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Theorem ellimc3 25620
Description: Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.)
Hypotheses
Ref Expression
ellimc3.f (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)
ellimc3.a (πœ‘ β†’ 𝐴 βŠ† β„‚)
ellimc3.b (πœ‘ β†’ 𝐡 ∈ β„‚)
Assertion
Ref Expression
ellimc3 (πœ‘ β†’ (𝐢 ∈ (𝐹 limβ„‚ 𝐡) ↔ (𝐢 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ+ βˆ€π‘§ ∈ 𝐴 ((𝑧 β‰  𝐡 ∧ (absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦) β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯))))
Distinct variable groups:   π‘₯,𝑦,𝑧,𝐴   π‘₯,𝐡,𝑦,𝑧   π‘₯,𝐢,𝑦,𝑧   πœ‘,π‘₯,𝑦,𝑧   π‘₯,𝐹,𝑦,𝑧

Proof of Theorem ellimc3
Dummy variables 𝑣 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ellimc3.f . . 3 (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)
2 ellimc3.a . . 3 (πœ‘ β†’ 𝐴 βŠ† β„‚)
3 ellimc3.b . . 3 (πœ‘ β†’ 𝐡 ∈ β„‚)
4 eqid 2732 . . 3 (TopOpenβ€˜β„‚fld) = (TopOpenβ€˜β„‚fld)
51, 2, 3, 4ellimc2 25618 . 2 (πœ‘ β†’ (𝐢 ∈ (𝐹 limβ„‚ 𝐡) ↔ (𝐢 ∈ β„‚ ∧ βˆ€π‘’ ∈ (TopOpenβ€˜β„‚fld)(𝐢 ∈ 𝑒 β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† 𝑒)))))
6 cnxmet 24509 . . . . . . . . 9 (abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚)
7 simplr 767 . . . . . . . . 9 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) β†’ 𝐢 ∈ β„‚)
8 simpr 485 . . . . . . . . 9 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) β†’ π‘₯ ∈ ℝ+)
9 blcntr 24139 . . . . . . . . 9 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 𝐢 ∈ β„‚ ∧ π‘₯ ∈ ℝ+) β†’ 𝐢 ∈ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯))
106, 7, 8, 9mp3an2i 1466 . . . . . . . 8 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) β†’ 𝐢 ∈ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯))
11 rpxr 12987 . . . . . . . . . . 11 (π‘₯ ∈ ℝ+ β†’ π‘₯ ∈ ℝ*)
1211adantl 482 . . . . . . . . . 10 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) β†’ π‘₯ ∈ ℝ*)
134cnfldtopn 24518 . . . . . . . . . . 11 (TopOpenβ€˜β„‚fld) = (MetOpenβ€˜(abs ∘ βˆ’ ))
1413blopn 24229 . . . . . . . . . 10 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 𝐢 ∈ β„‚ ∧ π‘₯ ∈ ℝ*) β†’ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) ∈ (TopOpenβ€˜β„‚fld))
156, 7, 12, 14mp3an2i 1466 . . . . . . . . 9 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) β†’ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) ∈ (TopOpenβ€˜β„‚fld))
16 eleq2 2822 . . . . . . . . . . 11 (𝑒 = (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) β†’ (𝐢 ∈ 𝑒 ↔ 𝐢 ∈ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)))
17 sseq2 4008 . . . . . . . . . . . . 13 (𝑒 = (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) β†’ ((𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† 𝑒 ↔ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)))
1817anbi2d 629 . . . . . . . . . . . 12 (𝑒 = (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) β†’ ((𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† 𝑒) ↔ (𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯))))
1918rexbidv 3178 . . . . . . . . . . 11 (𝑒 = (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) β†’ (βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† 𝑒) ↔ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯))))
2016, 19imbi12d 344 . . . . . . . . . 10 (𝑒 = (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) β†’ ((𝐢 ∈ 𝑒 β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† 𝑒)) ↔ (𝐢 ∈ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)))))
2120rspcv 3608 . . . . . . . . 9 ((𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) ∈ (TopOpenβ€˜β„‚fld) β†’ (βˆ€π‘’ ∈ (TopOpenβ€˜β„‚fld)(𝐢 ∈ 𝑒 β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† 𝑒)) β†’ (𝐢 ∈ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)))))
2215, 21syl 17 . . . . . . . 8 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) β†’ (βˆ€π‘’ ∈ (TopOpenβ€˜β„‚fld)(𝐢 ∈ 𝑒 β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† 𝑒)) β†’ (𝐢 ∈ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)))))
