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Theorem xmullem2 13307
Description: Lemma for xmulneg1 13311. (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 11317 . . . . . . . . . . . 12 -∞ ≠ +∞
2 eqeq1 2741 . . . . . . . . . . . . 13 (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞))
32necon3bbid 2978 . . . . . . . . . . . 12 (𝐴 = -∞ → (¬ 𝐴 = +∞ ↔ -∞ ≠ +∞))
41, 3mpbiri 258 . . . . . . . . . . 11 (𝐴 = -∞ → ¬ 𝐴 = +∞)
54con2i 139 . . . . . . . . . 10 (𝐴 = +∞ → ¬ 𝐴 = -∞)
65adantl 481 . . . . . . . . 9 ((0 < 𝐵𝐴 = +∞) → ¬ 𝐴 = -∞)
7 0xr 11308 . . . . . . . . . . . . 13 0 ∈ ℝ*
8 nltmnf 13171 . . . . . . . . . . . . 13 (0 ∈ ℝ* → ¬ 0 < -∞)
97, 8ax-mp 5 . . . . . . . . . . . 12 ¬ 0 < -∞
10 breq2 5147 . . . . . . . . . . . 12 (𝐴 = -∞ → (0 < 𝐴 ↔ 0 < -∞))
119, 10mtbiri 327 . . . . . . . . . . 11 (𝐴 = -∞ → ¬ 0 < 𝐴)
1211con2i 139 . . . . . . . . . 10 (0 < 𝐴 → ¬ 𝐴 = -∞)
1312adantr 480 . . . . . . . . 9 ((0 < 𝐴𝐵 = +∞) → ¬ 𝐴 = -∞)
146, 13jaoi 858 . . . . . . . 8 (((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) → ¬ 𝐴 = -∞)
1514a1i 11 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) → ¬ 𝐴 = -∞))
16 simpr 484 . . . . . . . . . 10 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → 𝐵 ∈ ℝ*)
17 xrltnsym 13179 . . . . . . . . . 10 ((𝐵 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝐵 < 0 → ¬ 0 < 𝐵))
1816, 7, 17sylancl 586 . . . . . . . . 9 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐵 < 0 → ¬ 0 < 𝐵))
1918adantrd 491 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 0 < 𝐵))
20 breq2 5147 . . . . . . . . . . 11 (𝐵 = -∞ → (0 < 𝐵 ↔ 0 < -∞))
219, 20mtbiri 327 . . . . . . . . . 10 (𝐵 = -∞ → ¬ 0 < 𝐵)
2221adantl 481 . . . . . . . . 9 ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 0 < 𝐵)
2322a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 0 < 𝐵))
2419, 23jaod 860 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 0 < 𝐵))
2515, 24orim12d 967 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐴 = -∞ ∨ ¬ 0 < 𝐵)))
26 ianor 984 . . . . . . 7 (¬ (0 < 𝐵𝐴 = -∞) ↔ (¬ 0 < 𝐵 ∨ ¬ 𝐴 = -∞))
27 orcom 871 . . . . . . 7 ((¬ 0 < 𝐵 ∨ ¬ 𝐴 = -∞) ↔ (¬ 𝐴 = -∞ ∨ ¬ 0 < 𝐵))
2826, 27bitri 275 . . . . . 6 (¬ (0 < 𝐵𝐴 = -∞) ↔ (¬ 𝐴 = -∞ ∨ ¬ 0 < 𝐵))
2925, 28imbitrrdi 252 . . . . 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 491 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((0 < 𝐵𝐴 = +∞) → ¬ 𝐵 < 0))
32 pnfnlt 13170 . . . . . . . . . . 11 (0 ∈ ℝ* → ¬ +∞ < 0)
337, 32ax-mp 5 . . . . . . . . . 10 ¬ +∞ < 0
34 simpr 484 . . . . . . . . . . 11 ((0 < 𝐴𝐵 = +∞) → 𝐵 = +∞)
3534breq1d 5153 . . . . . . . . . 10 ((0 < 𝐴𝐵 = +∞) → (𝐵 < 0 ↔ +∞ < 0))
3633, 35mtbiri 327 . . . . . . . . 9 ((0 < 𝐴𝐵 = +∞) → ¬ 𝐵 < 0)
3736a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((0 < 𝐴𝐵 = +∞) → ¬ 𝐵 < 0))
3831, 37jaod 860 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) → ¬ 𝐵 < 0))
394a1i 11 . . . . . . . . 9 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 = -∞ → ¬ 𝐴 = +∞))
4039adantld 490 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 𝐴 = +∞))
41 breq1 5146 . . . . . . . . . . . 12 (𝐴 = +∞ → (𝐴 < 0 ↔ +∞ < 0))
4233, 41mtbiri 327 . . . . . . . . . . 11 (𝐴 = +∞ → ¬ 𝐴 < 0)
4342con2i 139 . . . . . . . . . 10 (𝐴 < 0 → ¬ 𝐴 = +∞)
4443adantr 480 . . . . . . . . 9 ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 𝐴 = +∞)
4544a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 𝐴 = +∞))
4640, 45jaod 860 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 𝐴 = +∞))
4738, 46orim12d 967 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐵 < 0 ∨ ¬ 𝐴 = +∞)))
48 ianor 984 . . . . . 6 (¬ (𝐵 < 0 ∧ 𝐴 = +∞) ↔ (¬ 𝐵 < 0 ∨ ¬ 𝐴 = +∞))
4947, 48imbitrrdi 252 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (𝐵 < 0 ∧ 𝐴 = +∞)))
5029, 49jcad 512 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞))))
51 ioran 986 . . . 4 (¬ ((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ↔ (¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)))
