Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elbigolo1 Structured version   Visualization version   GIF version
Theorem elbigolo1 47243
Description: A function (into the positive reals) is of order G(x) iff the quotient of the function and G(x) (also a function into the positive reals) is an eventually upper bounded function. (Contributed by AV, 20-May-2020.) (Proof shortened by II, 16-Feb-2023.)
Assertion
Ref Expression
elbigolo1 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) β†’ (𝐹 ∈ (ΞŸβ€˜πΊ) ↔ (𝐹 /f 𝐺) ∈ ≀𝑂(1)))
Proof of Theorem elbigolo1
Dummy variables π‘š π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . . . . . . . . 12 (𝐹:π΄βŸΆβ„+ β†’ 𝐹:π΄βŸΆβ„+)
2 rpssre 12981 . . . . . . . . . . . . 13 ℝ+ βŠ† ℝ
32a1i 11 . . . . . . . . . . . 12 (𝐹:π΄βŸΆβ„+ β†’ ℝ+ βŠ† ℝ)
41, 3fssd 6736 . . . . . . . . . . 11 (𝐹:π΄βŸΆβ„+ β†’ 𝐹:π΄βŸΆβ„)
543ad2ant3 1136 . . . . . . . . . 10 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) β†’ 𝐹:π΄βŸΆβ„)
65adantr 482 . . . . . . . . 9 (((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) ∧ (π‘₯ ∈ ℝ ∧ π‘š ∈ ℝ)) β†’ 𝐹:π΄βŸΆβ„)
76ffvelcdmda 7087 . . . . . . . 8 ((((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) ∧ (π‘₯ ∈ ℝ ∧ π‘š ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) β†’ (πΉβ€˜π‘¦) ∈ ℝ)
8 simplrr 777 . . . . . . . 8 ((((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) ∧ (π‘₯ ∈ ℝ ∧ π‘š ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) β†’ π‘š ∈ ℝ)
9 simpl2 1193 . . . . . . . . . 10 (((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) ∧ (π‘₯ ∈ ℝ ∧ π‘š ∈ ℝ)) β†’ 𝐺:π΄βŸΆβ„+)
109ffvelcdmda 7087 . . . . . . . . 9 ((((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) ∧ (π‘₯ ∈ ℝ ∧ π‘š ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) β†’ (πΊβ€˜π‘¦) ∈ ℝ+)
1110rpregt0d 13022 . . . . . . . 8 ((((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) ∧ (π‘₯ ∈ ℝ ∧ π‘š ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) β†’ ((πΊβ€˜π‘¦) ∈ ℝ ∧ 0 < (πΊβ€˜π‘¦)))
127, 8, 113jca 1129 . . . . . . 7 ((((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) ∧ (π‘₯ ∈ ℝ ∧ π‘š ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) β†’ ((πΉβ€˜π‘¦) ∈ ℝ ∧ π‘š ∈ ℝ ∧ ((πΊβ€˜π‘¦) ∈ ℝ ∧ 0 < (πΊβ€˜π‘¦))))
13 ledivmul2 12093 . . . . . . . 8 (((πΉβ€˜π‘¦) ∈ ℝ ∧ π‘š ∈ ℝ ∧ ((πΊβ€˜π‘¦) ∈ ℝ ∧ 0 < (πΊβ€˜π‘¦))) β†’ (((πΉβ€˜π‘¦) / (πΊβ€˜π‘¦)) ≀ π‘š ↔ (πΉβ€˜π‘¦) ≀ (π‘š Β· (πΊβ€˜π‘¦))))
1413bicomd 222 . . . . . . 7 (((πΉβ€˜π‘¦) ∈ ℝ ∧ π‘š ∈ ℝ ∧ ((πΊβ€˜π‘¦) ∈ ℝ ∧ 0 < (πΊβ€˜π‘¦))) β†’ ((πΉβ€˜π‘¦) ≀ (π‘š Β· (πΊβ€˜π‘¦)) ↔ ((πΉβ€˜π‘¦) / (πΊβ€˜π‘¦)) ≀ π‘š))