2310, 22mpid 44 . . . . . . 7 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) β†’ (βˆ€π‘’ ∈ (TopOpenβ€˜β„‚fld)(𝐢 ∈ 𝑒 β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† 𝑒)) β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯))))
2413mopni2 24222 . . . . . . . . . . 11 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 𝑣 ∈ (TopOpenβ€˜β„‚fld) ∧ 𝐡 ∈ 𝑣) β†’ βˆƒπ‘¦ ∈ ℝ+ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) βŠ† 𝑣)
256, 24mp3an1 1448 . . . . . . . . . 10 ((𝑣 ∈ (TopOpenβ€˜β„‚fld) ∧ 𝐡 ∈ 𝑣) β†’ βˆƒπ‘¦ ∈ ℝ+ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) βŠ† 𝑣)
26 ssrin 4233 . . . . . . . . . . . . 13 ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) βŠ† 𝑣 β†’ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡})) βŠ† (𝑣 ∩ (𝐴 βˆ– {𝐡})))
27 imass2 6101 . . . . . . . . . . . . 13 (((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡})) βŠ† (𝑣 ∩ (𝐴 βˆ– {𝐡})) β†’ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))))
28 sstr2 3989 . . . . . . . . . . . . 13 ((𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) β†’ ((𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) β†’ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)))
2926, 27, 283syl 18 . . . . . . . . . . . 12 ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) βŠ† 𝑣 β†’ ((𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) β†’ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)))
3029com12 32 . . . . . . . . . . 11 ((𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) β†’ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) βŠ† 𝑣 β†’ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)))
3130reximdv 3170 . . . . . . . . . 10 ((𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) β†’ (βˆƒπ‘¦ ∈ ℝ+ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) βŠ† 𝑣 β†’ βˆƒπ‘¦ ∈ ℝ+ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)))
3225, 31syl5com 31 . . . . . . . . 9 ((𝑣 ∈ (TopOpenβ€˜β„‚fld) ∧ 𝐡 ∈ 𝑣) β†’ ((𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) β†’ βˆƒπ‘¦ ∈ ℝ+ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)))
3332impr 455 . . . . . . . 8 ((𝑣 ∈ (TopOpenβ€˜β„‚fld) ∧ (𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯))) β†’ βˆƒπ‘¦ ∈ ℝ+ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯))
3433rexlimiva 3147 . . . . . . 7 (βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)) β†’ βˆƒπ‘¦ ∈ ℝ+ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯))
3523, 34syl6 35 . . . . . 6 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) β†’ (βˆ€π‘’ ∈ (TopOpenβ€˜β„‚fld)(𝐢 ∈ 𝑒 β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† 𝑒)) β†’ βˆƒπ‘¦ ∈ ℝ+ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)))
3635ralrimdva 3154 . . . . 5 ((πœ‘ ∧ 𝐢 ∈ β„‚) β†’ (βˆ€π‘’ ∈ (TopOpenβ€˜β„‚fld)(𝐢 ∈ 𝑒 β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† 𝑒)) β†’ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ+ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)))
3713mopni2 24222 . . . . . . . . . 10 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 𝑒 ∈ (TopOpenβ€˜β„‚fld) ∧ 𝐢 ∈ 𝑒) β†’ βˆƒπ‘₯ ∈ ℝ+ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) βŠ† 𝑒)
386, 37mp3an1 1448 . . . . . . . . 9 ((𝑒 ∈ (TopOpenβ€˜β„‚fld) ∧ 𝐢 ∈ 𝑒) β†’ βˆƒπ‘₯ ∈ ℝ+ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) βŠ† 𝑒)
39 r19.29r 3116 . . . . . . . . . . 11 ((βˆƒπ‘₯ ∈ ℝ+ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) βŠ† 𝑒 ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ+ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)) β†’ βˆƒπ‘₯ ∈ ℝ+ ((𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) βŠ† 𝑒 ∧ βˆƒπ‘¦ ∈ ℝ+ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)))
403ad3antrrr 728 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) β†’ 𝐡 ∈ β„‚)
41 simpr 485 . . . . . . . . . . . . . . . . . 18 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) β†’ 𝑦 ∈ ℝ+)
4241rpxrd 13021 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) β†’ 𝑦 ∈ ℝ*)
