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Theorem xmullem2 12477
Description: Lemma for xmulneg1 12481. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xmullem2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))))

Proof of Theorem xmullem2
StepHypRef Expression
1 mnfnepnf 10499 . . . . . . . . . . . 12 -∞ ≠ +∞
2 eqeq1 2782 . . . . . . . . . . . . 13 (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞))
32necon3bbid 3004 . . . . . . . . . . . 12 (𝐴 = -∞ → (¬ 𝐴 = +∞ ↔ -∞ ≠ +∞))
41, 3mpbiri 250 . . . . . . . . . . 11 (𝐴 = -∞ → ¬ 𝐴 = +∞)
54con2i 137 . . . . . . . . . 10 (𝐴 = +∞ → ¬ 𝐴 = -∞)
65adantl 474 . . . . . . . . 9 ((0 < 𝐵𝐴 = +∞) → ¬ 𝐴 = -∞)
7 0xr 10489 . . . . . . . . . . . . 13 0 ∈ ℝ*
8 nltmnf 12344 . . . . . . . . . . . . 13 (0 ∈ ℝ* → ¬ 0 < -∞)
97, 8ax-mp 5 . . . . . . . . . . . 12 ¬ 0 < -∞
10 breq2 4934 . . . . . . . . . . . 12 (𝐴 = -∞ → (0 < 𝐴 ↔ 0 < -∞))
119, 10mtbiri 319 . . . . . . . . . . 11 (𝐴 = -∞ → ¬ 0 < 𝐴)
1211con2i 137 . . . . . . . . . 10 (0 < 𝐴 → ¬ 𝐴 = -∞)
1312adantr 473 . . . . . . . . 9 ((0 < 𝐴𝐵 = +∞) → ¬ 𝐴 = -∞)
146, 13jaoi 843 . . . . . . . 8 (((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) → ¬ 𝐴 = -∞)
1514a1i 11 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) → ¬ 𝐴 = -∞))
16 simpr 477 . . . . . . . . . 10 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → 𝐵 ∈ ℝ*)
17 xrltnsym 12350 . . . . . . . . . 10 ((𝐵 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝐵 < 0 → ¬ 0 < 𝐵))
1816, 7, 17sylancl 577 . . . . . . . . 9 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐵 < 0 → ¬ 0 < 𝐵))
1918adantrd 484 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 0 < 𝐵))
20 breq2 4934 . . . . . . . . . . 11 (𝐵 = -∞ → (0 < 𝐵 ↔ 0 < -∞))
219, 20mtbiri 319 . . . . . . . . . 10 (𝐵 = -∞ → ¬ 0 < 𝐵)
2221adantl 474 . . . . . . . . 9 ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 0 < 𝐵)
2322a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 0 < 𝐵))
2419, 23jaod 845 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 0 < 𝐵))
2515, 24orim12d 947 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐴 = -∞ ∨ ¬ 0 < 𝐵)))
26 ianor 964 . . . . . . 7 (¬ (0 < 𝐵𝐴 = -∞) ↔ (¬ 0 < 𝐵 ∨ ¬ 𝐴 = -∞))
27 orcom 856 . . . . . . 7 ((¬ 0 < 𝐵 ∨ ¬ 𝐴 = -∞) ↔ (¬ 𝐴 = -∞ ∨ ¬ 0 < 𝐵))
2826, 27bitri 267 . . . . . 6 (¬ (0 < 𝐵𝐴 = -∞) ↔ (¬ 𝐴 = -∞ ∨ ¬ 0 < 𝐵))
2925, 28syl6ibr 244 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (0 < 𝐵𝐴 = -∞)))
3018con2d 132 . . . . . . . . 9 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (0 < 𝐵 → ¬ 𝐵 < 0))
3130adantrd 484 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((0 < 𝐵𝐴 = +∞) → ¬ 𝐵 < 0))
32 pnfnlt 12343 . . . . . . . . . . 11 (0 ∈ ℝ* → ¬ +∞ < 0)
337, 32ax-mp 5 . . . . . . . . . 10 ¬ +∞ < 0
34 simpr 477 . . . . . . . . . . 11 ((0 < 𝐴𝐵 = +∞) → 𝐵 = +∞)
3534breq1d 4940 . . . . . . . . . 10 ((0 < 𝐴𝐵 = +∞) → (𝐵 < 0 ↔ +∞ < 0))
3633, 35mtbiri 319 . . . . . . . . 9 ((0 < 𝐴𝐵 = +∞) → ¬ 𝐵 < 0)
3736a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((0 < 𝐴𝐵 = +∞) → ¬ 𝐵 < 0))
3831, 37jaod 845 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) → ¬ 𝐵 < 0))
394a1i 11 . . . . . . . . 9 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 = -∞ → ¬ 𝐴 = +∞))
4039adantld 483 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 𝐴 = +∞))
41 breq1 4933 . . . . . . . . . . . 12 (𝐴 = +∞ → (𝐴 < 0 ↔ +∞ < 0))
4233, 41mtbiri 319 . . . . . . . . . . 11 (𝐴 = +∞ → ¬ 𝐴 < 0)
4342con2i 137 . . . . . . . . . 10 (𝐴 < 0 → ¬ 𝐴 = +∞)
4443adantr 473 . . . . . . . . 9 ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 𝐴 = +∞)
4544a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 𝐴 = +∞))
4640, 45jaod 845 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 𝐴 = +∞))
4738, 46orim12d 947 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐵 < 0 ∨ ¬ 𝐴 = +∞)))
48 ianor 964 . . . . . 6 (¬ (𝐵 < 0 ∧ 𝐴 = +∞) ↔ (¬ 𝐵 < 0 ∨ ¬ 𝐴 = +∞))
4947, 48syl6ibr 244 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (𝐵 < 0 ∧ 𝐴 = +∞)))
5029, 49jcad 505 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞))))
51 ioran 966 . . . 4 (¬ ((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ↔ (¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)))
5250, 51syl6ibr 244 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ ((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞))))
5321con2i 137 . . . . . . . . . 10 (0 < 𝐵 → ¬ 𝐵 = -∞)
5453adantr 473 . . . . . . . . 9 ((0 < 𝐵𝐴 = +∞) → ¬ 𝐵 = -∞)