1512, 14syl 17 . . . . . 6 ((((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) ∧ (π‘₯ ∈ ℝ ∧ π‘š ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) β†’ ((πΉβ€˜π‘¦) ≀ (π‘š Β· (πΊβ€˜π‘¦)) ↔ ((πΉβ€˜π‘¦) / (πΊβ€˜π‘¦)) ≀ π‘š))
16 id 22 . . . . . . . . . . . . 13 (𝐺:π΄βŸΆβ„+ β†’ 𝐺:π΄βŸΆβ„+)
172a1i 11 . . . . . . . . . . . . 13 (𝐺:π΄βŸΆβ„+ β†’ ℝ+ βŠ† ℝ)
1816, 17fssd 6736 . . . . . . . . . . . 12 (𝐺:π΄βŸΆβ„+ β†’ 𝐺:π΄βŸΆβ„)
19183ad2ant2 1135 . . . . . . . . . . 11 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) β†’ 𝐺:π΄βŸΆβ„)
20 reex 11201 . . . . . . . . . . . . 13 ℝ ∈ V
2120ssex 5322 . . . . . . . . . . . 12 (𝐴 βŠ† ℝ β†’ 𝐴 ∈ V)
22213ad2ant1 1134 . . . . . . . . . . 11 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) β†’ 𝐴 ∈ V)
235, 19, 223jca 1129 . . . . . . . . . 10 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) β†’ (𝐹:π΄βŸΆβ„ ∧ 𝐺:π΄βŸΆβ„ ∧ 𝐴 ∈ V))
2423adantr 482 . . . . . . . . 9 (((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) ∧ (π‘₯ ∈ ℝ ∧ π‘š ∈ ℝ)) β†’ (𝐹:π΄βŸΆβ„ ∧ 𝐺:π΄βŸΆβ„ ∧ 𝐴 ∈ V))
2524adantr 482 . . . . . . . 8 ((((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) ∧ (π‘₯ ∈ ℝ ∧ π‘š ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) β†’ (𝐹:π΄βŸΆβ„ ∧ 𝐺:π΄βŸΆβ„ ∧ 𝐴 ∈ V))
26 ffun 6721 . . . . . . . . . . . . . . . 16 (𝐺:π΄βŸΆβ„+ β†’ Fun 𝐺)
2726adantl 483 . . . . . . . . . . . . . . 15 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+) β†’ Fun 𝐺)
2821anim1ci 617 . . . . . . . . . . . . . . . 16 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+) β†’ (𝐺:π΄βŸΆβ„+ ∧ 𝐴 ∈ V))
29 fex 7228 . . . . . . . . . . . . . . . 16 ((𝐺:π΄βŸΆβ„+ ∧ 𝐴 ∈ V) β†’ 𝐺 ∈ V)
3028, 29syl 17 . . . . . . . . . . . . . . 15 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+) β†’ 𝐺 ∈ V)
31 0red 11217 . . . . . . . . . . . . . . 15 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+) β†’ 0 ∈ ℝ)
32 frn 6725 . . . . . . . . . . . . . . . . 17 (𝐺:π΄βŸΆβ„+ β†’ ran 𝐺 βŠ† ℝ+)
33 0nrp 13009 . . . . . . . . . . . . . . . . . . 19 Β¬ 0 ∈ ℝ+
34 id 22 . . . . . . . . . . . . . . . . . . . 20 (ran 𝐺 βŠ† ℝ+ β†’ ran 𝐺 βŠ† ℝ+)
3534ssneld 3985 . . . . . . . . . . . . . . . . . . 19 (ran 𝐺 βŠ† ℝ+ β†’ (Β¬ 0 ∈ ℝ+ β†’ Β¬ 0 ∈ ran 𝐺))
3633, 35mpi 20 . . . . . . . . . . . . . . . . . 18 (ran 𝐺 βŠ† ℝ+ β†’ Β¬ 0 ∈ ran 𝐺)
37 df-nel 3048 . . . . . . . . . . . . . . . . . 18 (0 βˆ‰ ran 𝐺 ↔ Β¬ 0 ∈ ran 𝐺)
3836, 37sylibr 233 . . . . . . . . . . . . . . . . 17 (ran 𝐺 βŠ† ℝ+ β†’ 0 βˆ‰ ran 𝐺)
3932, 38syl 17 . . . . . . . . . . . . . . . 16 (𝐺:π΄βŸΆβ„+ β†’ 0 βˆ‰ ran 𝐺)
4039adantl 483 . . . . . . . . . . . . . . 15 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+) β†’ 0 βˆ‰ ran 𝐺)
