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Theorem xmulneg1 12655
Description: Extended real version of mulneg1 11068. (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
StepHypRef Expression
1 xneg0 12598 . . . . . . . . 9 -𝑒0 = 0
21eqeq2i 2838 . . . . . . . 8 (-𝑒𝐴 = -𝑒0 ↔ -𝑒𝐴 = 0)
3 0xr 10680 . . . . . . . . 9 0 ∈ ℝ*
4 xneg11 12601 . . . . . . . . 9 ((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) → (-𝑒𝐴 = -𝑒0 ↔ 𝐴 = 0))
53, 4mpan2 687 . . . . . . . 8 (𝐴 ∈ ℝ* → (-𝑒𝐴 = -𝑒0 ↔ 𝐴 = 0))
62, 5syl5bbr 286 . . . . . . 7 (𝐴 ∈ ℝ* → (-𝑒𝐴 = 0 ↔ 𝐴 = 0))
76adantr 481 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (-𝑒𝐴 = 0 ↔ 𝐴 = 0))
87orbi1d 912 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((-𝑒𝐴 = 0 ∨ 𝐵 = 0) ↔ (𝐴 = 0 ∨ 𝐵 = 0)))
98ifbid 4491 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → if((-𝑒𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ -𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = -∞)) ∨ ((0 < -𝑒𝐴𝐵 = +∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = +∞)) ∨ ((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-𝑒𝐴 · 𝐵)))) = if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ -𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = -∞)) ∨ ((0 < -𝑒𝐴𝐵 = +∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = +∞)) ∨ ((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-𝑒𝐴 · 𝐵)))))
10 xnegpnf 12595 . . . . . . . . . . . . . 14 -𝑒+∞ = -∞
1110eqeq2i 2838 . . . . . . . . . . . . 13 (-𝑒𝐴 = -𝑒+∞ ↔ -𝑒𝐴 = -∞)
12 simpll 763 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → 𝐴 ∈ ℝ*)
13 pnfxr 10687 . . . . . . . . . . . . . 14 +∞ ∈ ℝ*
14 xneg11 12601 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-𝑒𝐴 = -𝑒+∞ ↔ 𝐴 = +∞))
1512, 13, 14sylancl 586 . . . . . . . . . . . . 13 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (-𝑒𝐴 = -𝑒+∞ ↔ 𝐴 = +∞))
1611, 15syl5bbr 286 . . . . . . . . . . . 12 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (-𝑒𝐴 = -∞ ↔ 𝐴 = +∞))
1716anbi2d 628 . . . . . . . . . . 11 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → ((0 < 𝐵 ∧ -𝑒𝐴 = -∞) ↔ (0 < 𝐵𝐴 = +∞)))
18 xnegmnf 12596 . . . . . . . . . . . . . 14 -𝑒-∞ = +∞
1918eqeq2i 2838 . . . . . . . . . . . . 13 (-𝑒𝐴 = -𝑒-∞ ↔ -𝑒𝐴 = +∞)
20 mnfxr 10690 . . . . . . . . . . . . . 14 -∞ ∈ ℝ*
21 xneg11 12601 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ* ∧ -∞ ∈ ℝ*) → (-𝑒𝐴 = -𝑒-∞ ↔ 𝐴 = -∞))
2212, 20, 21sylancl 586 . . . . . . . . . . . . 13 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (-𝑒𝐴 = -𝑒-∞ ↔ 𝐴 = -∞))
2319, 22syl5bbr 286 . . . . . . . . . . . 12 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (-𝑒𝐴 = +∞ ↔ 𝐴 = -∞))
2423anbi2d 628 . . . . . . . . . . 11 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → ((𝐵 < 0 ∧ -𝑒𝐴 = +∞) ↔ (𝐵 < 0 ∧ 𝐴 = -∞)))
2517, 24orbi12d 914 . . . . . . . . . 10 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (((0 < 𝐵 ∧ -𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = +∞)) ↔ ((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞))))
26 xlt0neg1 12605 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℝ* → (𝐴 < 0 ↔ 0 < -𝑒𝐴))
2726ad2antrr 722 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (𝐴 < 0 ↔ 0 < -𝑒𝐴))
2827bicomd 224 . . . . . . . . . . . . 13 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (0 < -𝑒𝐴𝐴 < 0))
2928anbi1d 629 . . . . . . . . . . . 12 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → ((0 < -𝑒𝐴𝐵 = -∞) ↔ (𝐴 < 0 ∧ 𝐵 = -∞)))
30 xlt0neg2 12606 . . . . . . . . . . . . . . 15 (𝐴 ∈ ℝ* → (0 < 𝐴 ↔ -𝑒𝐴 < 0))
3130ad2antrr 722 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (0 < 𝐴 ↔ -𝑒𝐴 < 0))
3231bicomd 224 . . . . . . . . . . . . 13 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (-𝑒𝐴 < 0 ↔ 0 < 𝐴))
3332anbi1d 629 . . . . . . . . . . . 12 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → ((-𝑒𝐴 < 0 ∧ 𝐵 = +∞) ↔ (0 < 𝐴𝐵 = +∞)))
3429, 33orbi12d 914 . . . . . . . . . . 11 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞)) ↔ ((𝐴 < 0 ∧ 𝐵 = -∞) ∨ (0 < 𝐴𝐵 = +∞))))
35 orcom 866 . . . . . . . . . . 11 (((𝐴 < 0 ∧ 𝐵 = -∞) ∨ (0 < 𝐴𝐵 = +∞)) ↔ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))
3634, 35syl6bb 288 . . . . . . . . . 10 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞)) ↔ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))))
3725, 36orbi12d 914 . . . . . . . . 9 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → ((((0 < 𝐵 ∧ -𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = +∞)) ∨ ((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞))) ↔ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))))
3837biimpar 478 . . . . . . . 8 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) → (((0 < 𝐵 ∧ -𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = +∞)) ∨ ((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞))))