5554a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((0 < 𝐵𝐴 = +∞) → ¬ 𝐵 = -∞))
56 pnfnemnf 10498 . . . . . . . . . . 11 +∞ ≠ -∞
57 eqeq1 2782 . . . . . . . . . . . 12 (𝐵 = +∞ → (𝐵 = -∞ ↔ +∞ = -∞))
5857necon3bbid 3004 . . . . . . . . . . 11 (𝐵 = +∞ → (¬ 𝐵 = -∞ ↔ +∞ ≠ -∞))
5956, 58mpbiri 250 . . . . . . . . . 10 (𝐵 = +∞ → ¬ 𝐵 = -∞)
6059adantl 474 . . . . . . . . 9 ((0 < 𝐴𝐵 = +∞) → ¬ 𝐵 = -∞)
6160a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((0 < 𝐴𝐵 = +∞) → ¬ 𝐵 = -∞))
6255, 61jaod 845 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) → ¬ 𝐵 = -∞))
6311adantl 474 . . . . . . . . 9 ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 0 < 𝐴)
6463a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 0 < 𝐴))
65 simpl 475 . . . . . . . . . 10 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → 𝐴 ∈ ℝ*)
66 xrltnsym 12350 . . . . . . . . . 10 ((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝐴 < 0 → ¬ 0 < 𝐴))
6765, 7, 66sylancl 577 . . . . . . . . 9 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 0 → ¬ 0 < 𝐴))
6867adantrd 484 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 0 < 𝐴))
6964, 68jaod 845 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 0 < 𝐴))
7062, 69orim12d 947 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐵 = -∞ ∨ ¬ 0 < 𝐴)))
71 ianor 964 . . . . . . 7 (¬ (0 < 𝐴𝐵 = -∞) ↔ (¬ 0 < 𝐴 ∨ ¬ 𝐵 = -∞))
72 orcom 856 . . . . . . 7 ((¬ 0 < 𝐴 ∨ ¬ 𝐵 = -∞) ↔ (¬ 𝐵 = -∞ ∨ ¬ 0 < 𝐴))
7371, 72bitri 267 . . . . . 6 (¬ (0 < 𝐴𝐵 = -∞) ↔ (¬ 𝐵 = -∞ ∨ ¬ 0 < 𝐴))
7470, 73syl6ibr 244 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (0 < 𝐴𝐵 = -∞)))
7542adantl 474 . . . . . . . . 9 ((0 < 𝐵𝐴 = +∞) → ¬ 𝐴 < 0)
7675a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((0 < 𝐵𝐴 = +∞) → ¬ 𝐴 < 0))
7767con2d 132 . . . . . . . . 9 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (0 < 𝐴 → ¬ 𝐴 < 0))
7877adantrd 484 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((0 < 𝐴𝐵 = +∞) → ¬ 𝐴 < 0))
7976, 78jaod 845 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) → ¬ 𝐴 < 0))
80 breq1 4933 . . . . . . . . . . . 12 (𝐵 = +∞ → (𝐵 < 0 ↔ +∞ < 0))
8133, 80mtbiri 319 . . . . . . . . . . 11 (𝐵 = +∞ → ¬ 𝐵 < 0)
8281con2i 137 . . . . . . . . . 10 (𝐵 < 0 → ¬ 𝐵 = +∞)
8382adantr 473 . . . . . . . . 9 ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 𝐵 = +∞)
8459con2i 137 . . . . . . . . . 10 (𝐵 = -∞ → ¬ 𝐵 = +∞)
8584adantl 474 . . . . . . . . 9 ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 𝐵 = +∞)
8683, 85jaoi 843 . . . . . . . 8 (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 𝐵 = +∞)
8786a1i 11 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 𝐵 = +∞))
8879, 87orim12d 947 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐴 < 0 ∨ ¬ 𝐵 = +∞)))
89 ianor 964 . . . . . 6 (¬ (𝐴 < 0 ∧ 𝐵 = +∞) ↔ (¬ 𝐴 < 0 ∨ ¬ 𝐵 = +∞))
9088, 89syl6ibr 244 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))
9174, 90jcad 505 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))
92 ioran 966 . . . 4 (¬ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)) ↔ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))
9391, 92syl6ibr 244 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))))
9452, 93jcad 505 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ ((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ ¬ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))))
95 or4 910 . 2 ((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) ↔ (((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))))
96 ioran 966 . 2 (¬ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) ↔ (¬ ((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ ¬ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))))
9794, 95, 963imtr4g 288 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 387  wo 833   = wceq 1507  wcel 2050  wne 2967   class class class wbr 4930  0cc0 10337  +∞cpnf 10473  -∞cmnf 10474  *cxr 10475   < clt 10476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281  ax-cnex 10393  ax-resscn 10394  ax-1cn 10395  ax-addrcl 10398  ax-rnegex 10408  ax-cnre 10410  ax-pre-lttri 10411  ax-pre-lttrn 10412
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-opab 4993  df-mpt 5010  df-id 5313  df-po 5327  df-so 5328  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-er 8091  df-en 8309  df-dom 8310  df-sdom 8311  df-pnf 10478  df-mnf 10479  df-xr 10480  df-ltxr 10481
This theorem is referenced by:  xmulneg1  12481
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