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Theorem ellimc3 25396
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 2733 . . 3 (TopOpenβ€˜β„‚fld) = (TopOpenβ€˜β„‚fld)
51, 2, 3, 4ellimc2 25394 . 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 24289 . . . . . . . . 9 (abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚)
7 simplr 768 . . . . . . . . 9 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) β†’ 𝐢 ∈ β„‚)
8 simpr 486 . . . . . . . . 9 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) β†’ π‘₯ ∈ ℝ+)
9 blcntr 23919 . . . . . . . . 9 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 𝐢 ∈ β„‚ ∧ π‘₯ ∈ ℝ+) β†’ 𝐢 ∈ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯))
106, 7, 8, 9mp3an2i 1467 . . . . . . . 8 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) β†’ 𝐢 ∈ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯))
11 rpxr 12983 . . . . . . . . . . 11 (π‘₯ ∈ ℝ+ β†’ π‘₯ ∈ ℝ*)
1211adantl 483 . . . . . . . . . 10 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) β†’ π‘₯ ∈ ℝ*)
134cnfldtopn 24298 . . . . . . . . . . 11 (TopOpenβ€˜β„‚fld) = (MetOpenβ€˜(abs ∘ βˆ’ ))
1413blopn 24009 . . . . . . . . . 10 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 𝐢 ∈ β„‚ ∧ π‘₯ ∈ ℝ*) β†’ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) ∈ (TopOpenβ€˜β„‚fld))
156, 7, 12, 14mp3an2i 1467 . . . . . . . . 9 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) β†’ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) ∈ (TopOpenβ€˜β„‚fld))
16 eleq2 2823 . . . . . . . . . . 11 (𝑒 = (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) β†’ (𝐢 ∈ 𝑒 ↔ 𝐢 ∈ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯)))
17 sseq2 4009 . . . . . . . . . . . . 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 630 . . . . . . . . . . . 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 3179 . . . . . . . . . . 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 345 . . . . . . . . . 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 3609 . . . . . . . . 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 24002 . . . . . . . . . . 11 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 𝑣 ∈ (TopOpenβ€˜β„‚fld) ∧ 𝐡 ∈ 𝑣) β†’ βˆƒπ‘¦ ∈ ℝ+ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) βŠ† 𝑣)
256, 24mp3an1 1449 . . . . . . . . . 10 ((𝑣 ∈ (TopOpenβ€˜β„‚fld) ∧ 𝐡 ∈ 𝑣) β†’ βˆƒπ‘¦ ∈ ℝ+ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) βŠ† 𝑣)
26 ssrin 4234 . . . . . . . . . . . . 13 ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) βŠ† 𝑣 β†’ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡})) βŠ† (𝑣 ∩ (𝐴 βˆ– {𝐡})))
27 imass2 6102 . . . . . . . . . . . . 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 3990 . . . . . . . . . . . . 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 3171 . . . . . . . . . 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 456 . . . . . . . 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 3148 . . . . . . 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 3155 . . . . 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 24002 . . . . . . . . . 10 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 𝑒 ∈ (TopOpenβ€˜β„‚fld) ∧ 𝐢 ∈ 𝑒) β†’ βˆƒπ‘₯ ∈ ℝ+ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) βŠ† 𝑒)
386, 37mp3an1 1449 . . . . . . . . 9 ((𝑒 ∈ (TopOpenβ€˜β„‚fld) ∧ 𝐢 ∈ 𝑒) β†’ βˆƒπ‘₯ ∈ ℝ+ (𝐢(ballβ€˜(abs ∘ βˆ’ ))π‘₯) βŠ† 𝑒)
39 r19.29r 3117 . . . . . . . . . . 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 729 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) β†’ 𝐡 ∈ β„‚)
41 simpr 486 . . . . . . . . . . . . . . . . . 18 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) β†’ 𝑦 ∈ ℝ+)
4241rpxrd 13017 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) β†’ 𝑦 ∈ ℝ*)
4313blopn 24009 . . . . . . . . . . . . . . . . 17 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 𝐡 ∈ β„‚ ∧ 𝑦 ∈ ℝ*) β†’ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∈ (TopOpenβ€˜β„‚fld))
446, 40, 42, 43mp3an2i 1467 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) β†’ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∈ (TopOpenβ€˜β„‚fld))
45 blcntr 23919 . . . . . . . . . . . . . . . . 17 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 𝐡 ∈ β„‚ ∧ 𝑦 ∈ ℝ+) β†’ 𝐡 ∈ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦))
466, 40, 41, 45mp3an2i 1467 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) β†’ 𝐡 ∈ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦))
