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Theorem dvrecap 14560
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 5268 . . . . . . . . 9 Fun (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))
2 funforn 5459 . . . . . . . . 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 5452 . . . . . . . 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 4020 . . . . . . . . . . . . . 14 (𝑀 = π‘₯ β†’ (𝑀 # 0 ↔ π‘₯ # 0))
87elrab 2907 . . . . . . . . . . . . 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 8764 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (𝐴 / π‘₯) ∈ β„‚)
1413ralrimiva 2562 . . . . . . . 8 (𝐴 ∈ β„‚ β†’ βˆ€π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} (𝐴 / π‘₯) ∈ β„‚)
15 eqid 2188 . . . . . . . . 9 (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)) = (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))
1615rnmptss 5692 . . . . . . . 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 5391 . . . . . . 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 5138 . . . . . . 7 dom (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)) = {π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ (𝐴 / π‘₯) ∈ V}
21 ssrab2 3254 . . . . . . . 8 {π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ (𝐴 / π‘₯) ∈ V} βŠ† {𝑀 ∈ β„‚ ∣ 𝑀 # 0}
22 ssrab2 3254 . . . . . . . 8 {𝑀 ∈ β„‚ ∣ 𝑀 # 0} βŠ† β„‚
2321, 22sstri 3178 . . . . . . 7 {π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ (𝐴 / π‘₯) ∈ V} βŠ† β„‚
2420, 23eqsstri 3201 . . . . . 6 dom (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)) βŠ† β„‚
25 cnex 7952 . . . . . . 7 β„‚ ∈ V
2625, 25elpm2 6697 . . . . . 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 14542 . . . . 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 3190 . . . . . . 7 (𝐴 ∈ β„‚ β†’ β„‚ βŠ† β„‚)
31 divclap 8652 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ β„‚ ∧ π‘₯ # 0) β†’ (𝐴 / π‘₯) ∈ β„‚)
32313expb 1205 . . . . . . . . 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 5686 . . . . . . 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 14537 . . . . . 6 (𝐴 ∈ β„‚ β†’ dom (β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))) βŠ† {𝑀 ∈ β„‚ ∣ 𝑀 # 0})
37 elrabi 2904 . . . . . . . 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 10669 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (𝑦↑2) ∈ β„‚)
41 breq1 4020 . . . . . . . . . . . . 13 (𝑀 = 𝑦 β†’ (𝑀 # 0 ↔ 𝑦 # 0))
4241elrab 2907 . . . . . . . . . . . 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 10604 . . . . . . . . . . 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 8764 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (𝐴 / (𝑦↑2)) ∈ β„‚)
4948negcld 8272 . . . . . . 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 2188 . . . . . . . . . . 11 (MetOpenβ€˜(abs ∘ βˆ’ )) = (MetOpenβ€˜(abs ∘ βˆ’ ))
5251cntoptop 14416 . . . . . . . . . 10 (MetOpenβ€˜(abs ∘ βˆ’ )) ∈ Top
53 0cn 7966 . . . . . . . . . . 11 0 ∈ β„‚
54 cnopnap 14477 . . . . . . . . . . 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 14018 . . . . . . . . . 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 2282 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ 𝑦 ∈ ((intβ€˜(MetOpenβ€˜(abs ∘ βˆ’ )))β€˜{𝑀 ∈ β„‚ ∣ 𝑀 # 0}))
5938sqvald 10668 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (𝑦↑2) = (𝑦 Β· 𝑦))
6059oveq2d 5906 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (𝐴 / (𝑦↑2)) = (𝐴 / (𝑦 Β· 𝑦)))
6139, 38, 38, 44, 44divdivap1d 8796 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ ((𝐴 / 𝑦) / 𝑦) = (𝐴 / (𝑦 Β· 𝑦)))
6260, 61eqtr4d 2224 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (𝐴 / (𝑦↑2)) = ((𝐴 / 𝑦) / 𝑦))
6362negeqd 8169 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ -(𝐴 / (𝑦↑2)) = -((𝐴 / 𝑦) / 𝑦))
6439, 38, 44divclapd 8764 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (𝐴 / 𝑦) ∈ β„‚)
6564, 38, 44divnegapd 8777 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ -((𝐴 / 𝑦) / 𝑦) = (-(𝐴 / 𝑦) / 𝑦))
6663, 65eqtrd 2221 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ -(𝐴 / (𝑦↑2)) = (-(𝐴 / 𝑦) / 𝑦))
6764negcld 8272 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ -(𝐴 / 𝑦) ∈ β„‚)
68 eqid 2188 . . . . . . . . . . . . 13 (𝑧 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) = (𝑧 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧))
6968cdivcncfap 14470 . . . . . . . . . . . 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 5898 . . . . . . . . . . 11 (𝑧 = 𝑦 β†’ (-(𝐴 / 𝑦) / 𝑧) = (-(𝐴 / 𝑦) / 𝑦))
7270, 50, 71cnmptlimc 14526 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (-(𝐴 / 𝑦) / 𝑦) ∈ ((𝑧 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) limβ„‚ 𝑦))
7366, 72eqeltrd 2265 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ -(𝐴 / (𝑦↑2)) ∈ ((𝑧 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (-(𝐴 / 𝑦) / 𝑧)) limβ„‚ 𝑦))
74 cncff 14447 . . . . . . . . . . . 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 14514 . . . . . . . . . 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 2904 . . . . . . . . . . . . . . . . . . . . 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 4020 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 = 𝑧 β†’ (𝑀 # 0 ↔ 𝑧 # 0))
8180elrab 2907 . . . . . . . . . . . . . . . . . . . 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 8285 . . . . . . . . . . . . . . . . 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 8764 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ ((𝐴 / 𝑦) / 𝑧) ∈ β„‚)
90 mulneg12 8371 . . . . . . . . . . . . . . . . 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 8389 . . . . . . . . . . . . . . . . 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 8301 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ -(𝑧 βˆ’ 𝑦) = (𝑦 βˆ’ 𝑧))
9493oveq1d 5905 . . . . . . . . . . . . . . . . 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 5898 . . . . . . . . . . . . . . . . . . . 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 8764 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ (𝐴 / 𝑧) ∈ β„‚)
9815, 95, 79, 97fvmptd3 5624 . . . . . . . . . . . . . . . . . . 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 8766 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ (𝑦 Β· (𝐴 / 𝑦)) = 𝐴)
101100oveq1d 5905 . . . . . . . . . . . . . . . . . . 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 8800 . . . . . . . . . . . . . . . . . . 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 2227 . . . . . . . . . . . . . . . . . 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 5898 . . . . . . . . . . . . . . . . . . . 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 5624 . . . . . . . . . . . . . . . . . . 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 8766 . . . . . . . . . . . . . . . . . . 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 2224 . . . . . . . . . . . . . . . . . 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 5908 . . . . . . . . . . . . . . . . 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 2231 . . . . . . . . . . . . . . . 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 8777 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ -((𝐴 / 𝑦) / 𝑧) = (-(𝐴 / 𝑦) / 𝑧))
112111oveq2d 5906 . . . . . . . . . . . . . . . 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 2229 . . . . . . . . . . . . . . 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 5905 . . . . . . . . . . . . . 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 8272 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ -(𝐴 / 𝑦) ∈ β„‚)
116115, 83, 88divclapd 8764 . . . . . . . . . . . . . . 15 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ (-(𝐴 / 𝑦) / 𝑧) ∈ β„‚)
117 breq1 4020 . . . . . . . . . . . . . . . . . . 19 (π‘˜ = 𝑧 β†’ (π‘˜ # 𝑦 ↔ 𝑧 # 𝑦))
118117elrab 2907 . . . . . . . . . . . . . . . . . 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 8618 . . . . . . . . . . . . . . 15 (((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) ∧ 𝑧 ∈ {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦}) β†’ (𝑧 βˆ’ 𝑦) # 0)
122116, 85, 121divcanap3d 8769 . . . . . . . . . . . . . 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 2221 . . . . . . . . . . . . 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 4107 . . . . . . . . . . . 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 3254 . . . . . . . . . . . . 13 {π‘˜ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ∣ π‘˜ # 𝑦} βŠ† {𝑀 ∈ β„‚ ∣ 𝑀 # 0}
126 resmpt 4969 . . . . . . . . . . . . 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 2239 . . . . . . . . . . 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 5905 . . . . . . . . . 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 2224 . . . . . . . . 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 2267 . . . . . . . 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 14415 . . . . . . . . . 10 (MetOpenβ€˜(abs ∘ βˆ’ )) ∈ (TopOnβ€˜β„‚)
133132toponrestid 13904 . . . . . . . . 9 (MetOpenβ€˜(abs ∘ βˆ’ )) = ((MetOpenβ€˜(abs ∘ βˆ’ )) β†Ύt β„‚)