41 suppdm 47191 . . . . . . . . . . . . . . 15 (((Fun 𝐺 ∧ 𝐺 ∈ V ∧ 0 ∈ ℝ) ∧ 0 βˆ‰ ran 𝐺) β†’ (𝐺 supp 0) = dom 𝐺)
4227, 30, 31, 40, 41syl31anc 1374 . . . . . . . . . . . . . 14 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+) β†’ (𝐺 supp 0) = dom 𝐺)
43 fdm 6727 . . . . . . . . . . . . . . 15 (𝐺:π΄βŸΆβ„+ β†’ dom 𝐺 = 𝐴)
4443adantl 483 . . . . . . . . . . . . . 14 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+) β†’ dom 𝐺 = 𝐴)
4542, 44eqtrd 2773 . . . . . . . . . . . . 13 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+) β†’ (𝐺 supp 0) = 𝐴)
46453adant3 1133 . . . . . . . . . . . 12 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) β†’ (𝐺 supp 0) = 𝐴)
4746eqcomd 2739 . . . . . . . . . . 11 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) β†’ 𝐴 = (𝐺 supp 0))
4847adantr 482 . . . . . . . . . 10 (((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) ∧ (π‘₯ ∈ ℝ ∧ π‘š ∈ ℝ)) β†’ 𝐴 = (𝐺 supp 0))
4948eleq2d 2820 . . . . . . . . 9 (((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) ∧ (π‘₯ ∈ ℝ ∧ π‘š ∈ ℝ)) β†’ (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ (𝐺 supp 0)))
5049biimpa 478 . . . . . . . 8 ((((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) ∧ (π‘₯ ∈ ℝ ∧ π‘š ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) β†’ 𝑦 ∈ (𝐺 supp 0))
51 refdivmptfv 47232 . . . . . . . 8 (((𝐹:π΄βŸΆβ„ ∧ 𝐺:π΄βŸΆβ„ ∧ 𝐴 ∈ V) ∧ 𝑦 ∈ (𝐺 supp 0)) β†’ ((𝐹 /f 𝐺)β€˜π‘¦) = ((πΉβ€˜π‘¦) / (πΊβ€˜π‘¦)))
5225, 50, 51syl2anc 585 . . . . . . 7 ((((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) ∧ (π‘₯ ∈ ℝ ∧ π‘š ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) β†’ ((𝐹 /f 𝐺)β€˜π‘¦) = ((πΉβ€˜π‘¦) / (πΊβ€˜π‘¦)))
5352breq1d 5159 . . . . . 6 ((((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) ∧ (π‘₯ ∈ ℝ ∧ π‘š ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) β†’ (((𝐹 /f 𝐺)β€˜π‘¦) ≀ π‘š ↔ ((πΉβ€˜π‘¦) / (πΊβ€˜π‘¦)) ≀ π‘š))
5415, 53bitr4d 282 . . . . 5 ((((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) ∧ (π‘₯ ∈ ℝ ∧ π‘š ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) β†’ ((πΉβ€˜π‘¦) ≀ (π‘š Β· (πΊβ€˜π‘¦)) ↔ ((𝐹 /f 𝐺)β€˜π‘¦) ≀ π‘š))
5554imbi2d 341 . . . 4 ((((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) ∧ (π‘₯ ∈ ℝ ∧ π‘š ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) β†’ ((π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘¦) ≀ (π‘š Β· (πΊβ€˜π‘¦))) ↔ (π‘₯ ≀ 𝑦 β†’ ((𝐹 /f 𝐺)β€˜π‘¦) ≀ π‘š)))
5655ralbidva 3176 . . 3 (((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) ∧ (π‘₯ ∈ ℝ ∧ π‘š ∈ ℝ)) β†’ (βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘¦) ≀ (π‘š Β· (πΊβ€˜π‘¦))) ↔ βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ ((𝐹 /f 𝐺)β€˜π‘¦) ≀ π‘š)))