3938iftrued 4477 . . . . . . 7 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) → if((((0 < 𝐵 ∧ -𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = +∞)) ∨ ((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-𝑒𝐴 · 𝐵)) = -∞)
40 xmullem2 12651 . . . . . . . . . . 11 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))))
4140adantr 481 . . . . . . . . . 10 (((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 = +∞)))))
4223anbi2d 628 . . . . . . . . . . . . 13 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → ((0 < 𝐵 ∧ -𝑒𝐴 = +∞) ↔ (0 < 𝐵𝐴 = -∞)))
4316anbi2d 628 . . . . . . . . . . . . 13 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → ((𝐵 < 0 ∧ -𝑒𝐴 = -∞) ↔ (𝐵 < 0 ∧ 𝐴 = +∞)))
4442, 43orbi12d 914 . . . . . . . . . . . 12 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (((0 < 𝐵 ∧ -𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = -∞)) ↔ ((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞))))
4528anbi1d 629 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → ((0 < -𝑒𝐴𝐵 = +∞) ↔ (𝐴 < 0 ∧ 𝐵 = +∞)))
4632anbi1d 629 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → ((-𝑒𝐴 < 0 ∧ 𝐵 = -∞) ↔ (0 < 𝐴𝐵 = -∞)))
4745, 46orbi12d 914 . . . . . . . . . . . . 13 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (((0 < -𝑒𝐴𝐵 = +∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = -∞)) ↔ ((𝐴 < 0 ∧ 𝐵 = +∞) ∨ (0 < 𝐴𝐵 = -∞))))
48 orcom 866 . . . . . . . . . . . . 13 (((𝐴 < 0 ∧ 𝐵 = +∞) ∨ (0 < 𝐴𝐵 = -∞)) ↔ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))
4947, 48syl6bb 288 . . . . . . . . . . . 12 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (((0 < -𝑒𝐴𝐵 = +∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = -∞)) ↔ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))))
5044, 49orbi12d 914 . . . . . . . . . . 11 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → ((((0 < 𝐵 ∧ -𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = -∞)) ∨ ((0 < -𝑒𝐴𝐵 = +∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = -∞))) ↔ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))))
5150notbid 319 . . . . . . . . . 10 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (¬ (((0 < 𝐵 ∧ -𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = -∞)) ∨ ((0 < -𝑒𝐴𝐵 = +∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = -∞))) ↔ ¬ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))))
5241, 51sylibrd 260 . . . . . . . . 9 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → ((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (((0 < 𝐵 ∧ -𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = -∞)) ∨ ((0 < -𝑒𝐴𝐵 = +∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = -∞)))))
5352imp 407 . . . . . . . 8 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) → ¬ (((0 < 𝐵 ∧ -𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = -∞)) ∨ ((0 < -𝑒𝐴𝐵 = +∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = -∞))))
5453iffalsed 4480 . . . . . . 7 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) → if((((0 < 𝐵 ∧ -𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = -∞)) ∨ ((0 < -𝑒𝐴𝐵 = +∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = +∞)) ∨ ((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-𝑒𝐴 · 𝐵))) = if((((0 < 𝐵 ∧ -𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = +∞)) ∨ ((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-𝑒𝐴 · 𝐵)))
55 iftrue 4475 . . . . . . . . . 10 ((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = +∞)
5655adantl 482 . . . . . . . . 9 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) → if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = +∞)
57 xnegeq 12593 . . . . . . . . 9 (if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = +∞ → -𝑒if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = -𝑒+∞)
5856, 57syl 17 . . . . . . . 8 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) → -𝑒if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = -𝑒+∞)
5958, 10syl6eq 2876 . . . . . . 7 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) → -𝑒if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = -∞)
6039, 54, 593eqtr4d 2870 . . . . . 6 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) → if((((0 < 𝐵 ∧ -𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = -∞)) ∨ ((0 < -𝑒𝐴𝐵 = +∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = +∞)) ∨ ((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-𝑒𝐴 · 𝐵))) = -𝑒if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))))
6150biimpar 478 . . . . . . . . . 10 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) → (((0 < 𝐵 ∧ -𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = -∞)) ∨ ((0 < -𝑒𝐴𝐵 = +∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = -∞))))