4313blopn 24229 . . . . . . . . . . . . . . . . 17 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 𝐡 ∈ β„‚ ∧ 𝑦 ∈ ℝ*) β†’ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∈ (TopOpenβ€˜β„‚fld))
446, 40, 42, 43mp3an2i 1466 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) β†’ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∈ (TopOpenβ€˜β„‚fld))
45 blcntr 24139 . . . . . . . . . . . . . . . . 17 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 𝐡 ∈ β„‚ ∧ 𝑦 ∈ ℝ+) β†’ 𝐡 ∈ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦))
466, 40, 41, 45mp3an2i 1466 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) β†’ 𝐡 ∈ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦))
47 eleq2 2822 . . . . . . . . . . . . . . . . . . 19 (𝑣 = (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) β†’ (𝐡 ∈ 𝑣 ↔ 𝐡 ∈ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦)))
48 ineq1 4205 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) β†’ (𝑣 ∩ (𝐴 βˆ– {𝐡})) = ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡})))
4948imaeq2d 6059 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) β†’ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) = (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))))
5049sseq1d 4013 . . . . . . . . . . . . . . . . . . 19 (𝑣 = (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) β†’ ((𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) ↔ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)))
5147, 50anbi12d 631 . . . . . . . . . . . . . . . . . 18 (𝑣 = (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) β†’ ((𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)) ↔ (𝐡 ∈ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∧ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯))))
5251rspcev 3612 . . . . . . . . . . . . . . . . 17 (((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∈ (TopOpenβ€˜β„‚fld) ∧ (𝐡 ∈ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∧ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯))) β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)))
5352expr 457 . . . . . . . . . . . . . . . 16 (((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∈ (TopOpenβ€˜β„‚fld) ∧ 𝐡 ∈ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦)) β†’ ((𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯))))
5444, 46, 53syl2anc 584 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) β†’ ((𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯))))
5554rexlimdva 3155 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) β†’ (βˆƒπ‘¦ ∈ ℝ+ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯))))
56 sstr2 3989 . . . . . . . . . . . . . . . . 17 ((𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) β†’ ((𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) βŠ† 𝑒 β†’ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† 𝑒))
5756com12 32 . . . . . . . . . . . . . . . 16 ((𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) βŠ† 𝑒 β†’ ((𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) β†’ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† 𝑒))
5857anim2d 612 . . . . . . . . . . . . . . 15 ((𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) βŠ† 𝑒 β†’ ((𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)) β†’ (𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† 𝑒)))
5958reximdv 3170 . . . . . . . . . . . . . 14 ((𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) βŠ† 𝑒 β†’ (βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)) β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† 𝑒)))
6055, 59syl9 77 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) β†’ ((𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) βŠ† 𝑒 β†’ (βˆƒπ‘¦ ∈ ℝ+ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† 𝑒))))
6160impd 411 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) β†’ (((𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) βŠ† 𝑒 ∧ βˆƒπ‘¦ ∈ ℝ+ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)) β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† 𝑒)))
6261rexlimdva 3155 . . . . . . . . . . 11 ((πœ‘ ∧ 𝐢 ∈ β„‚) β†’ (βˆƒπ‘₯ ∈ ℝ+ ((𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) βŠ† 𝑒 ∧ βˆƒπ‘¦ ∈ ℝ+ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)) β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† 𝑒)))
6339, 62syl5 34 . . . . . . . . . 10 ((πœ‘ ∧ 𝐢 ∈ β„‚) β†’ ((βˆƒπ‘₯ ∈ ℝ+ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) βŠ† 𝑒 ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ+ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)) β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† 𝑒)))
6463expd 416 . . . . . . . . 9 ((πœ‘ ∧ 𝐢 ∈ β„‚) β†’ (βˆƒπ‘₯ ∈ ℝ+ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) βŠ† 𝑒 β†’ (βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ+ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† 𝑒))))
6538, 64syl5 34 . . . . . . . 8 ((πœ‘ ∧ 𝐢 ∈ β„‚) β†’ ((𝑒 ∈ (TopOpenβ€˜β„‚fld) ∧ 𝐢 ∈ 𝑒) β†’ (βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ+ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† 𝑒))))
6665expdimp 453 . . . . . . 7 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ 𝑒 ∈ (TopOpenβ€˜β„‚fld)) β†’ (𝐢 ∈ 𝑒 β†’ (βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ+ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† 𝑒))))
6766com23 86 . . . . . 6 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ 𝑒 ∈ (TopOpenβ€˜β„‚fld)) β†’ (βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ+ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) β†’ (𝐢 ∈ 𝑒 β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† 𝑒))))
6867ralrimdva 3154 . . . . 5 ((πœ‘ ∧ 𝐢 ∈ β„‚) β†’ (βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ+ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) β†’ βˆ€π‘’ ∈ (TopOpenβ€˜β„‚fld)(𝐢 ∈ 𝑒 β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† 𝑒))))
6936, 68impbid 211 . . . 4 ((πœ‘ ∧ 𝐢 ∈ β„‚) β†’ (βˆ€π‘’ ∈ (TopOpenβ€˜β„‚fld)(𝐢 ∈ 𝑒 β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† 𝑒)) ↔ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ+ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)))
701ad2antrr 724 . . . . . . . . . 10 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) β†’ 𝐹:π΄βŸΆβ„‚)
7170ffund 6721 . . . . . . . . 9 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) β†’ Fun 𝐹)
72 inss2 4229 . . . . . . . . . 10 ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡})) βŠ† (𝐴 βˆ– {𝐡})
73 difss 4131 . . . . . . . . . . 11 (𝐴 βˆ– {𝐡}) βŠ† 𝐴
7470fdmd 6728 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) β†’ dom 𝐹 = 𝐴)
7573, 74sseqtrrid 4035 . . . . . . . . . 10 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) β†’ (𝐴 βˆ– {𝐡}) βŠ† dom 𝐹)
7672, 75sstrid 3993 . . . . . . . . 9 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) β†’ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡})) βŠ† dom 𝐹)
77 funimass4 6956 . . . . . . . . 9 ((Fun 𝐹 ∧ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡})) βŠ† dom 𝐹) β†’ ((𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) ↔ βˆ€π‘§ ∈ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))(πΉβ€˜π‘§) ∈ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)))
7871, 76, 77syl2anc 584 . . . . . . . 8 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) β†’ ((𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) ↔ βˆ€π‘§ ∈ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))(πΉβ€˜π‘§) ∈ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)))
796a1i 11 . . . . . . . . . . . . 13 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ (abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚))
80 simplrr 776 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ 𝑦 ∈ ℝ+)
8180rpxrd 13021 . . . . . . . . . . . . 13 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ 𝑦 ∈ ℝ*)
823ad3antrrr 728 . . . . . . . . . . . . 13 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ 𝐡 ∈ β„‚)
8373, 2sstrid 3993 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (𝐴 βˆ– {𝐡}) βŠ† β„‚)
8483ad2antrr 724 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) β†’ (𝐴 βˆ– {𝐡}) βŠ† β„‚)
8584sselda 3982 . . . . . . . . . . . . 13 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ 𝑧 ∈ β„‚)
86 elbl3 24118 . . . . . . . . . . . . 13 ((((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 𝑦 ∈ ℝ*) ∧ (𝐡 ∈ β„‚ ∧ 𝑧 ∈ β„‚)) β†’ (𝑧 ∈ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ↔ (𝑧(abs ∘ βˆ’ )𝐡) < 𝑦))
8779, 81, 82, 85, 86syl22anc 837 . . . . . . . . . . . 12 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ (𝑧 ∈ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ↔ (𝑧(abs ∘ βˆ’ )𝐡) < 𝑦))
88 eqid 2732 . . . . . . . . . . . . . . 15 (abs ∘ βˆ’ ) = (abs ∘ βˆ’ )
8988cnmetdval 24507 . . . . . . . . . . . . . 14 ((𝑧 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝑧(abs ∘ βˆ’ )𝐡) = (absβ€˜(𝑧 βˆ’ 𝐡)))
9085, 82, 89syl2anc 584 . . . . . . . . . . . . 13 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ (𝑧(abs ∘ βˆ’ )𝐡) = (absβ€˜(𝑧 βˆ’ 𝐡)))
9190breq1d 5158 . . . . . . . . . . . 12 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ ((𝑧(abs ∘ βˆ’ )𝐡) < 𝑦 ↔ (absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦))
9287, 91bitrd 278 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ (𝑧 ∈ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ↔ (absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦))
93 simplrl 775 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ π‘₯ ∈ ℝ+)
9493rpxrd 13021 . . . . . . . . . . . . 13 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ π‘₯ ∈ ℝ*)
95 simpllr 774 . . . . . . . . . . . . 13 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ 𝐢 ∈ β„‚)
96 eldifi 4126 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝐴 βˆ– {𝐡}) β†’ 𝑧 ∈ 𝐴)
97 ffvelcdm 7083 . . . . . . . . . . . . . 14 ((𝐹:π΄βŸΆβ„‚ ∧ 𝑧 ∈ 𝐴) β†’ (πΉβ€˜π‘§) ∈ β„‚)
9870, 96, 97syl2an 596 . . . . . . . . . . . . 13 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ (πΉβ€˜π‘§) ∈ β„‚)
99 elbl3 24118 . . . . . . . . . . . . 13 ((((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ π‘₯ ∈ ℝ*) ∧ (𝐢 ∈ β„‚ ∧ (πΉβ€˜π‘§) ∈ β„‚)) β†’ ((πΉβ€˜π‘§) ∈ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) ↔ ((πΉβ€˜π‘§)(abs ∘ βˆ’ )𝐢) < π‘₯))
10079, 94, 95, 98, 99syl22anc 837 . . . . . . . . . . . 12 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ ((πΉβ€˜π‘§) ∈ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) ↔ ((πΉβ€˜π‘§)(abs ∘ βˆ’ )𝐢) < π‘₯))
10188cnmetdval 24507 . . . . . . . . . . . . . 14 (((πΉβ€˜π‘§) ∈ β„‚ ∧ 𝐢 ∈ β„‚) β†’ ((πΉβ€˜π‘§)(abs ∘ βˆ’ )𝐢) = (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)))
10298, 95, 101syl2anc 584 . . . . . . . . . . . . 13 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ ((πΉβ€˜π‘§)(abs ∘ βˆ’ )𝐢) = (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)))
103102breq1d 5158 . . . . . . . . . . . 12 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ (((πΉβ€˜π‘§)(abs ∘ βˆ’ )𝐢) < π‘₯ ↔ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯))
104100, 103bitrd 278 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ ((πΉβ€˜π‘§) ∈ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) ↔ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯))
10592, 104imbi12d 344 . . . . . . . . . 10 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ ((𝑧 ∈ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) β†’ (πΉβ€˜π‘§) ∈ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)) ↔ ((absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯)))
106105ralbidva 3175 . . . . . . . . 9 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) β†’ (βˆ€π‘§ ∈ (𝐴 βˆ– {𝐡})(𝑧 ∈ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) β†’ (πΉβ€˜π‘§) ∈ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)) ↔ βˆ€π‘§ ∈ (𝐴 βˆ– {𝐡})((absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯)))
107 elin 3964 . . . . . . . . . . . . 13 (𝑧 ∈ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡})) ↔ (𝑧 ∈ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})))
108107biancomi 463 . . . . . . . . . . . 12 (𝑧 ∈ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡})) ↔ (𝑧 ∈ (𝐴 βˆ– {𝐡}) ∧ 𝑧 ∈ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦)))
109108imbi1i 349 . . . . . . . . . . 11 ((𝑧 ∈ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡})) β†’ (πΉβ€˜π‘§) ∈ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)) ↔ ((𝑧 ∈ (𝐴 βˆ– {𝐡}) ∧ 𝑧 ∈ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦)) β†’ (πΉβ€˜π‘§) ∈ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)))
110 impexp 451 . . . . . . . . . . 11 (((𝑧 ∈ (𝐴 βˆ– {𝐡}) ∧ 𝑧 ∈ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦)) β†’ (πΉβ€˜π‘§) ∈ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)) ↔ (𝑧 ∈ (𝐴 βˆ– {𝐡}) β†’ (𝑧 ∈ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) β†’ (πΉβ€˜π‘§) ∈ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯))))
111109, 110bitr2i 275 . . . . . . . . . 10 ((𝑧 ∈ (𝐴 βˆ– {𝐡}) β†’ (𝑧 ∈ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) β†’ (πΉβ€˜π‘§) ∈ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯))) ↔ (𝑧 ∈ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡})) β†’ (πΉβ€˜π‘§) ∈ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)))
112111ralbii2 3089 . . . . . . . . 9 (βˆ€π‘§ ∈ (𝐴 βˆ– {𝐡})(𝑧 ∈ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) β†’ (πΉβ€˜π‘§) ∈ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)) ↔ βˆ€π‘§ ∈ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))(πΉβ€˜π‘§) ∈ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯))
113 impexp 451 . . . . . . . . . . 11 (((𝑧 ∈ 𝐴 ∧ 𝑧 β‰  𝐡) β†’ ((absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯)) ↔ (𝑧 ∈ 𝐴 β†’ (𝑧 β‰  𝐡 β†’ ((absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯))))
114 eldifsn 4790 . . . . . . . . . . . 12 (𝑧 ∈ (𝐴 βˆ– {𝐡}) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧 β‰  𝐡))
115114imbi1i 349 . . . . . . . . . . 11 ((𝑧 ∈ (𝐴 βˆ– {𝐡}) β†’ ((absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯)) ↔ ((𝑧 ∈ 𝐴 ∧ 𝑧 β‰  𝐡) β†’ ((absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯)))
116 impexp 451 . . . . . . . . . . . 12 (((𝑧 β‰  𝐡 ∧ (absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦) β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯) ↔ (𝑧 β‰  𝐡 β†’ ((absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯)))
117116imbi2i 335 . . . . . . . . . . 11 ((𝑧 ∈ 𝐴 β†’ ((𝑧 β‰  𝐡 ∧ (absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦) β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯)) ↔ (𝑧 ∈ 𝐴 β†’ (𝑧 β‰  𝐡 β†’ ((absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯))))
118113, 115, 1173bitr4i 302 . . . . . . . . . 10 ((𝑧 ∈ (𝐴 βˆ– {𝐡}) β†’ ((absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯)) ↔ (𝑧 ∈ 𝐴 β†’ ((𝑧 β‰  𝐡 ∧ (absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦) β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯)))
119118ralbii2 3089 . . . . . . . . 9 (βˆ€π‘§ ∈ (𝐴 βˆ– {𝐡})((absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦 β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯) ↔ βˆ€π‘§ ∈ 𝐴 ((𝑧 β‰  𝐡 ∧ (absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦) β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯))
120106, 112, 1193bitr3g 312 . . . . . . . 8 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) β†’ (βˆ€π‘§ ∈ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))(πΉβ€˜π‘§) ∈ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) ↔ βˆ€π‘§ ∈ 𝐴 ((𝑧 β‰  𝐡 ∧ (absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦) β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯)))
12178, 120bitrd 278 . . . . . . 7 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) β†’ ((𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) ↔ βˆ€π‘§ ∈ 𝐴 ((𝑧 β‰  𝐡 ∧ (absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦) β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯)))
