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Theorem dvrecap 14365
Description: Derivative of the reciprocal function. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 28-Dec-2016.)
Assertion
Ref Expression
dvrecap (𝐴 ∈ β„‚ β†’ (β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))) = (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ -(𝐴 / (π‘₯↑2))))
Distinct variable group:   π‘₯,𝑀,𝐴

Proof of Theorem dvrecap
Dummy variables 𝑦 𝑧 π‘˜ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funmpt 5256 . . . . . . . . 9 Fun (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))
2 funforn 5447 . . . . . . . . 9 (Fun (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)) ↔ (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)):dom (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))–ontoβ†’ran (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)))
31, 2mpbi 145 . . . . . . . 8 (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)):dom (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))–ontoβ†’ran (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))
4 fof 5440 . . . . . . . 8 ((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)):dom (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))–ontoβ†’ran (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)) β†’ (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)):dom (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))⟢ran (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)))
53, 4ax-mp 5 . . . . . . 7 (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)):dom (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))⟢ran (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))
6 simpl 109 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ 𝐴 ∈ β„‚)
7 breq1 4008 . . . . . . . . . . . . . 14 (𝑀 = π‘₯ β†’ (𝑀 # 0 ↔ π‘₯ # 0))
87elrab 2895 . . . . . . . . . . . . 13 (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↔ (π‘₯ ∈ β„‚ ∧ π‘₯ # 0))
98biimpi 120 . . . . . . . . . . . 12 (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} β†’ (π‘₯ ∈ β„‚ ∧ π‘₯ # 0))
109adantl 277 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (π‘₯ ∈ β„‚ ∧ π‘₯ # 0))
1110simpld 112 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ π‘₯ ∈ β„‚)
1210simprd 114 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ π‘₯ # 0)
136, 11, 12divclapd 8750 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (𝐴 / π‘₯) ∈ β„‚)
1413ralrimiva 2550 . . . . . . . 8 (𝐴 ∈ β„‚ β†’ βˆ€π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} (𝐴 / π‘₯) ∈ β„‚)
15 eqid 2177 . . . . . . . . 9 (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)) = (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))
1615rnmptss 5680 . . . . . . . 8 (βˆ€π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} (𝐴 / π‘₯) ∈ β„‚ β†’ ran (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)) βŠ† β„‚)
1714, 16syl 14 . . . . . . 7 (𝐴 ∈ β„‚ β†’ ran (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)) βŠ† β„‚)
18 fss 5379 . . . . . . 7 (((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)):dom (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))⟢ran (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)) ∧ ran (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)) βŠ† β„‚) β†’ (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)):dom (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))βŸΆβ„‚)
195, 17, 18sylancr 414 . . . . . 6 (𝐴 ∈ β„‚ β†’ (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)):dom (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))βŸΆβ„‚)
2015dmmpt 5126 . . . . . . 7 dom (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)) = {π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ (𝐴 / π‘₯) ∈ V}
21 ssrab2 3242 . . . . . . . 8 {π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ (𝐴 / π‘₯) ∈ V} βŠ† {𝑀 ∈ β„‚ ∣ 𝑀 # 0}
22 ssrab2 3242 . . . . . . . 8 {𝑀 ∈ β„‚ ∣ 𝑀 # 0} βŠ† β„‚
2321, 22sstri 3166 . . . . . . 7 {π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ (𝐴 / π‘₯) ∈ V} βŠ† β„‚
2420, 23eqsstri 3189 . . . . . 6 dom (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)) βŠ† β„‚
25 cnex 7938 . . . . . . 7 β„‚ ∈ V
2625, 25elpm2 6683 . . . . . 6 ((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)) ∈ (β„‚ ↑pm β„‚) ↔ ((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)):dom (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))βŸΆβ„‚ ∧ dom (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)) βŠ† β„‚))
2719, 24, 26sylanblrc 416 . . . . 5 (𝐴 ∈ β„‚ β†’ (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)) ∈ (β„‚ ↑pm β„‚))
28 dvfcnpm 14347 . . . . 5 ((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)) ∈ (β„‚ ↑pm β„‚) β†’ (β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))):dom (β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)))βŸΆβ„‚)
2927, 28syl 14 . . . 4 (𝐴 ∈ β„‚ β†’ (β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))):dom (β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)))βŸΆβ„‚)
30 ssidd 3178 . . . . . . 7 (𝐴 ∈ β„‚ β†’ β„‚ βŠ† β„‚)
31 divclap 8638 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ β„‚ ∧ π‘₯ # 0) β†’ (𝐴 / π‘₯) ∈ β„‚)
32313expb 1204 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (π‘₯ ∈ β„‚ ∧ π‘₯ # 0)) β†’ (𝐴 / π‘₯) ∈ β„‚)
338, 32sylan2b 287 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (𝐴 / π‘₯) ∈ β„‚)
3433fmpttd 5674 . . . . . . 7 (𝐴 ∈ β„‚ β†’ (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)):{𝑀 ∈ β„‚ ∣ 𝑀 # 0}βŸΆβ„‚)
3522a1i 9 . . . . . . 7 (𝐴 ∈ β„‚ β†’ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} βŠ† β„‚)
3630, 34, 35dvbss 14342 . . . . . 6 (𝐴 ∈ β„‚ β†’ dom (β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))) βŠ† {𝑀 ∈ β„‚ ∣ 𝑀 # 0})
37 elrabi 2892 . . . . . . . 8 (𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} β†’ 𝑦 ∈ β„‚)
3837adantl 277 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ 𝑦 ∈ β„‚)
39 simpl 109 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ 𝐴 ∈ β„‚)
4038sqcld 10655 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (𝑦↑2) ∈ β„‚)
41 breq1 4008 . . . . . . . . . . . . 13 (𝑀 = 𝑦 β†’ (𝑀 # 0 ↔ 𝑦 # 0))
4241elrab 2895 . . . . . . . . . . . 12 (𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↔ (𝑦 ∈ β„‚ ∧ 𝑦 # 0))
4342simprbi 275 . . . . . . . . . . 11 (𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} β†’ 𝑦 # 0)
4443adantl 277 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ 𝑦 # 0)
45 sqap0 10590 . . . . . . . . . . 11 (𝑦 ∈ β„‚ β†’ ((𝑦↑2) # 0 ↔ 𝑦 # 0))
4638, 45syl 14 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ ((𝑦↑2) # 0 ↔ 𝑦 # 0))
4744, 46mpbird 167 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (𝑦↑2) # 0)
4839, 40, 47divclapd 8750 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (𝐴 / (𝑦↑2)) ∈ β„‚)
4948negcld 8258 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ -(𝐴 / (𝑦↑2)) ∈ β„‚)
50 simpr 110 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0})
51 eqid 2177 . . . . . . . . . . 11 (MetOpenβ€˜(abs ∘ βˆ’ )) = (MetOpenβ€˜(abs ∘ βˆ’ ))
5251cntoptop 14221 . . . . . . . . . 10 (MetOpenβ€˜(abs ∘ βˆ’ )) ∈ Top
53 0cn 7952 . . . . . . . . . . 11 0 ∈ β„‚
54 cnopnap 14282 . . . . . . . . . . 11 (0 ∈ β„‚ β†’ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∈ (MetOpenβ€˜(abs ∘ βˆ’ )))
5553, 54ax-mp 5 . . . . . . . . . 10 {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∈ (MetOpenβ€˜(abs ∘ βˆ’ ))
56 isopn3i 13823 . . . . . . . . . 10 (((MetOpenβ€˜(abs ∘ βˆ’ )) ∈ Top ∧ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∈ (MetOpenβ€˜(abs ∘ βˆ’ ))) β†’ ((intβ€˜(MetOpenβ€˜(abs ∘ βˆ’ )))β€˜{𝑀 ∈ β„‚ ∣ 𝑀 # 0}) = {𝑀 ∈ β„‚ ∣ 𝑀 # 0})
5752, 55, 56mp2an 426 . . . . . . . . 9 ((intβ€˜(MetOpenβ€˜(abs ∘ βˆ’ )))β€˜{𝑀 ∈ β„‚ ∣ 𝑀 # 0}) = {𝑀 ∈ β„‚ ∣ 𝑀 # 0}
5850, 57eleqtrrdi 2271 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ 𝑦 ∈ ((intβ€˜(MetOpenβ€˜(abs ∘ βˆ’ )))β€˜{𝑀 ∈ β„‚ ∣ 𝑀 # 0}))
5938sqvald 10654 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (𝑦↑2) = (𝑦 Β· 𝑦))
6059oveq2d 5894 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (𝐴 / (𝑦↑2)) = (𝐴 / (𝑦 Β· 𝑦)))
6139, 38, 38, 44, 44divdivap1d 8782 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ ((𝐴 / 𝑦) / 𝑦) = (𝐴 / (𝑦 Β· 𝑦)))
6260, 61eqtr4d 2213 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (𝐴 / (𝑦↑2)) = ((𝐴 / 𝑦) / 𝑦))
6362negeqd 8155 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ -(𝐴 / (𝑦↑2)) = -((𝐴 / 𝑦) / 𝑦))
6439, 38, 44divclapd 8750 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (𝐴 / 𝑦) ∈ β„‚)
6564, 38, 44divnegapd 8763 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ -((𝐴 / 𝑦) / 𝑦) = (-(𝐴 / 𝑦) / 𝑦))
6663, 65eqtrd 2210 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ -(𝐴 / (𝑦↑2)) = (-(𝐴 / 𝑦) / 𝑦))
6764negcld 8258 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ -(𝐴 / 𝑦) ∈ β„‚)
68 eqid 2177 . . . . . . . . . . . . 13 (𝑧 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) = (𝑧 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧))
6968cdivcncfap 14275 . . . . . . . . . . . 12 (-(𝐴 / 𝑦) ∈ β„‚ β†’ (𝑧 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) ∈ ({𝑀 ∈ β„‚ ∣ 𝑀 # 0}–cnβ†’β„‚))
7067, 69syl 14 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (𝑧 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) ∈ ({𝑀 ∈ β„‚ ∣ 𝑀 # 0}–cnβ†’β„‚))
71 oveq2 5886 . . . . . . . . . . 11 (𝑧 = 𝑦 β†’ (-(𝐴 / 𝑦) / 𝑧) = (-(𝐴 / 𝑦) / 𝑦))
7270, 50, 71cnmptlimc 14331 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (-(𝐴 / 𝑦) / 𝑦) ∈ ((𝑧 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) limβ„‚ 𝑦))
7366, 72eqeltrd 2254 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ -(𝐴 / (𝑦↑2)) ∈ ((𝑧 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) limβ„‚ 𝑦))
74 cncff 14252 . . . . . . . . . . . 12 ((𝑧 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) ∈ ({𝑀 ∈ β„‚ ∣ 𝑀 # 0}–cnβ†’β„‚) β†’ (𝑧 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)):{𝑀 ∈ β„‚ ∣ 𝑀 # 0}βŸΆβ„‚)
7570, 74syl 14 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (𝑧 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)):{𝑀 ∈ β„‚ ∣ 𝑀 # 0}βŸΆβ„‚)
7622a1i 9 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} βŠ† β„‚)
7775, 76limcdifap 14319 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ ((𝑧 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) limβ„‚ 𝑦) = (((𝑧 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) β†Ύ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) limβ„‚ 𝑦))
78 elrabi 2892 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦} β†’ 𝑧 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0})
7978adantl 277 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ 𝑧 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0})
80 breq1 4008 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 = 𝑧 β†’ (𝑀 # 0 ↔ 𝑧 # 0))
8180elrab 2895 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↔ (𝑧 ∈ β„‚ ∧ 𝑧 # 0))
8279, 81sylib 122 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ (𝑧 ∈ β„‚ ∧ 𝑧 # 0))
8382simpld 112 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ 𝑧 ∈ β„‚)
8437ad2antlr 489 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ 𝑦 ∈ β„‚)
8583, 84subcld 8271 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ (𝑧 βˆ’ 𝑦) ∈ β„‚)
8664adantr 276 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ (𝐴 / 𝑦) ∈ β„‚)
8781simprbi 275 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} β†’ 𝑧 # 0)
8879, 87syl 14 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ 𝑧 # 0)
8986, 83, 88divclapd 8750 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ ((𝐴 / 𝑦) / 𝑧) ∈ β„‚)
90 mulneg12 8357 . . . . . . . . . . . . . . . . 17 (((𝑧 βˆ’ 𝑦) ∈ β„‚ ∧ ((𝐴 / 𝑦) / 𝑧) ∈ β„‚) β†’ (-(𝑧 βˆ’ 𝑦) Β· ((𝐴 / 𝑦) / 𝑧)) = ((𝑧 βˆ’ 𝑦) Β· -((𝐴 / 𝑦) / 𝑧)))
9185, 89, 90syl2anc 411 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ (-(𝑧 βˆ’ 𝑦) Β· ((𝐴 / 𝑦) / 𝑧)) = ((𝑧 βˆ’ 𝑦) Β· -((𝐴 / 𝑦) / 𝑧)))
9284, 83, 89subdird 8375 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ ((𝑦 βˆ’ 𝑧) Β· ((𝐴 / 𝑦) / 𝑧)) = ((𝑦 Β· ((𝐴 / 𝑦) / 𝑧)) βˆ’ (𝑧 Β· ((𝐴 / 𝑦) / 𝑧))))
9383, 84negsubdi2d 8287 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ -(𝑧 βˆ’ 𝑦) = (𝑦 βˆ’ 𝑧))
9493oveq1d 5893 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ (-(𝑧 βˆ’ 𝑦) Β· ((𝐴 / 𝑦) / 𝑧)) = ((𝑦 βˆ’ 𝑧) Β· ((𝐴 / 𝑦) / 𝑧)))
95 oveq2 5886 . . . . . . . . . . . . . . . . . . . 20 (π‘₯ = 𝑧 β†’ (𝐴 / π‘₯) = (𝐴 / 𝑧))
96 simpll 527 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ 𝐴 ∈ β„‚)
9796, 83, 88divclapd 8750 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ (𝐴 / 𝑧) ∈ β„‚)
9815, 95, 79, 97fvmptd3 5612 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ ((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘§) = (𝐴 / 𝑧))
9943ad2antlr 489 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ 𝑦 # 0)
10096, 84, 99divcanap2d 8752 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ (𝑦 Β· (𝐴 / 𝑦)) = 𝐴)
101100oveq1d 5893 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ ((𝑦 Β· (𝐴 / 𝑦)) / 𝑧) = (𝐴 / 𝑧))
10284, 86, 83, 88divassapd 8786 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ ((𝑦 Β· (𝐴 / 𝑦)) / 𝑧) = (𝑦 Β· ((𝐴 / 𝑦) / 𝑧)))
10398, 101, 1023eqtr2d 2216 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ ((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘§) = (𝑦 Β· ((𝐴 / 𝑦) / 𝑧)))
104 oveq2 5886 . . . . . . . . . . . . . . . . . . . 20 (π‘₯ = 𝑦 β†’ (𝐴 / π‘₯) = (𝐴 / 𝑦))
10550adantr 276 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0})
10615, 104, 105, 86fvmptd3 5612 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ ((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘¦) = (𝐴 / 𝑦))
10786, 83, 88divcanap2d 8752 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ (𝑧 Β· ((𝐴 / 𝑦) / 𝑧)) = (𝐴 / 𝑦))
108106, 107eqtr4d 2213 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ ((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘¦) = (𝑧 Β· ((𝐴 / 𝑦) / 𝑧)))
109103, 108oveq12d 5896 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ (((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘§) βˆ’ ((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘¦)) = ((𝑦 Β· ((𝐴 / 𝑦) / 𝑧)) βˆ’ (𝑧 Β· ((𝐴 / 𝑦) / 𝑧))))
11092, 94, 1093eqtr4d 2220 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ (-(𝑧 βˆ’ 𝑦) Β· ((𝐴 / 𝑦) / 𝑧)) = (((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘§) βˆ’ ((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘¦)))
11186, 83, 88divnegapd 8763 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ -((𝐴 / 𝑦) / 𝑧) = (-(𝐴 / 𝑦) / 𝑧))
112111oveq2d 5894 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ ((𝑧 βˆ’ 𝑦) Β· -((𝐴 / 𝑦) / 𝑧)) = ((𝑧 βˆ’ 𝑦) Β· (-(𝐴 / 𝑦) / 𝑧)))
11391, 110, 1123eqtr3d 2218 . . . . . . . . . . . . . . 15 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ (((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘§) βˆ’ ((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘¦)) = ((𝑧 βˆ’ 𝑦) Β· (-(𝐴 / 𝑦) / 𝑧)))
114113oveq1d 5893 . . . . . . . . . . . . . 14 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ ((((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘§) βˆ’ ((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘¦)) / (𝑧 βˆ’ 𝑦)) = (((𝑧 βˆ’ 𝑦) Β· (-(𝐴 / 𝑦) / 𝑧)) / (𝑧 βˆ’ 𝑦)))
11586negcld 8258 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ -(𝐴 / 𝑦) ∈ β„‚)
116115, 83, 88divclapd 8750 . . . . . . . . . . . . . . 15 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ (-(𝐴 / 𝑦) / 𝑧) ∈ β„‚)
117 breq1 4008 . . . . . . . . . . . . . . . . . . 19 (π‘˜ = 𝑧 β†’ (π‘˜ # 𝑦 ↔ 𝑧 # 𝑦))
118117elrab 2895 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦} ↔ (𝑧 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∧ 𝑧 # 𝑦))
119118simprbi 275 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦} β†’ 𝑧 # 𝑦)
120119adantl 277 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ 𝑧 # 𝑦)
12183, 84, 120subap0d 8604 . . . . . . . . . . . . . . 15 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ (𝑧 βˆ’ 𝑦) # 0)
122116, 85, 121divcanap3d 8755 . . . . . . . . . . . . . 14 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ (((𝑧 βˆ’ 𝑦) Β· (-(𝐴 / 𝑦) / 𝑧)) / (𝑧 βˆ’ 𝑦)) = (-(𝐴 / 𝑦) / 𝑧))
123114, 122eqtrd 2210 . . . . . . . . . . . . 13 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ ((((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘§) βˆ’ ((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘¦)) / (𝑧 βˆ’ 𝑦)) = (-(𝐴 / 𝑦) / 𝑧))
124123mpteq2dva 4095 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦} ↦ ((((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘§) βˆ’ ((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘¦)) / (𝑧 βˆ’ 𝑦))) = (𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦} ↦ (-(𝐴 / 𝑦) / 𝑧)))
125 ssrab2 3242 . . . . . . . . . . . . 13 {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦} βŠ† {𝑀 ∈ β„‚ ∣ 𝑀 # 0}
126 resmpt 4957 . . . . . . . . . . . . 13 ({π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦} βŠ† {𝑀 ∈ β„‚ ∣ 𝑀 # 0} β†’ ((𝑧 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) β†Ύ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) = (𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦} ↦ (-(𝐴 / 𝑦) / 𝑧)))
127125, 126ax-mp 5 . . . . . . . . . . . 12 ((𝑧 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) β†Ύ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) = (𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦} ↦ (-(𝐴 / 𝑦) / 𝑧))
128124, 127eqtr4di 2228 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦} ↦ ((((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘§) βˆ’ ((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘¦)) / (𝑧 βˆ’ 𝑦))) = ((𝑧 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) β†Ύ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}))
129128oveq1d 5893 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ ((𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦} ↦ ((((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘§) βˆ’ ((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘¦)) / (𝑧 βˆ’ 𝑦))) limβ„‚ 𝑦) = (((𝑧 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) β†Ύ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) limβ„‚ 𝑦))
13077, 129eqtr4d 2213 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ ((𝑧 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) limβ„‚ 𝑦) = ((𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦} ↦ ((((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘§) βˆ’ ((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘¦)) / (𝑧 βˆ’ 𝑦))) limβ„‚ 𝑦))
13173, 130eleqtrd 2256 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ -(𝐴 / (𝑦↑2)) ∈ ((𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦} ↦ ((((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘§) βˆ’ ((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘¦)) / (𝑧 βˆ’ 𝑦))) limβ„‚ 𝑦))
13251cntoptopon 14220 . . . . . . . . . 10 (MetOpenβ€˜(abs ∘ βˆ’ )) ∈ (TopOnβ€˜β„‚)
133132toponrestid 13709 . . . . . . . . 9 (MetOpenβ€˜(abs ∘ βˆ’ )) = ((MetOpenβ€˜(abs ∘ βˆ’ )) β†Ύt β„‚)
134 eqid 2177 . . . . . . . . 9 (𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦} ↦ ((((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘§) βˆ’ ((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘¦)) / (𝑧 βˆ’ 𝑦))) = (𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦} ↦ ((((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘§) βˆ’ ((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘¦)) / (𝑧 βˆ’ 𝑦)))
135 ssidd 3178 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ β„‚ βŠ† β„‚)
13634adantr 276 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)):{𝑀 ∈ β„‚ ∣ 𝑀 # 0}βŸΆβ„‚)
137133, 51, 134, 135, 136, 76eldvap 14339 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (𝑦(β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)))-(𝐴 / (𝑦↑2)) ↔ (𝑦 ∈ ((intβ€˜(MetOpenβ€˜(abs ∘ βˆ’ )))β€˜{𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ -(𝐴 / (𝑦↑2)) ∈ ((𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦} ↦ ((((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘§) βˆ’ ((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))β€˜π‘¦)) / (𝑧 βˆ’ 𝑦))) limβ„‚ 𝑦))))
13858, 131, 137mpbir2and 944 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ 𝑦(β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)))-(𝐴 / (𝑦↑2)))
139 breldmg 4835 . . . . . . 7 ((𝑦 ∈ β„‚ ∧ -(𝐴 / (𝑦↑2)) ∈ β„‚ ∧ 𝑦(β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)))-(𝐴 / (𝑦↑2))) β†’ 𝑦 ∈ dom (β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))))
14038, 49, 138, 139syl3anc 1238 . . . . . 6 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ 𝑦 ∈ dom (β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))))
14136, 140eqelssd 3176 . . . . 5 (𝐴 ∈ β„‚ β†’ dom (β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))) = {𝑀 ∈ β„‚ ∣ 𝑀 # 0})
142141feq2d 5355 . . . 4 (𝐴 ∈ β„‚ β†’ ((β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))):dom (β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)))βŸΆβ„‚ ↔ (β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))):{𝑀 ∈ β„‚ ∣ 𝑀 # 0}βŸΆβ„‚))
14329, 142mpbid 147 . . 3 (𝐴 ∈ β„‚ β†’ (β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))):{𝑀 ∈ β„‚ ∣ 𝑀 # 0}βŸΆβ„‚)
144143ffnd 5368 . 2 (𝐴 ∈ β„‚ β†’ (β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))) Fn {𝑀 ∈ β„‚ ∣ 𝑀 # 0})
14511sqcld 10655 . . . . . 6 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (π‘₯↑2) ∈ β„‚)
146 sqap0 10590 . . . . . . . 8 (π‘₯ ∈ β„‚ β†’ ((π‘₯↑2) # 0 ↔ π‘₯ # 0))
14711, 146syl 14 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ ((π‘₯↑2) # 0 ↔ π‘₯ # 0))
14812, 147mpbird 167 . . . . . 6 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (π‘₯↑2) # 0)
1496, 145, 148divclapd 8750 . . . . 5 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (𝐴 / (π‘₯↑2)) ∈ β„‚)
150149negcld 8258 . . . 4 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ -(𝐴 / (π‘₯↑2)) ∈ β„‚)
151150ralrimiva 2550 . . 3 (𝐴 ∈ β„‚ β†’ βˆ€π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}-(𝐴 / (π‘₯↑2)) ∈ β„‚)