47 eleq2 2823 . . . . . . . . . . . . . . . . . . 19 (𝑣 = (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) β†’ (𝐡 ∈ 𝑣 ↔ 𝐡 ∈ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦)))
48 ineq1 4206 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) β†’ (𝑣 ∩ (𝐴 βˆ– {𝐡})) = ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡})))
4948imaeq2d 6060 . . . . . . . . . . . . . . . . . . . 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 4014 . . . . . . . . . . . . . . . . . . 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 632 . . . . . . . . . . . . . . . . . 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 3613 . . . . . . . . . . . . . . . . 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 458 . . . . . . . . . . . . . . . 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 585 . . . . . . . . . . . . . . 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 3156 . . . . . . . . . . . . . 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 3990 . . . . . . . . . . . . . . . . 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 613 . . . . . . . . . . . . . . 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 3171 . . . . . . . . . . . . . 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 412 . . . . . . . . . . . 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 3156 . . . . . . . . . . 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 417 . . . . . . . . 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 454 . . . . . . 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 3155 . . . . 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 725 . . . . . . . . . 10 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) β†’ 𝐹:π΄βŸΆβ„‚)
7170ffund 6722 . . . . . . . . 9 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) β†’ Fun 𝐹)
72 inss2 4230 . . . . . . . . . 10 ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡})) βŠ† (𝐴 βˆ– {𝐡})
73 difss 4132 . . . . . . . . . . 11 (𝐴 βˆ– {𝐡}) βŠ† 𝐴
7470fdmd 6729 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) β†’ dom 𝐹 = 𝐴)
7573, 74sseqtrrid 4036 . . . . . . . . . 10 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) β†’ (𝐴 βˆ– {𝐡}) βŠ† dom 𝐹)
7672, 75sstrid 3994 . . . . . . . . 9 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) β†’ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡})) βŠ† dom 𝐹)
77 funimass4 6957 . . . . . . . . 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 585 . . . . . . . 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 777 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ 𝑦 ∈ ℝ+)
8180rpxrd 13017 . . . . . . . . . . . . 13 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ 𝑦 ∈ ℝ*)
823ad3antrrr 729 . . . . . . . . . . . . 13 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ 𝐡 ∈ β„‚)
8373, 2sstrid 3994 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (𝐴 βˆ– {𝐡}) βŠ† β„‚)
8483ad2antrr 725 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) β†’ (𝐴 βˆ– {𝐡}) βŠ† β„‚)
8584sselda 3983 . . . . . . . . . . . . 13 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ 𝑧 ∈ β„‚)
86 elbl3 23898 . . . . . . . . . . . . 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 838 . . . . . . . . . . . 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 2733 . . . . . . . . . . . . . . 15 (abs ∘ βˆ’ ) = (abs ∘ βˆ’ )
8988cnmetdval 24287 . . . . . . . . . . . . . 14 ((𝑧 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝑧(abs ∘ βˆ’ )𝐡) = (absβ€˜(𝑧 βˆ’ 𝐡)))
9085, 82, 89syl2anc 585 . . . . . . . . . . . . 13 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ (𝑧(abs ∘ βˆ’ )𝐡) = (absβ€˜(𝑧 βˆ’ 𝐡)))
9190breq1d 5159 . . . . . . . . . . . 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 279 . . . . . . . . . . 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 776 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ π‘₯ ∈ ℝ+)
9493rpxrd 13017 . . . . . . . . . . . . 13 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ π‘₯ ∈ ℝ*)
95 simpllr 775 . . . . . . . . . . . . 13 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ 𝐢 ∈ β„‚)
96 eldifi 4127 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝐴 βˆ– {𝐡}) β†’ 𝑧 ∈ 𝐴)
97 ffvelcdm 7084 . . . . . . . . . . . . . 14 ((𝐹:π΄βŸΆβ„‚ ∧ 𝑧 ∈ 𝐴) β†’ (πΉβ€˜π‘§) ∈ β„‚)
9870, 96, 97syl2an 597 . . . . . . . . . . . . 13 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ (πΉβ€˜π‘§) ∈ β„‚)
