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Theorem rlim 15439
Description: Express the predicate: The limit of complex number function 𝐹 is 𝐢, or 𝐹 converges to 𝐢, in the real sense. This means that for any real π‘₯, no matter how small, there always exists a number 𝑦 such that the absolute difference of any number in the function beyond 𝑦 and the limit is less than π‘₯. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Apr-2015.)
Hypotheses
Ref Expression
rlim.1 (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)
rlim.2 (πœ‘ β†’ 𝐴 βŠ† ℝ)
rlim.4 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ (πΉβ€˜π‘§) = 𝐡)
Assertion
Ref Expression
rlim (πœ‘ β†’ (𝐹 β‡π‘Ÿ 𝐢 ↔ (𝐢 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ 𝐴 (𝑦 ≀ 𝑧 β†’ (absβ€˜(𝐡 βˆ’ 𝐢)) < π‘₯))))
Distinct variable groups:   𝑧,𝐴   π‘₯,𝑦,𝑧,𝐢   π‘₯,𝐹,𝑦,𝑧   πœ‘,π‘₯,𝑦,𝑧
Allowed substitution hints:   𝐴(π‘₯,𝑦)   𝐡(π‘₯,𝑦,𝑧)

Proof of Theorem rlim
Dummy variables 𝑀 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rlimrel 15437 . . . . 5 Rel β‡π‘Ÿ
21brrelex2i 5734 . . . 4 (𝐹 β‡π‘Ÿ 𝐢 β†’ 𝐢 ∈ V)
32a1i 11 . . 3 (πœ‘ β†’ (𝐹 β‡π‘Ÿ 𝐢 β†’ 𝐢 ∈ V))
4 elex 3493 . . . . 5 (𝐢 ∈ β„‚ β†’ 𝐢 ∈ V)
54ad2antrl 727 . . . 4 ((𝐹 ∈ (β„‚ ↑pm ℝ) ∧ (𝐢 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯))) β†’ 𝐢 ∈ V)
65a1i 11 . . 3 (πœ‘ β†’ ((𝐹 ∈ (β„‚ ↑pm ℝ) ∧ (𝐢 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯))) β†’ 𝐢 ∈ V))
7 rlim.1 . . . . 5 (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)
8 rlim.2 . . . . 5 (πœ‘ β†’ 𝐴 βŠ† ℝ)
9 cnex 11191 . . . . . 6 β„‚ ∈ V
10 reex 11201 . . . . . 6 ℝ ∈ V
11 elpm2r 8839 . . . . . 6 (((β„‚ ∈ V ∧ ℝ ∈ V) ∧ (𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† ℝ)) β†’ 𝐹 ∈ (β„‚ ↑pm ℝ))
129, 10, 11mpanl12 701 . . . . 5 ((𝐹:π΄βŸΆβ„‚ ∧ 𝐴 βŠ† ℝ) β†’ 𝐹 ∈ (β„‚ ↑pm ℝ))
137, 8, 12syl2anc 585 . . . 4 (πœ‘ β†’ 𝐹 ∈ (β„‚ ↑pm ℝ))
14 eleq1 2822 . . . . . . . . 9 (𝑓 = 𝐹 β†’ (𝑓 ∈ (β„‚ ↑pm ℝ) ↔ 𝐹 ∈ (β„‚ ↑pm ℝ)))
15 eleq1 2822 . . . . . . . . 9 (𝑀 = 𝐢 β†’ (𝑀 ∈ β„‚ ↔ 𝐢 ∈ β„‚))
1614, 15bi2anan9 638 . . . . . . . 8 ((𝑓 = 𝐹 ∧ 𝑀 = 𝐢) β†’ ((𝑓 ∈ (β„‚ ↑pm ℝ) ∧ 𝑀 ∈ β„‚) ↔ (𝐹 ∈ (β„‚ ↑pm ℝ) ∧ 𝐢 ∈ β„‚)))
17 simpl 484 . . . . . . . . . . . 12 ((𝑓 = 𝐹 ∧ 𝑀 = 𝐢) β†’ 𝑓 = 𝐹)
1817dmeqd 5906 . . . . . . . . . . 11 ((𝑓 = 𝐹 ∧ 𝑀 = 𝐢) β†’ dom 𝑓 = dom 𝐹)
19 fveq1 6891 . . . . . . . . . . . . . . 15 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘§) = (πΉβ€˜π‘§))
20 oveq12 7418 . . . . . . . . . . . . . . 15 (((π‘“β€˜π‘§) = (πΉβ€˜π‘§) ∧ 𝑀 = 𝐢) β†’ ((π‘“β€˜π‘§) βˆ’ 𝑀) = ((πΉβ€˜π‘§) βˆ’ 𝐢))
2119, 20sylan 581 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹 ∧ 𝑀 = 𝐢) β†’ ((π‘“β€˜π‘§) βˆ’ 𝑀) = ((πΉβ€˜π‘§) βˆ’ 𝐢))
2221fveq2d 6896 . . . . . . . . . . . . 13 ((𝑓 = 𝐹 ∧ 𝑀 = 𝐢) β†’ (absβ€˜((π‘“β€˜π‘§) βˆ’ 𝑀)) = (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)))
2322breq1d 5159 . . . . . . . . . . . 12 ((𝑓 = 𝐹 ∧ 𝑀 = 𝐢) β†’ ((absβ€˜((π‘“β€˜π‘§) βˆ’ 𝑀)) < π‘₯ ↔ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯))
2423imbi2d 341 . . . . . . . . . . 11 ((𝑓 = 𝐹 ∧ 𝑀 = 𝐢) β†’ ((𝑦 ≀ 𝑧 β†’ (absβ€˜((π‘“β€˜π‘§) βˆ’ 𝑀)) < π‘₯) ↔ (𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯)))
2518, 24raleqbidv 3343 . . . . . . . . . 10 ((𝑓 = 𝐹 ∧ 𝑀 = 𝐢) β†’ (βˆ€π‘§ ∈ dom 𝑓(𝑦 ≀ 𝑧 β†’ (absβ€˜((π‘“β€˜π‘§) βˆ’ 𝑀)) < π‘₯) ↔ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯)))
2625rexbidv 3179 . . . . . . . . 9 ((𝑓 = 𝐹 ∧ 𝑀 = 𝐢) β†’ (βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝑓(𝑦 ≀ 𝑧 β†’ (absβ€˜((π‘“β€˜π‘§) βˆ’ 𝑀)) < π‘₯) ↔ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯)))
2726ralbidv 3178 . . . . . . . 8 ((𝑓 = 𝐹 ∧ 𝑀 = 𝐢) β†’ (βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝑓(𝑦 ≀ 𝑧 β†’ (absβ€˜((π‘“β€˜π‘§) βˆ’ 𝑀)) < π‘₯) ↔ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯)))
2816, 27anbi12d 632 . . . . . . 7 ((𝑓 = 𝐹 ∧ 𝑀 = 𝐢) β†’ (((𝑓 ∈ (β„‚ ↑pm ℝ) ∧ 𝑀 ∈ β„‚) ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝑓(𝑦 ≀ 𝑧 β†’ (absβ€˜((π‘“β€˜π‘§) βˆ’ 𝑀)) < π‘₯)) ↔ ((𝐹 ∈ (β„‚ ↑pm ℝ) ∧ 𝐢 ∈ β„‚) ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯))))
29 df-rlim 15433 . . . . . . 7 β‡π‘Ÿ = {βŸ¨π‘“, π‘€βŸ© ∣ ((𝑓 ∈ (β„‚ ↑pm ℝ) ∧ 𝑀 ∈ β„‚) ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝑓(𝑦 ≀ 𝑧 β†’ (absβ€˜((π‘“β€˜π‘§) βˆ’ 𝑀)) < π‘₯))}
3028, 29brabga 5535 . . . . . 6 ((𝐹 ∈ (β„‚ ↑pm ℝ) ∧ 𝐢 ∈ V) β†’ (𝐹 β‡π‘Ÿ 𝐢 ↔ ((𝐹 ∈ (β„‚ ↑pm ℝ) ∧ 𝐢 ∈ β„‚) ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯))))
31 anass 470 . . . . . 6 (((𝐹 ∈ (β„‚ ↑pm ℝ) ∧ 𝐢 ∈ β„‚) ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯)) ↔ (𝐹 ∈ (β„‚ ↑pm ℝ) ∧ (𝐢 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯))))
3230, 31bitrdi 287 . . . . 5 ((𝐹 ∈ (β„‚ ↑pm ℝ) ∧ 𝐢 ∈ V) β†’ (𝐹 β‡π‘Ÿ 𝐢 ↔ (𝐹 ∈ (β„‚ ↑pm ℝ) ∧ (𝐢 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯)))))
3332ex 414 . . . 4 (𝐹 ∈ (β„‚ ↑pm ℝ) β†’ (𝐢 ∈ V β†’ (𝐹 β‡π‘Ÿ 𝐢 ↔ (𝐹 ∈ (β„‚ ↑pm ℝ) ∧ (𝐢 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯))))))
3413, 33syl 17 . . 3 (πœ‘ β†’ (𝐢 ∈ V β†’ (𝐹 β‡π‘Ÿ 𝐢 ↔ (𝐹 ∈ (β„‚ ↑pm ℝ) ∧ (𝐢 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯))))))
353, 6, 34pm5.21ndd 381 . 2 (πœ‘ β†’ (𝐹 β‡π‘Ÿ 𝐢 ↔ (𝐹 ∈ (β„‚ ↑pm ℝ) ∧ (𝐢 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯)))))
3613biantrurd 534 . 2 (πœ‘ β†’ ((𝐢 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯)) ↔ (𝐹 ∈ (β„‚ ↑pm ℝ) ∧ (𝐢 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯)))))
377fdmd 6729 . . . . . . 7 (πœ‘ β†’ dom 𝐹 = 𝐴)
3837raleqdv 3326 . . . . . 6 (πœ‘ β†’ (βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯) ↔ βˆ€π‘§ ∈ 𝐴 (𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯)))
39 rlim.4 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ (πΉβ€˜π‘§) = 𝐡)
4039fvoveq1d 7431 . . . . . . . . 9 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) = (absβ€˜(𝐡 βˆ’ 𝐢)))
4140breq1d 5159 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ ((absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯ ↔ (absβ€˜(𝐡 βˆ’ 𝐢)) < π‘₯))
4241imbi2d 341 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ ((𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯) ↔ (𝑦 ≀ 𝑧 β†’ (absβ€˜(𝐡 βˆ’ 𝐢)) < π‘₯)))
4342ralbidva 3176 . . . . . 6 (πœ‘ β†’ (βˆ€π‘§ ∈ 𝐴 (𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯) ↔ βˆ€π‘§ ∈ 𝐴 (𝑦 ≀ 𝑧 β†’ (absβ€˜(𝐡 βˆ’ 𝐢)) < π‘₯)))
4438, 43bitrd 279 . . . . 5 (πœ‘ β†’ (βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯) ↔ βˆ€π‘§ ∈ 𝐴 (𝑦 ≀ 𝑧 β†’ (absβ€˜(𝐡 βˆ’ 𝐢)) < π‘₯)))
4544rexbidv 3179 . . . 4 (πœ‘ β†’ (βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯) ↔ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ 𝐴 (𝑦 ≀ 𝑧 β†’ (absβ€˜(𝐡 βˆ’ 𝐢)) < π‘₯)))
4645ralbidv 3178 . . 3 (πœ‘ β†’ (βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯) ↔ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ 𝐴 (𝑦 ≀ 𝑧 β†’ (absβ€˜(𝐡 βˆ’ 𝐢)) < π‘₯)))
4746anbi2d 630 . 2 (πœ‘ β†’ ((𝐢 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ dom 𝐹(𝑦 ≀ 𝑧 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯)) ↔ (𝐢 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ 𝐴 (𝑦 ≀ 𝑧 β†’ (absβ€˜(𝐡 βˆ’ 𝐢)) < π‘₯))))
4835, 36, 473bitr2d 307 1 (πœ‘ β†’ (𝐹 β‡π‘Ÿ 𝐢 ↔ (𝐢 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ βˆ€π‘§ ∈ 𝐴 (𝑦 ≀ 𝑧 β†’ (absβ€˜(𝐡 βˆ’ 𝐢)) < π‘₯))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071  Vcvv 3475   βŠ† wss 3949   class class class wbr 5149  dom cdm 5677  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ↑pm cpm 8821  β„‚cc 11108  β„cr 11109   < clt 11248   ≀ cle 11249   βˆ’ cmin 11444  β„+crp 12974  abscabs 15181   β‡π‘Ÿ crli 15429
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-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  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-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-pm 8823  df-rlim 15433
This theorem is referenced by:  rlim2  15440  rlimcl  15447  rlimclim  15490  rlimres  15502  caurcvgr  15620
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