57562rexbidva 3218 . 2 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) β†’ (βˆƒπ‘₯ ∈ ℝ βˆƒπ‘š ∈ ℝ βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘¦) ≀ (π‘š Β· (πΊβ€˜π‘¦))) ↔ βˆƒπ‘₯ ∈ ℝ βˆƒπ‘š ∈ ℝ βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ ((𝐹 /f 𝐺)β€˜π‘¦) ≀ π‘š)))
58 simp1 1137 . . 3 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) β†’ 𝐴 βŠ† ℝ)
59 ssidd 4006 . . 3 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) β†’ 𝐴 βŠ† 𝐴)
60 elbigo2 47238 . . 3 (((𝐺:π΄βŸΆβ„ ∧ 𝐴 βŠ† ℝ) ∧ (𝐹:π΄βŸΆβ„ ∧ 𝐴 βŠ† 𝐴)) β†’ (𝐹 ∈ (ΞŸβ€˜πΊ) ↔ βˆƒπ‘₯ ∈ ℝ βˆƒπ‘š ∈ ℝ βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘¦) ≀ (π‘š Β· (πΊβ€˜π‘¦)))))
6119, 58, 5, 59, 60syl22anc 838 . 2 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) β†’ (𝐹 ∈ (ΞŸβ€˜πΊ) ↔ βˆƒπ‘₯ ∈ ℝ βˆƒπ‘š ∈ ℝ βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘¦) ≀ (π‘š Β· (πΊβ€˜π‘¦)))))
62 refdivmptf 47228 . . . . 5 ((𝐹:π΄βŸΆβ„ ∧ 𝐺:π΄βŸΆβ„ ∧ 𝐴 ∈ V) β†’ (𝐹 /f 𝐺):(𝐺 supp 0)βŸΆβ„)
6323, 62syl 17 . . . 4 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) β†’ (𝐹 /f 𝐺):(𝐺 supp 0)βŸΆβ„)
6447feq2d 6704 . . . 4 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) β†’ ((𝐹 /f 𝐺):π΄βŸΆβ„ ↔ (𝐹 /f 𝐺):(𝐺 supp 0)βŸΆβ„))
6563, 64mpbird 257 . . 3 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) β†’ (𝐹 /f 𝐺):π΄βŸΆβ„)
66 ello12 15460 . . 3 (((𝐹 /f 𝐺):π΄βŸΆβ„ ∧ 𝐴 βŠ† ℝ) β†’ ((𝐹 /f 𝐺) ∈ ≀𝑂(1) ↔ βˆƒπ‘₯ ∈ ℝ βˆƒπ‘š ∈ ℝ βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ ((𝐹 /f 𝐺)β€˜π‘¦) ≀ π‘š)))
6765, 58, 66syl2anc 585 . 2 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) β†’ ((𝐹 /f 𝐺) ∈ ≀𝑂(1) ↔ βˆƒπ‘₯ ∈ ℝ βˆƒπ‘š ∈ ℝ βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ ((𝐹 /f 𝐺)β€˜π‘¦) ≀ π‘š)))
6857, 61, 673bitr4d 311 1 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) β†’ (𝐹 ∈ (ΞŸβ€˜πΊ) ↔ (𝐹 /f 𝐺) ∈ ≀𝑂(1)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   βˆ‰ wnel 3047  βˆ€wral 3062  βˆƒwrex 3071  Vcvv 3475   βŠ† wss 3949   class class class wbr 5149  dom cdm 5677  ran crn 5678  Fun wfun 6538  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   supp csupp 8146  β„cr 11109  0cc0 11110   Β· cmul 11115   < clt 11248   ≀ cle 11249   / cdiv 11871  β„+crp 12974  β‰€π‘‚(1)clo1 15431   /f cfdiv 47223  ΞŸcbigo 47233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-po 5589  df-so 5590  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-supp 8147  df-er 8703  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-rp 12975  df-ico 13330  df-lo1 15435  df-fdiv 47224  df-bigo 47234
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator