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Theorem xmullem2 12300
Description: Lemma for xmulneg1 12304. (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 10297 . . . . . . . . . . . 12 -∞ ≠ +∞
2 eqeq1 2775 . . . . . . . . . . . . 13 (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞))
32necon3bbid 2980 . . . . . . . . . . . 12 (𝐴 = -∞ → (¬ 𝐴 = +∞ ↔ -∞ ≠ +∞))
41, 3mpbiri 248 . . . . . . . . . . 11 (𝐴 = -∞ → ¬ 𝐴 = +∞)
54con2i 136 . . . . . . . . . 10 (𝐴 = +∞ → ¬ 𝐴 = -∞)
65adantl 467 . . . . . . . . 9 ((0 < 𝐵𝐴 = +∞) → ¬ 𝐴 = -∞)
7 0xr 10288 . . . . . . . . . . . . 13 0 ∈ ℝ*
8 nltmnf 12168 . . . . . . . . . . . . 13 (0 ∈ ℝ* → ¬ 0 < -∞)
97, 8ax-mp 5 . . . . . . . . . . . 12 ¬ 0 < -∞
10 breq2 4790 . . . . . . . . . . . 12 (𝐴 = -∞ → (0 < 𝐴 ↔ 0 < -∞))
119, 10mtbiri 316 . . . . . . . . . . 11 (𝐴 = -∞ → ¬ 0 < 𝐴)
1211con2i 136 . . . . . . . . . 10 (0 < 𝐴 → ¬ 𝐴 = -∞)
1312adantr 466 . . . . . . . . 9 ((0 < 𝐴𝐵 = +∞) → ¬ 𝐴 = -∞)
146, 13jaoi 846 . . . . . . . 8 (((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) → ¬ 𝐴 = -∞)
1514a1i 11 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) → ¬ 𝐴 = -∞))
16 simpr 471 . . . . . . . . . 10 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → 𝐵 ∈ ℝ*)
17 xrltnsym 12175 . . . . . . . . . 10 ((𝐵 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝐵 < 0 → ¬ 0 < 𝐵))
1816, 7, 17sylancl 574 . . . . . . . . 9 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐵 < 0 → ¬ 0 < 𝐵))
1918adantrd 479 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 0 < 𝐵))
20 breq2 4790 . . . . . . . . . . 11 (𝐵 = -∞ → (0 < 𝐵 ↔ 0 < -∞))
219, 20mtbiri 316 . . . . . . . . . 10 (𝐵 = -∞ → ¬ 0 < 𝐵)
2221adantl 467 . . . . . . . . 9 ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 0 < 𝐵)
2322a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 0 < 𝐵))
2419, 23jaod 848 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 0 < 𝐵))
2515, 24orim12d 949 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐴 = -∞ ∨ ¬ 0 < 𝐵)))
26 ianor 966 . . . . . . 7 (¬ (0 < 𝐵𝐴 = -∞) ↔ (¬ 0 < 𝐵 ∨ ¬ 𝐴 = -∞))
27 orcom 859 . . . . . . 7 ((¬ 0 < 𝐵 ∨ ¬ 𝐴 = -∞) ↔ (¬ 𝐴 = -∞ ∨ ¬ 0 < 𝐵))
2826, 27bitri 264 . . . . . 6 (¬ (0 < 𝐵𝐴 = -∞) ↔ (¬ 𝐴 = -∞ ∨ ¬ 0 < 𝐵))
2925, 28syl6ibr 242 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (0 < 𝐵𝐴 = -∞)))
3018con2d 131 . . . . . . . . 9 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (0 < 𝐵 → ¬ 𝐵 < 0))
3130adantrd 479 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((0 < 𝐵𝐴 = +∞) → ¬ 𝐵 < 0))
32 pnfnlt 12167 . . . . . . . . . . 11 (0 ∈ ℝ* → ¬ +∞ < 0)
337, 32ax-mp 5 . . . . . . . . . 10 ¬ +∞ < 0
34 simpr 471 . . . . . . . . . . 11 ((0 < 𝐴𝐵 = +∞) → 𝐵 = +∞)
3534breq1d 4796 . . . . . . . . . 10 ((0 < 𝐴𝐵 = +∞) → (𝐵 < 0 ↔ +∞ < 0))
3633, 35mtbiri 316 . . . . . . . . 9 ((0 < 𝐴𝐵 = +∞) → ¬ 𝐵 < 0)
3736a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((0 < 𝐴𝐵 = +∞) → ¬ 𝐵 < 0))
3831, 37jaod 848 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) → ¬ 𝐵 < 0))
394a1i 11 . . . . . . . . 9 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 = -∞ → ¬ 𝐴 = +∞))
4039adantld 478 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 𝐴 = +∞))
41 breq1 4789 . . . . . . . . . . . 12 (𝐴 = +∞ → (𝐴 < 0 ↔ +∞ < 0))
4233, 41mtbiri 316 . . . . . . . . . . 11 (𝐴 = +∞ → ¬ 𝐴 < 0)
4342con2i 136 . . . . . . . . . 10 (𝐴 < 0 → ¬ 𝐴 = +∞)
4443adantr 466 . . . . . . . . 9 ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 𝐴 = +∞)
4544a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 𝐴 = +∞))
4640, 45jaod 848 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 𝐴 = +∞))
4738, 46orim12d 949 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐵 < 0 ∨ ¬ 𝐴 = +∞)))
48 ianor 966 . . . . . 6 (¬ (𝐵 < 0 ∧ 𝐴 = +∞) ↔ (¬ 𝐵 < 0 ∨ ¬ 𝐴 = +∞))
4947, 48syl6ibr 242 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (𝐵 < 0 ∧ 𝐴 = +∞)))
5029, 49jcad 502 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞))))
51 ioran 968 . . . 4 (¬ ((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ↔ (¬ (0 < 𝐵𝐴 = -∞) ∧ ¬ (𝐵 < 0 ∧ 𝐴 = +∞)))
5250, 51syl6ibr 242 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ ((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞))))
5321con2i 136 . . . . . . . . . 10 (0 < 𝐵 → ¬ 𝐵 = -∞)