152 eqid 2177 . . . 4 (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ -(𝐴 / (π‘₯↑2))) = (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ -(𝐴 / (π‘₯↑2)))
153152fnmpt 5344 . . 3 (βˆ€π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}-(𝐴 / (π‘₯↑2)) ∈ β„‚ β†’ (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ -(𝐴 / (π‘₯↑2))) Fn {𝑀 ∈ β„‚ ∣ 𝑀 # 0})
154151, 153syl 14 . 2 (𝐴 ∈ β„‚ β†’ (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ -(𝐴 / (π‘₯↑2))) Fn {𝑀 ∈ β„‚ ∣ 𝑀 # 0})
15529ffund 5371 . . . . 5 (𝐴 ∈ β„‚ β†’ Fun (β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))))
156155adantr 276 . . . 4 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ Fun (β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))))
157 funbrfv 5557 . . . 4 (Fun (β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))) β†’ (𝑦(β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)))-(𝐴 / (𝑦↑2)) β†’ ((β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)))β€˜π‘¦) = -(𝐴 / (𝑦↑2))))
158156, 138, 157sylc 62 . . 3 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ ((β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)))β€˜π‘¦) = -(𝐴 / (𝑦↑2)))
159 oveq1 5885 . . . . . 6 (π‘₯ = 𝑦 β†’ (π‘₯↑2) = (𝑦↑2))
160159oveq2d 5894 . . . . 5 (π‘₯ = 𝑦 β†’ (𝐴 / (π‘₯↑2)) = (𝐴 / (𝑦↑2)))
161160negeqd 8155 . . . 4 (π‘₯ = 𝑦 β†’ -(𝐴 / (π‘₯↑2)) = -(𝐴 / (𝑦↑2)))
162152, 161, 50, 49fvmptd3 5612 . . 3 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ ((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ -(𝐴 / (π‘₯↑2)))β€˜π‘¦) = -(𝐴 / (𝑦↑2)))
163158, 162eqtr4d 2213 . 2 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ ((β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)))β€˜π‘¦) = ((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ -(𝐴 / (π‘₯↑2)))β€˜π‘¦))
164144, 154, 163eqfnfvd 5619 1 (𝐴 ∈ β„‚ β†’ (β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))) = (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ -(𝐴 / (π‘₯↑2))))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  βˆ€wral 2455  {crab 2459  Vcvv 2739   βŠ† wss 3131   class class class wbr 4005   ↦ cmpt 4066  dom cdm 4628  ran crn 4629   β†Ύ cres 4630   ∘ ccom 4632  Fun wfun 5212   Fn wfn 5213  βŸΆwf 5214  β€“ontoβ†’wfo 5216  β€˜cfv 5218  (class class class)co 5878   ↑pm cpm 6652  β„‚cc 7812  0cc0 7814   Β· cmul 7819   βˆ’ cmin 8131  -cneg 8132   # cap 8541   / cdiv 8632  2c2 8973  β†‘cexp 10522  abscabs 11009  MetOpencmopn 13619  Topctop 13685  intcnt 13781  β€“cnβ†’ccncf 14245   limβ„‚ climc 14311   D cdv 14312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-mulrcl 7913  ax-addcom 7914  ax-mulcom 7915  ax-addass 7916  ax-mulass 7917  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-1rid 7921  ax-0id 7922  ax-rnegex 7923  ax-precex 7924  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-apti 7929  ax-pre-ltadd 7930  ax-pre-mulgt0 7931  ax-pre-mulext 7932  ax-arch 7933  ax-caucvg 7934
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-frec 6395  df-map 6653  df-pm 6654  df-sup 6986  df-inf 6987  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-reap 8535  df-ap 8542  df-div 8633  df-inn 8923  df-2 8981  df-3 8982  df-4 8983  df-n0 9180  df-z 9257  df-uz 9532  df-q 9623  df-rp 9657  df-xneg 9775  df-xadd 9776  df-seqfrec 10449  df-exp 10523  df-cj 10854  df-re 10855  df-im 10856  df-rsqrt 11010  df-abs 11011  df-rest 12696  df-topgen 12715  df-psmet 13621  df-xmet 13622  df-met 13623  df-bl 13624  df-mopn 13625  df-top 13686  df-topon 13699  df-bases 13731  df-ntr 13784  df-cn 13876  df-cnp 13877  df-cncf 14246  df-limced 14313  df-dvap 14314
This theorem is referenced by: (None)
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