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Theorem ellimc3 25266
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 25264 . 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 24159 . . . . . . . . 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 23789 . . . . . . . . 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 12932 . . . . . . . . . . 11 (π‘₯ ∈ ℝ+ β†’ π‘₯ ∈ ℝ*)
1211adantl 483 . . . . . . . . . 10 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) β†’ π‘₯ ∈ ℝ*)
134cnfldtopn 24168 . . . . . . . . . . 11 (TopOpenβ€˜β„‚fld) = (MetOpenβ€˜(abs ∘ βˆ’ ))
1413blopn 23879 . . . . . . . . . 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 3974 . . . . . . . . . . . . 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 3172 . . . . . . . . . . 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 3579 . . . . . . . . 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 23872 . . . . . . . . . . 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 4197 . . . . . . . . . . . . 13 ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) βŠ† 𝑣 β†’ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡})) βŠ† (𝑣 ∩ (𝐴 βˆ– {𝐡})))
27 imass2 6058 . . . . . . . . . . . . 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 3955 . . . . . . . . . . . . 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 3164 . . . . . . . . . 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 3141 . . . . . . 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 3148 . . . . 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 23872 . . . . . . . . . 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 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 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 12966 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ π‘₯ ∈ ℝ+) ∧ 𝑦 ∈ ℝ+) β†’ 𝑦 ∈ ℝ*)
4313blopn 23879 . . . . . . . . . . . . . . . . 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 23789 . . . . . . . . . . . . . . . . 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 4169 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = (𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) β†’ (𝑣 ∩ (𝐴 βˆ– {𝐡})) = ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡})))
4948imaeq2d 6017 . . . . . . . . . . . . . . . . . . . 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 3979 . . . . . . . . . . . . . . . . . . 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 3583 . . . . . . . . . . . . . . . . 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 3149 . . . . . . . . . . . . . 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 3955 . . . . . . . . . . . . . . . . 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 3164 . . . . . . . . . . . . . 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 3149 . . . . . . . . . . 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 3148 . . . . 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 6676 . . . . . . . . 9 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) β†’ Fun 𝐹)
72 inss2 4193 . . . . . . . . . 10 ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡})) βŠ† (𝐴 βˆ– {𝐡})
73 difss 4095 . . . . . . . . . . 11 (𝐴 βˆ– {𝐡}) βŠ† 𝐴
7470fdmd 6683 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) β†’ dom 𝐹 = 𝐴)
7573, 74sseqtrrid 4001 . . . . . . . . . 10 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) β†’ (𝐴 βˆ– {𝐡}) βŠ† dom 𝐹)
7672, 75sstrid 3959 . . . . . . . . 9 (((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) β†’ ((𝐡(ballβ€˜(abs ∘ βˆ’ ))𝑦) ∩ (𝐴 βˆ– {𝐡})) βŠ† dom 𝐹)
77 funimass4 6911 . . . . . . . . 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 12966 . . . . . . . . . . . . 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 3959 . . . . . . . . . . . . . . 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 3948 . . . . . . . . . . . . 13 ((((πœ‘ ∧ 𝐢 ∈ β„‚) ∧ (π‘₯ ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) ∧ 𝑧 ∈ (𝐴 βˆ– {𝐡})) β†’ 𝑧 ∈ β„‚)
86 elbl3 23768 . . . . . . . . . . . . 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 24157 . . . . . . . . . . . . . 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 5119 . . . . . . . . . . . 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 12966 . . . . . . . . . . . . 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 4090 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝐴 βˆ– {𝐡}) β†’ 𝑧 ∈ 𝐴)
97 ffvelcdm 7036 . . . . . . . . . . . . . 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 23768 . . . . . . . . . . . . 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 24157 . . . . . . . . . . . . . 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 5119 . . . . . . . . . . . 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 3169 . . . . . . . . 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 3930 . . . . . . . . . . . . 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 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 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 4751 . . . . . . . . . . . 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 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 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 3170 . . . . 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 3169 . . . 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 2940  βˆ€wral 3061  βˆƒwrex 3070   βˆ– cdif 3911   ∩ cin 3913   βŠ† wss 3914  {csn 4590   class class class wbr 5109  dom cdm 5637   β€œ cima 5640   ∘ ccom 5641  Fun wfun 6494  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361  β„‚cc 11057  β„*cxr 11196   < clt 11197   βˆ’ cmin 11393  β„+crp 12923  abscabs 15128  TopOpenctopn 17311  βˆžMetcxmet 20804  ballcbl 20806  β„‚fldccnfld 20819   limβ„‚ climc 25249
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-map 8773  df-pm 8774  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-fi 9355  df-sup 9386  df-inf 9387  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-7 12229  df-8 12230  df-9 12231  df-n0 12422  df-z 12508  df-dec 12627  df-uz 12772  df-q 12882  df-rp 12924  df-xneg 13041  df-xadd 13042  df-xmul 13043  df-fz 13434  df-seq 13916  df-exp 13977  df-cj 14993  df-re 14994  df-im 14995  df-sqrt 15129  df-abs 15130  df-struct 17027  df-slot 17062  df-ndx 17074  df-base 17092  df-plusg 17154  df-mulr 17155  df-starv 17156  df-tset 17160  df-ple 17161  df-ds 17163  df-unif 17164  df-rest 17312  df-topn 17313  df-topgen 17333  df-psmet 20811  df-xmet 20812  df-met 20813  df-bl 20814  df-mopn 20815  df-cnfld 20820  df-top 22266  df-topon 22283  df-topsp 22305  df-bases 22319  df-cnp 22602  df-xms 23696  df-ms 23697  df-limc 25253
This theorem is referenced by:  dveflem  25366  dvferm1  25372  dvferm2  25374  lhop1  25401  ftc1lem6  25428  ulmdvlem3  25784  unblimceq0  35023  ftc1cnnc  36200  mullimc  43947  ellimcabssub0  43948  limcdm0  43949  mullimcf  43954  constlimc  43955  idlimc  43957  limcperiod  43959  limcrecl  43960  limcleqr  43975  neglimc  43978  addlimc  43979  0ellimcdiv  43980  limclner  43982  fperdvper  44250  ioodvbdlimc1lem2  44263  ioodvbdlimc2lem  44265
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