5453adantr 466 . . . . . . . . 9 ((0 < 𝐵𝐴 = +∞) → ¬ 𝐵 = -∞)
5554a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((0 < 𝐵𝐴 = +∞) → ¬ 𝐵 = -∞))
56 pnfnemnf 10296 . . . . . . . . . . 11 +∞ ≠ -∞
57 eqeq1 2775 . . . . . . . . . . . 12 (𝐵 = +∞ → (𝐵 = -∞ ↔ +∞ = -∞))
5857necon3bbid 2980 . . . . . . . . . . 11 (𝐵 = +∞ → (¬ 𝐵 = -∞ ↔ +∞ ≠ -∞))
5956, 58mpbiri 248 . . . . . . . . . 10 (𝐵 = +∞ → ¬ 𝐵 = -∞)
6059adantl 467 . . . . . . . . 9 ((0 < 𝐴𝐵 = +∞) → ¬ 𝐵 = -∞)
6160a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((0 < 𝐴𝐵 = +∞) → ¬ 𝐵 = -∞))
6255, 61jaod 848 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) → ¬ 𝐵 = -∞))
6311adantl 467 . . . . . . . . 9 ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 0 < 𝐴)
6463a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 0 < 𝐴))
65 simpl 468 . . . . . . . . . 10 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → 𝐴 ∈ ℝ*)
66 xrltnsym 12175 . . . . . . . . . 10 ((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝐴 < 0 → ¬ 0 < 𝐴))
6765, 7, 66sylancl 574 . . . . . . . . 9 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 < 0 → ¬ 0 < 𝐴))
6867adantrd 479 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 0 < 𝐴))
6964, 68jaod 848 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 0 < 𝐴))
7062, 69orim12d 949 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐵 = -∞ ∨ ¬ 0 < 𝐴)))
71 ianor 966 . . . . . . 7 (¬ (0 < 𝐴𝐵 = -∞) ↔ (¬ 0 < 𝐴 ∨ ¬ 𝐵 = -∞))
72 orcom 859 . . . . . . 7 ((¬ 0 < 𝐴 ∨ ¬ 𝐵 = -∞) ↔ (¬ 𝐵 = -∞ ∨ ¬ 0 < 𝐴))
7371, 72bitri 264 . . . . . 6 (¬ (0 < 𝐴𝐵 = -∞) ↔ (¬ 𝐵 = -∞ ∨ ¬ 0 < 𝐴))
7470, 73syl6ibr 242 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (0 < 𝐴𝐵 = -∞)))
7542adantl 467 . . . . . . . . 9 ((0 < 𝐵𝐴 = +∞) → ¬ 𝐴 < 0)
7675a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((0 < 𝐵𝐴 = +∞) → ¬ 𝐴 < 0))
7767con2d 131 . . . . . . . . 9 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (0 < 𝐴 → ¬ 𝐴 < 0))
7877adantrd 479 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((0 < 𝐴𝐵 = +∞) → ¬ 𝐴 < 0))
7976, 78jaod 848 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) → ¬ 𝐴 < 0))
80 breq1 4789 . . . . . . . . . . . 12 (𝐵 = +∞ → (𝐵 < 0 ↔ +∞ < 0))
8133, 80mtbiri 316 . . . . . . . . . . 11 (𝐵 = +∞ → ¬ 𝐵 < 0)
8281con2i 136 . . . . . . . . . 10 (𝐵 < 0 → ¬ 𝐵 = +∞)
8382adantr 466 . . . . . . . . 9 ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ 𝐵 = +∞)
8459con2i 136 . . . . . . . . . 10 (𝐵 = -∞ → ¬ 𝐵 = +∞)
8584adantl 467 . . . . . . . . 9 ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ 𝐵 = +∞)
8683, 85jaoi 846 . . . . . . . 8 (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 𝐵 = +∞)
8786a1i 11 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ 𝐵 = +∞))
8879, 87orim12d 949 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ 𝐴 < 0 ∨ ¬ 𝐵 = +∞)))
89 ianor 966 . . . . . 6 (¬ (𝐴 < 0 ∧ 𝐵 = +∞) ↔ (¬ 𝐴 < 0 ∨ ¬ 𝐵 = +∞))
9088, 89syl6ibr 242 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))
9174, 90jcad 502 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞))))
92 ioran 968 . . . 4 (¬ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)) ↔ (¬ (0 < 𝐴𝐵 = -∞) ∧ ¬ (𝐴 < 0 ∧ 𝐵 = +∞)))
9391, 92syl6ibr 242 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))))
9452, 93jcad 502 . 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 912 . 2 ((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) ↔ (((0 < 𝐵𝐴 = +∞) ∨ (0 < 𝐴𝐵 = +∞)) ∨ ((𝐵 < 0 ∧ 𝐴 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))))
96 ioran 968 . 2 (¬ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) ↔ (¬ ((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∧ ¬ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))))
9794, 95, 963imtr4g 285 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 382  wo 836   = wceq 1631  wcel 2145  wne 2943   class class class wbr 4786  0cc0 10138  +∞cpnf 10273  -∞cmnf 10274  *cxr 10275   < clt 10276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-i2m1 10206  ax-1ne0 10207  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-po 5170  df-so 5171  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281
This theorem is referenced by:  xmulneg1  12304
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