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Theorem xmullem2 13100
Description: Lemma for xmulneg1 13104. (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 11132 . . . . . . . . . . . 12 -∞ ≠ +∞
2 eqeq1 2740 . . . . . . . . . . . . 13 (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞))
32necon3bbid 2978 . . . . . . . . . . . 12 (𝐴 = -∞ → (¬ 𝐴 = +∞ ↔ -∞ ≠ +∞))
41, 3mpbiri 257 . . . . . . . . . . 11 (𝐴 = -∞ → ¬ 𝐴 = +∞)
54con2i 139 . . . . . . . . . 10 (𝐴 = +∞ → ¬ 𝐴 = -∞)
65adantl 482 . . . . . . . . 9 ((0 < 𝐵𝐴 = +∞) → ¬ 𝐴 = -∞)
7 0xr 11123 . . . . . . . . . . . . 13 0 ∈ ℝ*
8 nltmnf 12966 . . . . . . . . . . . . 13 (0 ∈ ℝ* → ¬ 0 < -∞)
97, 8ax-mp 5 . . . . . . . . . . . 12 ¬ 0 < -∞
10 breq2 5096 . . . . . . . . . . . 12 (𝐴 = -∞ → (0 < 𝐴 ↔ 0 < -∞))
119, 10mtbiri 326 . . . . . . . . . . 11 (𝐴 = -∞ → ¬ 0 < 𝐴)
1211con2i 139 . . . . . . . . . 10 (0 < 𝐴 → ¬ 𝐴 = -∞)
1312adantr 481 . . . . . . . . 9 ((0 < 𝐴𝐵 = +∞) → ¬ 𝐴 = -∞)
146, 13jaoi 854 . . . . . . . 8 (((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) → ¬ 𝐴 = -∞)
1514a1i 11 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) → ¬ 𝐴 = -∞))
16 simpr 485 . . . . . . . . . 10 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → 𝐵 ∈ ℝ*)
17 xrltnsym 12972 . . . . . . . . . 10 ((𝐵 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝐵 < 0 → ¬ 0 < 𝐵))
1816, 7, 17sylancl 586 . . . . . . . . 9 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐵 < 0 → ¬ 0 < 𝐵))
1918adantrd 492 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 0 < 𝐵))
20 breq2 5096 . . . . . . . . . . 11 (𝐵 = -∞ → (0 < 𝐵 ↔ 0 < -∞))
219, 20mtbiri 326 . . . . . . . . . 10 (𝐵 = -∞ → ¬ 0 < 𝐵)
2221adantl 482 . . . . . . . . 9 ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 0 < 𝐵)
2322a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 0 < 𝐵))
2419, 23jaod 856 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 0 < 𝐵))
2515, 24orim12d 962 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐴 = -∞ ∨ ¬ 0 < 𝐵)))
26 ianor 979 . . . . . . 7 (¬ (0 < 𝐵𝐴 = -∞) ↔ (¬ 0 < 𝐵 ∨ ¬ 𝐴 = -∞))
27 orcom 867 . . . . . . 7 ((¬ 0 < 𝐵 ∨ ¬ 𝐴 = -∞) ↔ (¬ 𝐴 = -∞ ∨ ¬ 0 < 𝐵))
2826, 27bitri 274 . . . . . 6 (¬ (0 < 𝐵𝐴 = -∞) ↔ (¬ 𝐴 = -∞ ∨ ¬ 0 < 𝐵))
2925, 28syl6ibr 251 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (0 < 𝐵𝐴 = -∞)))
3018con2d 134 . . . . . . . . 9 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (0 < 𝐵 → ¬ 𝐵 < 0))
3130adantrd 492 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((0 < 𝐵𝐴 = +∞) → ¬ 𝐵 < 0))
32 pnfnlt 12965 . . . . . . . . . . 11 (0 ∈ ℝ* → ¬ +∞ < 0)
337, 32ax-mp 5 . . . . . . . . . 10 ¬ +∞ < 0
34 simpr 485 . . . . . . . . . . 11 ((0 < 𝐴𝐵 = +∞) → 𝐵 = +∞)
3534breq1d 5102 . . . . . . . . . 10 ((0 < 𝐴𝐵 = +∞) → (𝐵 < 0 ↔ +∞ < 0))
3633, 35mtbiri 326 . . . . . . . . 9 ((0 < 𝐴𝐵 = +∞) → ¬ 𝐵 < 0)
3736a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((0 < 𝐴𝐵 = +∞) → ¬ 𝐵 < 0))
3831, 37jaod 856 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) → ¬ 𝐵 < 0))
394a1i 11 . . . . . . . . 9 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 = -∞ → ¬ 𝐴 = +∞))
4039adantld 491 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 𝐴 = +∞))
41 breq1 5095 . . . . . . . . . . . 12 (𝐴 = +∞ → (𝐴 < 0 ↔ +∞ < 0))
4233, 41mtbiri 326 . . . . . . . . . . 11 (𝐴 = +∞ → ¬ 𝐴 < 0)
4342con2i 139 . . . . . . . . . 10 (𝐴 < 0 → ¬ 𝐴 = +∞)
4443adantr 481 . . . . . . . . 9 ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 𝐴 = +∞)
4544a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 𝐴 = +∞))
4640, 45jaod 856 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 𝐴 = +∞))
4738, 46orim12d 962 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐵 < 0 ∨ ¬ 𝐴 = +∞)))
48 ianor 979 . . . . . 6 (¬ (𝐵 < 0 ∧ 𝐴 = +∞) ↔ (¬ 𝐵 < 0 ∨ ¬ 𝐴 = +∞))
4947, 48syl6ibr 251 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (𝐵 < 0 ∧ 𝐴 = +∞)))
5029, 49jcad 513 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞))))
51 ioran 981 . . . 4 (¬ ((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ↔ (¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)))
5250, 51syl6ibr 251 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ ((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞))))
5321con2i 139 . . . . . . . . . 10 (0 < 𝐵 → ¬ 𝐵 = -∞)