5250, 51imbitrrdi 252 . . 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 480 . . . . . . . . 9 ((0 < 𝐵𝐴 = +∞) → ¬ 𝐵 = -∞)
5554a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((0 < 𝐵𝐴 = +∞) → ¬ 𝐵 = -∞))
56 pnfnemnf 11316 . . . . . . . . . . 11 +∞ ≠ -∞
57 eqeq1 2741 . . . . . . . . . . . 12 (𝐵 = +∞ → (𝐵 = -∞ ↔ +∞ = -∞))
5857necon3bbid 2978 . . . . . . . . . . 11 (𝐵 = +∞ → (¬ 𝐵 = -∞ ↔ +∞ ≠ -∞))
5956, 58mpbiri 258 . . . . . . . . . 10 (𝐵 = +∞ → ¬ 𝐵 = -∞)
6059adantl 481 . . . . . . . . 9 ((0 < 𝐴𝐵 = +∞) → ¬ 𝐵 = -∞)
6160a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((0 < 𝐴𝐵 = +∞) → ¬ 𝐵 = -∞))
6255, 61jaod 860 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) → ¬ 𝐵 = -∞))
6311adantl 481 . . . . . . . . 9 ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 0 < 𝐴)
6463a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 0 < 𝐴))
65 simpl 482 . . . . . . . . . 10 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → 𝐴 ∈ ℝ*)
66 xrltnsym 13179 . . . . . . . . . 10 ((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝐴 < 0 → ¬ 0 < 𝐴))
6765, 7, 66sylancl 586 . . . . . . . . 9 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 0 → ¬ 0 < 𝐴))
6867adantrd 491 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 0 < 𝐴))
6964, 68jaod 860 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 0 < 𝐴))
7062, 69orim12d 967 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐵 = -∞ ∨ ¬ 0 < 𝐴)))
71 ianor 984 . . . . . . 7 (¬ (0 < 𝐴𝐵 = -∞) ↔ (¬ 0 < 𝐴 ∨ ¬ 𝐵 = -∞))
72 orcom 871 . . . . . . 7 ((¬ 0 < 𝐴 ∨ ¬ 𝐵 = -∞) ↔ (¬ 𝐵 = -∞ ∨ ¬ 0 < 𝐴))
7371, 72bitri 275 . . . . . 6 (¬ (0 < 𝐴𝐵 = -∞) ↔ (¬ 𝐵 = -∞ ∨ ¬ 0 < 𝐴))
7470, 73imbitrrdi 252 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (0 < 𝐴𝐵 = -∞)))
7542adantl 481 . . . . . . . . 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 491 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((0 < 𝐴𝐵 = +∞) → ¬ 𝐴 < 0))
7976, 78jaod 860 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) → ¬ 𝐴 < 0))
80 breq1 5146 . . . . . . . . . . . 12 (𝐵 = +∞ → (𝐵 < 0 ↔ +∞ < 0))
8133, 80mtbiri 327 . . . . . . . . . . 11 (𝐵 = +∞ → ¬ 𝐵 < 0)
8281con2i 139 . . . . . . . . . 10 (𝐵 < 0 → ¬ 𝐵 = +∞)
8382adantr 480 . . . . . . . . 9 ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 𝐵 = +∞)
8459con2i 139 . . . . . . . . . 10 (𝐵 = -∞ → ¬ 𝐵 = +∞)
8584adantl 481 . . . . . . . . 9 ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 𝐵 = +∞)
8683, 85jaoi 858 . . . . . . . 8 (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 𝐵 = +∞)
8786a1i 11 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 𝐵 = +∞))
8879, 87orim12d 967 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐴 < 0 ∨ ¬ 𝐵 = +∞)))
89 ianor 984 . . . . . 6 (¬ (𝐴 < 0 ∧ 𝐵 = +∞) ↔ (¬ 𝐴 < 0 ∨ ¬ 𝐵 = +∞))
9088, 89imbitrrdi 252 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))
9174, 90jcad 512 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))
92 ioran 986 . . . 4 (¬ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)) ↔ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))
9391, 92imbitrrdi 252 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))))
9452, 93jcad 512 . 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 927 . 2 ((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) ↔ (((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))))
96 ioran 986 . 2 (¬ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) ↔ (¬ ((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ ¬ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))))
9794, 95, 963imtr4g 296 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 395  wo 848   = wceq 1540  wcel 2108  wne 2940   class class class wbr 5143  0cc0 11155  +∞cpnf 11292  -∞cmnf 11293  *cxr 11294   < clt 11295
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-addrcl 11216  ax-rnegex 11226  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300
This theorem is referenced by:  xmulneg1  13311
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