6261iftrued 4477 . . . . . . . . 9 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) → if((((0 < 𝐵 ∧ -𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = -∞)) ∨ ((0 < -𝑒𝐴𝐵 = +∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = +∞)) ∨ ((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-𝑒𝐴 · 𝐵))) = +∞)
6341con2d 136 . . . . . . . . . . . . . 14 (((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 = -∞)))))
6463imp 407 . . . . . . . . . . . . 13 ((((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 = -∞))))
6564iffalsed 4480 . . . . . . . . . . . 12 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) → if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))
66 iftrue 4475 . . . . . . . . . . . . 13 ((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) → if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)) = -∞)
6766adantl 482 . . . . . . . . . . . 12 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) → if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)) = -∞)
6865, 67eqtrd 2860 . . . . . . . . . . 11 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) → if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = -∞)
69 xnegeq 12593 . . . . . . . . . . 11 (if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = -∞ → -𝑒if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = -𝑒-∞)
7068, 69syl 17 . . . . . . . . . 10 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) → -𝑒if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = -𝑒-∞)
7170, 18syl6eq 2876 . . . . . . . . 9 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) → -𝑒if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = +∞)
7262, 71eqtr4d 2863 . . . . . . . 8 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) → if((((0 < 𝐵 ∧ -𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = -∞)) ∨ ((0 < -𝑒𝐴𝐵 = +∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = +∞)) ∨ ((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-𝑒𝐴 · 𝐵))) = -𝑒if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))))
7372adantlr 711 . . . . . . 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 = +∞)))) → if((((0 < 𝐵 ∧ -𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = -∞)) ∨ ((0 < -𝑒𝐴𝐵 = +∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = +∞)) ∨ ((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-𝑒𝐴 · 𝐵))) = -𝑒if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))))
7437notbid 319 . . . . . . . . . . 11 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (¬ (((0 < 𝐵 ∧ -𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = +∞)) ∨ ((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞))) ↔ ¬ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))))
7574biimpar 478 . . . . . . . . . 10 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ ¬ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) → ¬ (((0 < 𝐵 ∧ -𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = +∞)) ∨ ((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞))))
7675adantr 481 . . . . . . . . 9 (((((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 ∧ -𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = +∞)) ∨ ((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞))))
7776iffalsed 4480 . . . . . . . 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 = +∞)))) → if((((0 < 𝐵 ∧ -𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = +∞)) ∨ ((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-𝑒𝐴 · 𝐵)) = (-𝑒𝐴 · 𝐵))
7851biimpar 478 . . . . . . . . . 10 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ ¬ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))) → ¬ (((0 < 𝐵 ∧ -𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = -∞)) ∨ ((0 < -𝑒𝐴𝐵 = +∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = -∞))))
7978adantlr 711 . . . . . . . . 9 (((((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 ∧ -𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = -∞)) ∨ ((0 < -𝑒𝐴𝐵 = +∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = -∞))))
8079iffalsed 4480 . . . . . . . 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 = +∞)))) → if((((0 < 𝐵 ∧ -𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = -∞)) ∨ ((0 < -𝑒𝐴𝐵 = +∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = +∞)) ∨ ((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-𝑒𝐴 · 𝐵))) = if((((0 < 𝐵 ∧ -𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = +∞)) ∨ ((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-𝑒𝐴 · 𝐵)))
81 iffalse 4478 . . . . . . . . . . . 12 (¬ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))
8281ad2antlr 723 . . . . . . . . . . 11 (((((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 = +∞)))) → if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))
83 iffalse 4478 . . . . . . . . . . . 12 (¬ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) → if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)) = (𝐴 · 𝐵))