5453adantr 481 . . . . . . . . 9 ((0 < 𝐵𝐴 = +∞) → ¬ 𝐵 = -∞)
5554a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((0 < 𝐵𝐴 = +∞) → ¬ 𝐵 = -∞))
56 pnfnemnf 11131 . . . . . . . . . . 11 +∞ ≠ -∞
57 eqeq1 2740 . . . . . . . . . . . 12 (𝐵 = +∞ → (𝐵 = -∞ ↔ +∞ = -∞))
5857necon3bbid 2978 . . . . . . . . . . 11 (𝐵 = +∞ → (¬ 𝐵 = -∞ ↔ +∞ ≠ -∞))
5956, 58mpbiri 257 . . . . . . . . . 10 (𝐵 = +∞ → ¬ 𝐵 = -∞)
6059adantl 482 . . . . . . . . 9 ((0 < 𝐴𝐵 = +∞) → ¬ 𝐵 = -∞)
6160a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((0 < 𝐴𝐵 = +∞) → ¬ 𝐵 = -∞))
6255, 61jaod 856 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) → ¬ 𝐵 = -∞))
6311adantl 482 . . . . . . . . 9 ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 0 < 𝐴)
6463a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 0 < 𝐴))
65 simpl 483 . . . . . . . . . 10 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → 𝐴 ∈ ℝ*)
66 xrltnsym 12972 . . . . . . . . . 10 ((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝐴 < 0 → ¬ 0 < 𝐴))
6765, 7, 66sylancl 586 . . . . . . . . 9 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 0 → ¬ 0 < 𝐴))
6867adantrd 492 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 0 < 𝐴))
6964, 68jaod 856 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 0 < 𝐴))
7062, 69orim12d 962 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐵 = -∞ ∨ ¬ 0 < 𝐴)))
71 ianor 979 . . . . . . 7 (¬ (0 < 𝐴𝐵 = -∞) ↔ (¬ 0 < 𝐴 ∨ ¬ 𝐵 = -∞))
72 orcom 867 . . . . . . 7 ((¬ 0 < 𝐴 ∨ ¬ 𝐵 = -∞) ↔ (¬ 𝐵 = -∞ ∨ ¬ 0 < 𝐴))
7371, 72bitri 274 . . . . . 6 (¬ (0 < 𝐴𝐵 = -∞) ↔ (¬ 𝐵 = -∞ ∨ ¬ 0 < 𝐴))
7470, 73syl6ibr 251 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (0 < 𝐴𝐵 = -∞)))
7542adantl 482 . . . . . . . . 9 ((0 < 𝐵𝐴 = +∞) → ¬ 𝐴 < 0)
7675a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((0 < 𝐵𝐴 = +∞) → ¬ 𝐴 < 0))
7767con2d 134 . . . . . . . . 9 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (0 < 𝐴 → ¬ 𝐴 < 0))
7877adantrd 492 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((0 < 𝐴𝐵 = +∞) → ¬ 𝐴 < 0))
7976, 78jaod 856 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) → ¬ 𝐴 < 0))
80 breq1 5095 . . . . . . . . . . . 12 (𝐵 = +∞ → (𝐵 < 0 ↔ +∞ < 0))
8133, 80mtbiri 326 . . . . . . . . . . 11 (𝐵 = +∞ → ¬ 𝐵 < 0)
8281con2i 139 . . . . . . . . . 10 (𝐵 < 0 → ¬ 𝐵 = +∞)
8382adantr 481 . . . . . . . . 9 ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 𝐵 = +∞)
8459con2i 139 . . . . . . . . . 10 (𝐵 = -∞ → ¬ 𝐵 = +∞)
8584adantl 482 . . . . . . . . 9 ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 𝐵 = +∞)
8683, 85jaoi 854 . . . . . . . 8 (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 𝐵 = +∞)
8786a1i 11 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 𝐵 = +∞))
8879, 87orim12d 962 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐴 < 0 ∨ ¬ 𝐵 = +∞)))
89 ianor 979 . . . . . 6 (¬ (𝐴 < 0 ∧ 𝐵 = +∞) ↔ (¬ 𝐴 < 0 ∨ ¬ 𝐵 = +∞))
9088, 89syl6ibr 251 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))
9174, 90jcad 513 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))
92 ioran 981 . . . 4 (¬ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)) ↔ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))
9391, 92syl6ibr 251 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))))
9452, 93jcad 513 . 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 924 . 2 ((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) ↔ (((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))))
96 ioran 981 . 2 (¬ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) ↔ (¬ ((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ ¬ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))))
9794, 95, 963imtr4g 295 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 396  wo 844   = wceq 1540  wcel 2105  wne 2940   class class class wbr 5092  0cc0 10972  +∞cpnf 11107  -∞cmnf 11108  *cxr 11109   < clt 11110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-addrcl 11033  ax-rnegex 11043  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-po 5532  df-so 5533  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-er 8569  df-en 8805  df-dom 8806  df-sdom 8807  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115
This theorem is referenced by:  xmulneg1  13104
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