134 eqid 2188 . . . . . . . . 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 3190 . . . . . . . . 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 14534 . . . . . . . 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 945 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ 𝑦(β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯)))-(𝐴 / (𝑦↑2)))
139 breldmg 4847 . . . . . . 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 1248 . . . . . 6 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ 𝑦 ∈ dom (β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))))
14136, 140eqelssd 3188 . . . . 5 (𝐴 ∈ β„‚ β†’ dom (β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))) = {𝑀 ∈ β„‚ ∣ 𝑀 # 0})
142141feq2d 5367 . . . 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 5380 . 2 (𝐴 ∈ β„‚ β†’ (β„‚ D (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ (𝐴 / π‘₯))) Fn {𝑀 ∈ β„‚ ∣ 𝑀 # 0})
14511sqcld 10669 . . . . . 6 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (π‘₯↑2) ∈ β„‚)
146 sqap0 10604 . . . . . . . 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 8764 . . . . 5 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ (𝐴 / (π‘₯↑2)) ∈ β„‚)
150149negcld 8272 . . . 4 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ -(𝐴 / (π‘₯↑2)) ∈ β„‚)
151150ralrimiva 2562 . . 3 (𝐴 ∈ β„‚ β†’ βˆ€π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}-(𝐴 / (π‘₯↑2)) ∈ β„‚)
152 eqid 2188 . . . 4 (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ -(𝐴 / (π‘₯↑2))) = (π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ -(𝐴 / (π‘₯↑2)))
153152fnmpt 5356 . . 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 5383 . . . . 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 5569 . . . 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 5897 . . . . . 6 (π‘₯ = 𝑦 β†’ (π‘₯↑2) = (𝑦↑2))
160159oveq2d 5906 . . . . 5 (π‘₯ = 𝑦 β†’ (𝐴 / (π‘₯↑2)) = (𝐴 / (𝑦↑2)))
161160negeqd 8169 . . . 4 (π‘₯ = 𝑦 β†’ -(𝐴 / (π‘₯↑2)) = -(𝐴 / (𝑦↑2)))
162152, 161, 50, 49fvmptd3 5624 . . 3 ((𝐴 ∈ β„‚ ∧ 𝑦 ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0}) β†’ ((π‘₯ ∈ {𝑀 ∈ β„‚ ∣ 𝑀 # 0} ↦ -(𝐴 / (π‘₯↑2)))β€˜π‘¦) = -(𝐴 / (𝑦↑2)))
163158, 162eqtr4d 2224 . 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 5631 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 1363   ∈ wcel 2159  βˆ€wral 2467  {crab 2471  Vcvv 2751   βŠ† wss 3143   class class class wbr 4017   ↦ cmpt 4078  dom cdm 4640  ran crn 4641   β†Ύ cres 4642   ∘ ccom 4644  Fun wfun 5224   Fn wfn 5225  βŸΆwf 5226  β€“ontoβ†’wfo 5228  β€˜cfv 5230  (class class class)co 5890   ↑pm cpm 6666  β„‚cc 7826  0cc0 7828   Β· cmul 7833   βˆ’ cmin 8145  -cneg 8146   # cap 8555   / cdiv 8646  2c2 8987  β†‘cexp 10536  abscabs 11023  MetOpencmopn 13814  Topctop 13880  intcnt 13976  β€“cnβ†’ccncf 14440   limβ„‚ climc 14506   D cdv 14507
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-coll 4132  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-iinf 4601  ax-cnex 7919  ax-resscn 7920  ax-1cn 7921  ax-1re 7922  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-mulrcl 7927  ax-addcom 7928  ax-mulcom 7929  ax-addass 7930  ax-mulass 7931  ax-distr 7932  ax-i2m1 7933  ax-0lt1 7934  ax-1rid 7935  ax-0id 7936  ax-rnegex 7937  ax-precex 7938  ax-cnre 7939  ax-pre-ltirr 7940  ax-pre-ltwlin 7941  ax-pre-lttrn 7942  ax-pre-apti 7943  ax-pre-ltadd 7944  ax-pre-mulgt0 7945  ax-pre-mulext 7946  ax-arch 7947  ax-caucvg 7948
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-reu 2474  df-rmo 2475  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-if 3549  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-tr 4116  df-id 4307  df-po 4310  df-iso 4311  df-iord 4380  df-on 4382  df-ilim 4383  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-isom 5239  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-recs 6323  df-frec 6409  df-map 6667  df-pm 6668  df-sup 7000  df-inf 7001  df-pnf 8011  df-mnf 8012  df-xr 8013  df-ltxr 8014  df-le 8015  df-sub 8147  df-neg 8148  df-reap 8549  df-ap 8556  df-div 8647  df-inn 8937  df-2 8995  df-3 8996  df-4 8997  df-n0 9194  df-z 9271  df-uz 9546  df-q 9637  df-rp 9671  df-xneg 9789  df-xadd 9790  df-seqfrec 10463  df-exp 10537  df-cj 10868  df-re 10869  df-im 10870  df-rsqrt 11024  df-abs 11025  df-rest 12711  df-topgen 12730  df-psmet 13816  df-xmet 13817  df-met 13818  df-bl 13819  df-mopn 13820  df-top 13881  df-topon 13894  df-bases 13926  df-ntr 13979  df-cn 14071  df-cnp 14072  df-cncf 14441  df-limced 14508  df-dvap 14509
This theorem is referenced by: (None)
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