122121anassrs 468 . . . . . 6 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) β†’ ((𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) ↔ βˆ€π‘§ ∈ 𝐴 ((𝑧 β‰  𝐡 ∧ (absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦) β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯)))
123122rexbidva 3176 . . . . 5 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) β†’ (βˆƒπ‘¦ ∈ ℝ+ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) ↔ βˆƒπ‘¦ ∈ ℝ+ βˆ€π‘§ ∈ 𝐴 ((𝑧 β‰  𝐡 ∧ (absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦) β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯)))
124123ralbidva 3175 . . . 4 ((πœ‘ ∧ 𝐢 ∈ β„‚) β†’ (βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ+ (𝐹 β€œ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡}))) βŠ† (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) ↔ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ+ βˆ€π‘§ ∈ 𝐴 ((𝑧 β‰  𝐡 ∧ (absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦) β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯)))
12569, 124bitrd 278 . . 3 ((πœ‘ ∧ 𝐢 ∈ β„‚) β†’ (βˆ€π‘’ ∈ (TopOpenβ€˜β„‚fld)(𝐢 ∈ 𝑒 β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† 𝑒)) ↔ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ+ βˆ€π‘§ ∈ 𝐴 ((𝑧 β‰  𝐡 ∧ (absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦) β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯)))
126125pm5.32da 579 . 2 (πœ‘ β†’ ((𝐢 ∈ β„‚ ∧ βˆ€π‘’ ∈ (TopOpenβ€˜β„‚fld)(𝐢 ∈ 𝑒 β†’ βˆƒπ‘£ ∈ (TopOpenβ€˜β„‚fld)(𝐡 ∈ 𝑣 ∧ (𝐹 β€œ (𝑣 ∩ (𝐴 βˆ– {𝐡}))) βŠ† 𝑒))) ↔ (𝐢 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ+ βˆ€π‘§ ∈ 𝐴 ((𝑧 β‰  𝐡 ∧ (absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦) β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯))))
1275, 126bitrd 278 1 (πœ‘ β†’ (𝐢 ∈ (𝐹 limβ„‚ 𝐡) ↔ (𝐢 ∈ β„‚ ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ+ βˆ€π‘§ ∈ 𝐴 ((𝑧 β‰  𝐡 ∧ (absβ€˜(𝑧 βˆ’ 𝐡)) < 𝑦) β†’ (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)) < π‘₯))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070   βˆ– cdif 3945   ∩ cin 3947   βŠ† wss 3948  {csn 4628   class class class wbr 5148  dom cdm 5676   β€œ cima 5679   ∘ ccom 5680  Fun wfun 6537  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411  β„‚cc 11110  β„*cxr 11251   < clt 11252   βˆ’ cmin 11448  β„+crp 12978  abscabs 15185  TopOpenctopn 17371  βˆžMetcxmet 21129  ballcbl 21131  β„‚fldccnfld 21144   limβ„‚ climc 25603
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  ax-pre-sup 11190
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-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  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-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  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-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fi 9408  df-sup 9439  df-inf 9440  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-q 12937  df-rp 12979  df-xneg 13096  df-xadd 13097  df-xmul 13098  df-fz 13489  df-seq 13971  df-exp 14032  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-struct 17084  df-slot 17119  df-ndx 17131  df-base 17149  df-plusg 17214  df-mulr 17215  df-starv 17216  df-tset 17220  df-ple 17221  df-ds 17223  df-unif 17224  df-rest 17372  df-topn 17373  df-topgen 17393  df-psmet 21136  df-xmet 21137  df-met 21138  df-bl 21139  df-mopn 21140  df-cnfld 21145  df-top 22616  df-topon 22633  df-topsp 22655  df-bases 22669  df-cnp 22952  df-xms 24046  df-ms 24047  df-limc 25607
This theorem is referenced by:  dveflem  25720  dvferm1  25726  dvferm2  25728  lhop1  25755  ftc1lem6  25782  ulmdvlem3  26138  unblimceq0  35686  ftc1cnnc  36863  mullimc  44631  ellimcabssub0  44632  limcdm0  44633  mullimcf  44638  constlimc  44639  idlimc  44641  limcperiod  44643  limcrecl  44644  limcleqr  44659  neglimc  44662  addlimc  44663  0ellimcdiv  44664  limclner  44666  fperdvper  44934  ioodvbdlimc1lem2  44947  ioodvbdlimc2lem  44949
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