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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elbigolo1 Structured version   Visualization version   GIF version
Theorem elbigolo1 47321
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 12983 . . . . . . . . . . . . 13 ℝ+ βŠ† ℝ
32a1i 11 . . . . . . . . . . . 12 (𝐹:π΄βŸΆβ„+ β†’ ℝ+ βŠ† ℝ)
41, 3fssd 6735 . . . . . . . . . . 11 (𝐹:π΄βŸΆβ„+ β†’ 𝐹:π΄βŸΆβ„)
543ad2ant3 1135 . . . . . . . . . 10 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) β†’ 𝐹:π΄βŸΆβ„)
65adantr 481 . . . . . . . . 9 (((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) ∧ (π‘₯ ∈ ℝ ∧ π‘š ∈ ℝ)) β†’ 𝐹:π΄βŸΆβ„)
76ffvelcdmda 7086 . . . . . . . 8 ((((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) ∧ (π‘₯ ∈ ℝ ∧ π‘š ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) β†’ (πΉβ€˜π‘¦) ∈ ℝ)
8 simplrr 776 . . . . . . . 8 ((((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) ∧ (π‘₯ ∈ ℝ ∧ π‘š ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) β†’ π‘š ∈ ℝ)
9 simpl2 1192 . . . . . . . . . 10 (((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) ∧ (π‘₯ ∈ ℝ ∧ π‘š ∈ ℝ)) β†’ 𝐺:π΄βŸΆβ„+)
109ffvelcdmda 7086 . . . . . . . . 9 ((((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) ∧ (π‘₯ ∈ ℝ ∧ π‘š ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) β†’ (πΊβ€˜π‘¦) ∈ ℝ+)
1110rpregt0d 13024 . . . . . . . 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 1128 . . . . . . 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 12095 . . . . . . . 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 6735 . . . . . . . . . . . 12 (𝐺:π΄βŸΆβ„+ β†’ 𝐺:π΄βŸΆβ„)
19183ad2ant2 1134 . . . . . . . . . . 11 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) β†’ 𝐺:π΄βŸΆβ„)
20 reex 11203 . . . . . . . . . . . . 13 ℝ ∈ V
2120ssex 5321 . . . . . . . . . . . 12 (𝐴 βŠ† ℝ β†’ 𝐴 ∈ V)
22213ad2ant1 1133 . . . . . . . . . . 11 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) β†’ 𝐴 ∈ V)
235, 19, 223jca 1128 . . . . . . . . . 10 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) β†’ (𝐹:π΄βŸΆβ„ ∧ 𝐺:π΄βŸΆβ„ ∧ 𝐴 ∈ V))
2423adantr 481 . . . . . . . . 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 481 . . . . . . . 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 6720 . . . . . . . . . . . . . . . 16 (𝐺:π΄βŸΆβ„+ β†’ Fun 𝐺)
2726adantl 482 . . . . . . . . . . . . . . 15 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+) β†’ Fun 𝐺)
2821anim1ci 616 . . . . . . . . . . . . . . . 16 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+) β†’ (𝐺:π΄βŸΆβ„+ ∧ 𝐴 ∈ V))
29 fex 7230 . . . . . . . . . . . . . . . 16 ((𝐺:π΄βŸΆβ„+ ∧ 𝐴 ∈ V) β†’ 𝐺 ∈ V)
3028, 29syl 17 . . . . . . . . . . . . . . 15 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+) β†’ 𝐺 ∈ V)
31 0red 11219 . . . . . . . . . . . . . . 15 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+) β†’ 0 ∈ ℝ)
32 frn 6724 . . . . . . . . . . . . . . . . 17 (𝐺:π΄βŸΆβ„+ β†’ ran 𝐺 βŠ† ℝ+)
33 0nrp 13011 . . . . . . . . . . . . . . . . . . 19 Β¬ 0 ∈ ℝ+
34 id 22 . . . . . . . . . . . . . . . . . . . 20 (ran 𝐺 βŠ† ℝ+ β†’ ran 𝐺 βŠ† ℝ+)
3534ssneld 3984 . . . . . . . . . . . . . . . . . . 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 3047 . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . 15 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+) β†’ 0 βˆ‰ ran 𝐺)
41 suppdm 47269 . . . . . . . . . . . . . . 15 (((Fun 𝐺 ∧ 𝐺 ∈ V ∧ 0 ∈ ℝ) ∧ 0 βˆ‰ ran 𝐺) β†’ (𝐺 supp 0) = dom 𝐺)
4227, 30, 31, 40, 41syl31anc 1373 . . . . . . . . . . . . . 14 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+) β†’ (𝐺 supp 0) = dom 𝐺)
43 fdm 6726 . . . . . . . . . . . . . . 15 (𝐺:π΄βŸΆβ„+ β†’ dom 𝐺 = 𝐴)
4443adantl 482 . . . . . . . . . . . . . 14 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+) β†’ dom 𝐺 = 𝐴)
4542, 44eqtrd 2772 . . . . . . . . . . . . 13 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+) β†’ (𝐺 supp 0) = 𝐴)
46453adant3 1132 . . . . . . . . . . . 12 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) β†’ (𝐺 supp 0) = 𝐴)
4746eqcomd 2738 . . . . . . . . . . 11 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) β†’ 𝐴 = (𝐺 supp 0))
4847adantr 481 . . . . . . . . . 10 (((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) ∧ (π‘₯ ∈ ℝ ∧ π‘š ∈ ℝ)) β†’ 𝐴 = (𝐺 supp 0))
4948eleq2d 2819 . . . . . . . . 9 (((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) ∧ (π‘₯ ∈ ℝ ∧ π‘š ∈ ℝ)) β†’ (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ (𝐺 supp 0)))
5049biimpa 477 . . . . . . . 8 ((((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) ∧ (π‘₯ ∈ ℝ ∧ π‘š ∈ ℝ)) ∧ 𝑦 ∈ 𝐴) β†’ 𝑦 ∈ (𝐺 supp 0))
51 refdivmptfv 47310 . . . . . . . 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 584 . . . . . . 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 5158 . . . . . 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 281 . . . . 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 340 . . . 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 3175 . . 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 3217 . 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 1136 . . 3 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) β†’ 𝐴 βŠ† ℝ)
59 ssidd 4005 . . 3 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) β†’ 𝐴 βŠ† 𝐴)
60 elbigo2 47316 . . 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 837 . 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 47306 . . . . 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 6703 . . . 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 256 . . 3 ((𝐴 βŠ† ℝ ∧ 𝐺:π΄βŸΆβ„+ ∧ 𝐹:π΄βŸΆβ„+) β†’ (𝐹 /f 𝐺):π΄βŸΆβ„)
66 ello12 15462 . . 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 584 . 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 310 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   βˆ‰ wnel 3046  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474   βŠ† wss 3948   class class class wbr 5148  dom cdm 5676  ran crn 5677  Fun wfun 6537  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411   supp csupp 8148  β„cr 11111  0cc0 11112   Β· cmul 11117   < clt 11250   ≀ cle 11251   / cdiv 11873  β„+crp 12976  β‰€π‘‚(1)clo1 15433   /f cfdiv 47301  ΞŸcbigo 47311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-supp 8149  df-er 8705  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-rp 12977  df-ico 13332  df-lo1 15437  df-fdiv 47302  df-bigo 47312
This theorem is referenced by: (None)
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