* → (0 < 𝐴 ↔ -_𝑒𝐴 < 0))
31	30	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (0 < 𝐴 ↔ -_𝑒𝐴 < 0))
32	31	bicomd 223	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (-_𝑒𝐴 < 0 ↔ 0 < 𝐴))
33	32	anbi1d 631	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → ((-_𝑒𝐴 < 0 ∧ 𝐵 = +∞) ↔ (0 < 𝐴 ∧ 𝐵 = +∞)))
34	29, 33	orbi12d 918	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞)) ↔ ((𝐴 < 0 ∧ 𝐵 = -∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞))))
35		orcom 870	. . . . . . . . . . 11 ⊢ (((𝐴 < 0 ∧ 𝐵 = -∞) ∨ (0 < 𝐴 ∧ 𝐵 = +∞)) ↔ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))
36	34, 35	bitrdi 287	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞)) ↔ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))))
37	25, 36	orbi12d 918	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → ((((0 < 𝐵 ∧ -_𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = +∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞))) ↔ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))))
38	37	biimpar 477	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) → (((0 < 𝐵 ∧ -_𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = +∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞))))
39	38	iftrued 4513	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) → if((((0 < 𝐵 ∧ -_𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = +∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-_𝑒𝐴 · 𝐵)) = -∞)
40		xmullem2 13286	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))))
41	40	adantr 480	. . . . . . . . . 10 ⊢ (((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 = +∞)))))
42	23	anbi2d 630	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → ((0 < 𝐵 ∧ -_𝑒𝐴 = +∞) ↔ (0 < 𝐵 ∧ 𝐴 = -∞)))
43	16	anbi2d 630	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → ((𝐵 < 0 ∧ -_𝑒𝐴 = -∞) ↔ (𝐵 < 0 ∧ 𝐴 = +∞)))
44	42, 43	orbi12d 918	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (((0 < 𝐵 ∧ -_𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = -∞)) ↔ ((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞))))
45	28	anbi1d 631	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → ((0 < -_𝑒𝐴 ∧ 𝐵 = +∞) ↔ (𝐴 < 0 ∧ 𝐵 = +∞)))
46	32	anbi1d 631	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → ((-_𝑒𝐴 < 0 ∧ 𝐵 = -∞) ↔ (0 < 𝐴 ∧ 𝐵 = -∞)))
47	45, 46	orbi12d 918	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (((0 < -_𝑒𝐴 ∧ 𝐵 = +∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = -∞)) ↔ ((𝐴 < 0 ∧ 𝐵 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = -∞))))
48		orcom 870	. . . . . . . . . . . . 13 ⊢ (((𝐴 < 0 ∧ 𝐵 = +∞) ∨ (0 < 𝐴 ∧ 𝐵 = -∞)) ↔ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))
49	47, 48	bitrdi 287	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (((0 < -_𝑒𝐴 ∧ 𝐵 = +∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = -∞)) ↔ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))))
50	44, 49	orbi12d 918	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → ((((0 < 𝐵 ∧ -_𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = -∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = +∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = -∞))) ↔ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))))
51	50	notbid 318	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (¬ (((0 < 𝐵 ∧ -_𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = -∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = +∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = -∞))) ↔ ¬ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))))
52	41, 51	sylibrd 259	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (((0 < 𝐵 ∧ -_𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = -∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = +∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = -∞)))))
53	52	imp 406	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) → ¬ (((0 < 𝐵 ∧ -_𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = -∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = +∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = -∞))))
54	53	iffalsed 4516	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) → if((((0 < 𝐵 ∧ -_𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = -∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = +∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -_𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = +∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-_𝑒𝐴 · 𝐵))) = if((((0 < 𝐵 ∧ -_𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = +∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-_𝑒𝐴 · 𝐵)))
55		iftrue 4511	. . . . . . . . . 10 ⊢ ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = +∞)
56	55	adantl 481	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) → if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = +∞)
57		xnegeq 13228	. . . . . . . . 9 ⊢ (if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = +∞ → -_𝑒if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = -_𝑒+∞)
58	56, 57	syl 17	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) → -_𝑒if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = -_𝑒+∞)
59	58, 10	eqtrdi 2787	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) → -_𝑒if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = -∞)
60	39, 54, 59	3eqtr4d 2781	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) → if((((0 < 𝐵 ∧ -_𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = -∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = +∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -_𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = +∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-_𝑒𝐴 · 𝐵))) = -_𝑒if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))))
61	50	biimpar 477	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) → (((0 < 𝐵 ∧ -_𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = -∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = +∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = -∞))))
62	61	iftrued 4513	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) → if((((0 < 𝐵 ∧ -_𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = -∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = +∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -_𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = +∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-_𝑒𝐴 · 𝐵))) = +∞)
63	41	con2d 134	. . . . . . . . . . . . . 14 ⊢ (((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 = -∞)))))
64	63	imp 406	. . . . . . . . . . . . 13 ⊢ ((((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 = -∞))))
65	64	iffalsed 4516	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) → if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))
66		iftrue 4511	. . . . . . . . . . . . 13 ⊢ ((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) → if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)) = -∞)
67	66	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) → if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)) = -∞)
68	65, 67	eqtrd 2771	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) → if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = -∞)
69		xnegeq 13228	. . . . . . . . . . 11 ⊢ (if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = -∞ → -_𝑒if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = -_𝑒-∞)
70	68, 69	syl 17	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) → -_𝑒if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = -_𝑒-∞)
71	70, 18	eqtrdi 2787	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) → -_𝑒if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = +∞)
72	62, 71	eqtr4d 2774	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) → if((((0 < 𝐵 ∧ -_𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = -∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = +∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -_𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = +∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-_𝑒𝐴 · 𝐵))) = -_𝑒if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))))
73	72	adantlr 715	. . . . . . 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 = +∞)))) → if((((0 < 𝐵 ∧ -_𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = -∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = +∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -_𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = +∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-_𝑒𝐴 · 𝐵))) = -_𝑒if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))))
74	37	notbid 318	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (¬ (((0 < 𝐵 ∧ -_𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = +∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞))) ↔ ¬ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))))
75	74	biimpar 477	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ ¬ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) → ¬ (((0 < 𝐵 ∧ -_𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = +∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞))))
76	75	adantr 480	. . . . . . . . 9 ⊢ (((((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 ∧ -_𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = +∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞))))
77	76	iffalsed 4516	. . . . . . . 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 = +∞)))) → if((((0 < 𝐵 ∧ -_𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = +∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-_𝑒𝐴 · 𝐵)) = (-_𝑒𝐴 · 𝐵))
78	51	biimpar 477	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ ¬ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) → ¬ (((0 < 𝐵 ∧ -_𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = -∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = +∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = -∞))))
79	78	adantlr 715	. . . . . . . . 9 ⊢ (((((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 ∧ -_𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = -∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = +∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = -∞))))
80	79	iffalsed 4516	. . . . . . . 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 = +∞)))) → if((((0 < 𝐵 ∧ -_𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = -∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = +∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -_𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = +∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-_𝑒𝐴 · 𝐵))) = if((((0 < 𝐵 ∧ -_𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = +∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-_𝑒𝐴 · 𝐵)))