8483adantl 482 . . . . . . . . . . 11 (((((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 = +∞)))) → if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)) = (𝐴 · 𝐵))
8582, 84eqtrd 2860 . . . . . . . . . 10 (((((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 = +∞)))) → if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = (𝐴 · 𝐵))
86 xnegeq 12593 . . . . . . . . . 10 (if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = (𝐴 · 𝐵) → -𝑒if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = -𝑒(𝐴 · 𝐵))
8785, 86syl 17 . . . . . . . . 9 (((((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 = +∞)))) → -𝑒if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = -𝑒(𝐴 · 𝐵))
88 xmullem 12650 . . . . . . . . . . . 12 (((((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)
8988recnd 10661 . . . . . . . . . . 11 (((((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 ∈ ℂ)
90 ancom 461 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ↔ (𝐵 ∈ ℝ*𝐴 ∈ ℝ*))
91 orcom 866 . . . . . . . . . . . . . . . 16 ((𝐴 = 0 ∨ 𝐵 = 0) ↔ (𝐵 = 0 ∨ 𝐴 = 0))
9291notbii 321 . . . . . . . . . . . . . . 15 (¬ (𝐴 = 0 ∨ 𝐵 = 0) ↔ ¬ (𝐵 = 0 ∨ 𝐴 = 0))
9390, 92anbi12i 626 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ↔ ((𝐵 ∈ ℝ*𝐴 ∈ ℝ*) ∧ ¬ (𝐵 = 0 ∨ 𝐴 = 0)))
94 orcom 866 . . . . . . . . . . . . . . 15 ((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) ↔ (((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) ∨ ((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞))))
9594notbii 321 . . . . . . . . . . . . . 14 (¬ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) ↔ ¬ (((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) ∨ ((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞))))
9693, 95anbi12i 626 . . . . . . . . . . . . 13 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ ¬ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) ↔ (((𝐵 ∈ ℝ*𝐴 ∈ ℝ*) ∧ ¬ (𝐵 = 0 ∨ 𝐴 = 0)) ∧ ¬ (((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) ∨ ((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)))))
97 orcom 866 . . . . . . . . . . . . . 14 ((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) ↔ (((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)) ∨ ((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞))))
9897notbii 321 . . . . . . . . . . . . 13 (¬ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) ↔ ¬ (((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)) ∨ ((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞))))
99 xmullem 12650 . . . . . . . . . . . . 13 (((((𝐵 ∈ ℝ*𝐴 ∈ ℝ*) ∧ ¬ (𝐵 = 0 ∨ 𝐴 = 0)) ∧ ¬ (((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) ∨ ((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)))) ∧ ¬ (((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)) ∨ ((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)))) → 𝐵 ∈ ℝ)
10096, 98, 99syl2anb 597 . . . . . . . . . . . 12 (((((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)
101100recnd 10661 . . . . . . . . . . 11 (((((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 ∈ ℂ)
10289, 101mulneg1d 11085 . . . . . . . . . 10 (((((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 · 𝐵) = -(𝐴 · 𝐵))
103 rexneg 12597 . . . . . . . . . . . 12 (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴)
10488, 103syl 17 . . . . . . . . . . 11 (((((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 = +∞)))) → -𝑒𝐴 = -𝐴)
105104oveq1d 7166 . . . . . . . . . 10 (((((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 = +∞)))) → (-𝑒𝐴 · 𝐵) = (-𝐴 · 𝐵))
10688, 100remulcld 10663 . . . . . . . . . . 11 (((((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 · 𝐵) ∈ ℝ)
107 rexneg 12597 . . . . . . . . . . 11 ((𝐴 · 𝐵) ∈ ℝ → -𝑒(𝐴 · 𝐵) = -(𝐴 · 𝐵))
108106, 107syl 17 . . . . . . . . . 10 (((((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 = +∞)))) → -𝑒(𝐴 · 𝐵) = -(𝐴 · 𝐵))
109102, 105, 1083eqtr4d 2870 . . . . . . . . 9 (((((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 = +∞)))) → (-𝑒𝐴 · 𝐵) = -𝑒(𝐴 · 𝐵))
11087, 109eqtr4d 2863 . . . . . . . 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 = +∞)))) → -𝑒if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = (-𝑒𝐴 · 𝐵))
11177, 80, 1103eqtr4d 2870 . . . . . . 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 = +∞)))) → if((((0 < 𝐵 ∧ -𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = -∞)) ∨ ((0 < -𝑒𝐴𝐵 = +∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = +∞)) ∨ ((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-𝑒𝐴 · 𝐵))) = -𝑒if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))))