99 elbl3 23898 . . . . . . . . . . . . 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 838 . . . . . . . . . . . 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 24287 . . . . . . . . . . . . . 14 (((πΉβ€˜π‘§) ∈ β„‚ ∧ 𝐢 ∈ β„‚) β†’ ((πΉβ€˜π‘§)(abs ∘ βˆ’ )𝐢) = (absβ€˜((πΉβ€˜π‘§) βˆ’ 𝐢)))
10298, 95, 101syl2anc 585 . . . . . . . . . . . . 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 5159 . . . . . . . . . . . 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 279 . . . . . . . . . . 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 345 . . . . . . . . . 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 3176 . . . . . . . . 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 3965 . . . . . . . . . . . . 13 (𝑧 ∈ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡})) ↔ (𝑧 ∈ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})))
108107biancomi 464 . . . . . . . . . . . 12 (𝑧 ∈ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡})) ↔ (𝑧 ∈ (𝐴 βˆ– {𝐡}) ∧ 𝑧 ∈ (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦)))
109108imbi1i 350 . . . . . . . . . . 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 452 . . . . . . . . . . 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 276 . . . . . . . . . 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 3090 . . . . . . . . 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 452 . . . . . . . . . . 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 4791 . . . . . . . . . . . 12 (𝑧 ∈ (𝐴 βˆ– {𝐡}) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧 β‰  𝐡))
115114imbi1i 350 . . . . . . . . . . 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 452 . . . . . . . . . . . 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 336 . . . . . . . . . . 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 303 . . . . . . . . . 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 3090 . . . . . . . . 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 313 . . . . . . . 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 279 . . . . . . 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 469 . . . . . 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 3177 . . . . 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 3176 . . . 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 279 . . 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 580 . 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 279 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 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071   βˆ– cdif 3946   ∩ cin 3948   βŠ† wss 3949  {csn 4629   class class class wbr 5149  dom cdm 5677   β€œ cima 5680   ∘ ccom 5681  Fun wfun 6538  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  β„‚cc 11108  β„*cxr 11247   < clt 11248   βˆ’ cmin 11444  β„+crp 12974  abscabs 15181  TopOpenctopn 17367  βˆžMetcxmet 20929  ballcbl 20931  β„‚fldccnfld 20944   limβ„‚ climc 25379
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  ax-pre-sup 11188
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-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  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-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  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-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fi 9406  df-sup 9437  df-inf 9438  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-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-fz 13485  df-seq 13967  df-exp 14028  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-struct 17080  df-slot 17115  df-ndx 17127  df-base 17145  df-plusg 17210  df-mulr 17211  df-starv 17212  df-tset 17216  df-ple 17217  df-ds 17219  df-unif 17220  df-rest 17368  df-topn 17369  df-topgen 17389  df-psmet 20936  df-xmet 20937  df-met 20938  df-bl 20939  df-mopn 20940  df-cnfld 20945  df-top 22396  df-topon 22413  df-topsp 22435  df-bases 22449  df-cnp 22732  df-xms 23826  df-ms 23827  df-limc 25383
This theorem is referenced by:  dveflem  25496  dvferm1  25502  dvferm2  25504  lhop1  25531  ftc1lem6  25558  ulmdvlem3  25914  unblimceq0  35431  ftc1cnnc  36608  mullimc  44380  ellimcabssub0  44381  limcdm0  44382  mullimcf  44387  constlimc  44388  idlimc  44390  limcperiod  44392  limcrecl  44393  limcleqr  44408  neglimc  44411  addlimc  44412  0ellimcdiv  44413  limclner  44415  fperdvper  44683  ioodvbdlimc1lem2  44696  ioodvbdlimc2lem  44698
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