81		iffalse 4514	. . . . . . . . . . . 12 ⊢ (¬ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))
82	81	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((((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 = +∞)))) → if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))
83		iffalse 4514	. . . . . . . . . . . 12 ⊢ (¬ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) → if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)) = (𝐴 · 𝐵))
84	83	adantl 481	. . . . . . . . . . 11 ⊢ (((((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 = +∞)))) → if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)) = (𝐴 · 𝐵))
85	82, 84	eqtrd 2771	. . . . . . . . . 10 ⊢ (((((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 = +∞)))) → if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = (𝐴 · 𝐵))
86		xnegeq 13228	. . . . . . . . . 10 ⊢ (if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = (𝐴 · 𝐵) → -_𝑒if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = -_𝑒(𝐴 · 𝐵))
87	85, 86	syl 17	. . . . . . . . 9 ⊢ (((((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 = +∞)))) → -_𝑒if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = -_𝑒(𝐴 · 𝐵))
88		xmullem 13285	. . . . . . . . . . . 12 ⊢ (((((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)
89	88	recnd 11268	. . . . . . . . . . 11 ⊢ (((((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 ∈ ℂ)
90		ancom 460	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ↔ (𝐵 ∈ ℝ^ ∧ 𝐴 ∈ ℝ^*))
91		orcom 870	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 = 0 ∨ 𝐵 = 0) ↔ (𝐵 = 0 ∨ 𝐴 = 0))
92	91	notbii 320	. . . . . . . . . . . . . . 15 ⊢ (¬ (𝐴 = 0 ∨ 𝐵 = 0) ↔ ¬ (𝐵 = 0 ∨ 𝐴 = 0))
93	90, 92	anbi12i 628	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ↔ ((𝐵 ∈ ℝ^ ∧ 𝐴 ∈ ℝ^*) ∧ ¬ (𝐵 = 0 ∨ 𝐴 = 0)))
94		orcom 870	. . . . . . . . . . . . . . 15 ⊢ ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) ↔ (((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) ∨ ((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞))))
95	94	notbii 320	. . . . . . . . . . . . . 14 ⊢ (¬ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) ↔ ¬ (((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) ∨ ((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞))))
96	93, 95	anbi12i 628	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ ¬ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) ↔ (((𝐵 ∈ ℝ^ ∧ 𝐴 ∈ ℝ^*) ∧ ¬ (𝐵 = 0 ∨ 𝐴 = 0)) ∧ ¬ (((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) ∨ ((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)))))
97		orcom 870	. . . . . . . . . . . . . 14 ⊢ ((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) ↔ (((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)) ∨ ((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞))))
98	97	notbii 320	. . . . . . . . . . . . 13 ⊢ (¬ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) ↔ ¬ (((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)) ∨ ((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞))))
99		xmullem 13285	. . . . . . . . . . . . 13 ⊢ (((((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) ∧ ¬ (𝐵 = 0 ∨ 𝐴 = 0)) ∧ ¬ (((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) ∨ ((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)))) ∧ ¬ (((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)) ∨ ((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)))) → 𝐵 ∈ ℝ)
100	96, 98, 99	syl2anb 598	. . . . . . . . . . . 12 ⊢ (((((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)
101	100	recnd 11268	. . . . . . . . . . 11 ⊢ (((((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 ∈ ℂ)
102	89, 101	mulneg1d 11695	. . . . . . . . . 10 ⊢ (((((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 · 𝐵) = -(𝐴 · 𝐵))
103		rexneg 13232	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 = -𝐴)
104	88, 103	syl 17	. . . . . . . . . . 11 ⊢ (((((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 = +∞)))) → -_𝑒𝐴 = -𝐴)
105	104	oveq1d 7425	. . . . . . . . . 10 ⊢ (((((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 = +∞)))) → (-_𝑒𝐴 · 𝐵) = (-𝐴 · 𝐵))
106	88, 100	remulcld 11270	. . . . . . . . . . 11 ⊢ (((((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 · 𝐵) ∈ ℝ)
107		rexneg 13232	. . . . . . . . . . 11 ⊢ ((𝐴 · 𝐵) ∈ ℝ → -_𝑒(𝐴 · 𝐵) = -(𝐴 · 𝐵))
108	106, 107	syl 17	. . . . . . . . . 10 ⊢ (((((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 = +∞)))) → -_𝑒(𝐴 · 𝐵) = -(𝐴 · 𝐵))
109	102, 105, 108	3eqtr4d 2781	. . . . . . . . 9 ⊢ (((((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 = +∞)))) → (-_𝑒𝐴 · 𝐵) = -_𝑒(𝐴 · 𝐵))
110	87, 109	eqtr4d 2774	. . . . . . . 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 = +∞)))) → -_𝑒if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = (-_𝑒𝐴 · 𝐵))
111	77, 80, 110	3eqtr4d 2781	. . . . . . 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 = +∞)))) → if((((0 < 𝐵 ∧ -_𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = -∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = +∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -_𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = +∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-_𝑒𝐴 · 𝐵))) = -_𝑒if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))))