11273, 111pm2.61dan 809 . . . . . 6 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) ∧ ¬ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))) → if((((0 < 𝐵 ∧ -𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = -∞)) ∨ ((0 < -𝑒𝐴𝐵 = +∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = +∞)) ∨ ((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-𝑒𝐴 · 𝐵))) = -𝑒if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))))
11360, 112pm2.61dan 809 . . . . 5 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → if((((0 < 𝐵 ∧ -𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = -∞)) ∨ ((0 < -𝑒𝐴𝐵 = +∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = +∞)) ∨ ((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-𝑒𝐴 · 𝐵))) = -𝑒if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))))
114113ifeq2da 4500 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ -𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = -∞)) ∨ ((0 < -𝑒𝐴𝐵 = +∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = +∞)) ∨ ((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-𝑒𝐴 · 𝐵)))) = if((𝐴 = 0 ∨ 𝐵 = 0), 0, -𝑒if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))))
1159, 114eqtrd 2860 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → if((-𝑒𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ -𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = -∞)) ∨ ((0 < -𝑒𝐴𝐵 = +∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = +∞)) ∨ ((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-𝑒𝐴 · 𝐵)))) = if((𝐴 = 0 ∨ 𝐵 = 0), 0, -𝑒if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))))
116 xnegeq 12593 . . . . 5 (if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))) = 0 → -𝑒if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))) = -𝑒0)
117116, 1syl6eq 2876 . . . 4 (if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))) = 0 → -𝑒if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))) = 0)
118 xnegeq 12593 . . . 4 (if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))) = if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) → -𝑒if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))) = -𝑒if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))))
119117, 118ifsb 4482 . . 3 -𝑒if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))) = if((𝐴 = 0 ∨ 𝐵 = 0), 0, -𝑒if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))))
120115, 119syl6eqr 2878 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → if((-𝑒𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ -𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = -∞)) ∨ ((0 < -𝑒𝐴𝐵 = +∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = +∞)) ∨ ((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-𝑒𝐴 · 𝐵)))) = -𝑒if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))))
121 xnegcl 12599 . . 3 (𝐴 ∈ ℝ* → -𝑒𝐴 ∈ ℝ*)
122 xmulval 12611 . . 3 ((-𝑒𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (-𝑒𝐴 ·e 𝐵) = if((-𝑒𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ -𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = -∞)) ∨ ((0 < -𝑒𝐴𝐵 = +∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = +∞)) ∨ ((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-𝑒𝐴 · 𝐵)))))
123121, 122sylan 580 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (-𝑒𝐴 ·e 𝐵) = if((-𝑒𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ -𝑒𝐴 = +∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = -∞)) ∨ ((0 < -𝑒𝐴𝐵 = +∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ -𝑒𝐴 = -∞) ∨ (𝐵 < 0 ∧ -𝑒𝐴 = +∞)) ∨ ((0 < -𝑒𝐴𝐵 = -∞) ∨ (-𝑒𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (-𝑒𝐴 · 𝐵)))))
124 xmulval 12611 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 ·e 𝐵) = if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))))
125 xnegeq 12593 . . 3 ((𝐴 ·e 𝐵) = if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))) → -𝑒(𝐴 ·e 𝐵) = -𝑒if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))))
126124, 125syl 17 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → -𝑒(𝐴 ·e 𝐵) = -𝑒if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))))
127120, 123, 1263eqtr4d 2870 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 207  wa 396  wo 843   = wceq 1530  wcel 2107  ifcif 4469   class class class wbr 5062  (class class class)co 7151  cr 10528  0cc0 10529   · cmul 10534  +∞cpnf 10664  -∞cmnf 10665  *cxr 10666   < clt 10667  -cneg 10863  -𝑒cxne 12497   ·e cxmu 12499
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-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605
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-reu 3149  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-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-xneg 12500  df-xmul 12502
This theorem is referenced by:  xmulneg2  12656  xmulpnf1n  12664  xmulm1  12667  xmulass  12673  xadddi  12681  xadddi2  12683  xrsmulgzz  30581
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