112	73, 111	pm2.61dan 812	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ ¬ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) → if((((0 < 𝐵 ∧ -_𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = -∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = +∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -_𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = +∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-_𝑒𝐴 · 𝐵))) = -_𝑒if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))))
113	60, 112	pm2.61dan 812	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → if((((0 < 𝐵 ∧ -_𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = -∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = +∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -_𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = +∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-_𝑒𝐴 · 𝐵))) = -_𝑒if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))))
114	113	ifeq2da 4538	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ -_𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = -∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = +∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -_𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = +∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-_𝑒𝐴 · 𝐵)))) = if((𝐴 = 0 ∨ 𝐵 = 0), 0, -_𝑒if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))))
115	9, 114	eqtrd 2771	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → if((-_𝑒𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ -_𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = -∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = +∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -_𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = +∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-_𝑒𝐴 · 𝐵)))) = if((𝐴 = 0 ∨ 𝐵 = 0), 0, -_𝑒if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))))
116		xnegeq 13228	. . . . 5 ⊢ (if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))) = 0 → -_𝑒if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))) = -_𝑒0)
117	116, 1	eqtrdi 2787	. . . 4 ⊢ (if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))) = 0 → -_𝑒if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))) = 0)
118		xnegeq 13228	. . . 4 ⊢ (if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))) = if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) → -_𝑒if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))) = -_𝑒if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))))
119	117, 118	ifsb 4519	. . 3 ⊢ -_𝑒if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))) = if((𝐴 = 0 ∨ 𝐵 = 0), 0, -_𝑒if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))))
120	115, 119	eqtr4di 2789	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → if((-_𝑒𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ -_𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = -∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = +∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -_𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = +∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-_𝑒𝐴 · 𝐵)))) = -_𝑒if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))))
121		xnegcl 13234	. . 3 ⊢ (𝐴 ∈ ℝ^* → -_𝑒𝐴 ∈ ℝ^*)
122		xmulval 13246	. . 3 ⊢ ((-_𝑒𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 ·_e 𝐵) = if((-_𝑒𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ -_𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = -∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = +∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -_𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = +∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-_𝑒𝐴 · 𝐵)))))
123	121, 122	sylan 580	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 ·_e 𝐵) = if((-_𝑒𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ -_𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = -∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = +∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -_𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -_𝑒𝐴 = +∞)) ∨ ((0 < -_𝑒𝐴 ∧ 𝐵 = -∞) ∨ (-_𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-_𝑒𝐴 · 𝐵)))))
124		xmulval 13246	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ·_e 𝐵) = if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))))
125		xnegeq 13228	. . 3 ⊢ ((𝐴 ·_e 𝐵) = if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))) → -_𝑒(𝐴 ·_e 𝐵) = -_𝑒if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))))
126	124, 125	syl 17	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 ·_e 𝐵) = -_𝑒if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))))
127	120, 123, 126	3eqtr4d 2781	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 ·_e 𝐵) = -_𝑒(𝐴 ·_e 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ifcif 4505 class class class wbr 5124 (class class class)co 7410 ℝcr 11133 0cc0 11134 · cmul 11139 +∞cpnf 11271 -∞cmnf 11272 ℝ^*cxr 11273 < clt 11274 -cneg 11472 -_𝑒cxne 13130 ·_e cxmu 13132

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-br 5125 df-opab 5187 df-mpt 5207 df-id 5553 df-po 5566 df-so 5567 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-xneg 13133 df-xmul 13135

This theorem is referenced by: xmulneg2 13291 xmulpnf1n 13299 xmulm1 13302 xmulass 13308 xadddi 13316 xadddi2 13318 xrsmulgzz 33006

W3C validator