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Theorem efrlimOLD 26843
Description: Obsolete version of efrlim 26842 as of 19-Apr-2025. (Contributed by Mario Carneiro, 1-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
efrlimOLD.1 𝑆 = (0(ballβ€˜(abs ∘ βˆ’ ))(1 / ((absβ€˜π΄) + 1)))
Assertion
Ref Expression
efrlimOLD (𝐴 ∈ β„‚ β†’ (π‘˜ ∈ ℝ+ ↦ ((1 + (𝐴 / π‘˜))β†‘π‘π‘˜)) β‡π‘Ÿ (expβ€˜π΄))
Distinct variable group:   𝐴,π‘˜
Allowed substitution hint:   𝑆(π‘˜)

Proof of Theorem efrlimOLD
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rge0ssre 13434 . . . . . . . 8 (0[,)+∞) βŠ† ℝ
2 ax-resscn 11164 . . . . . . . 8 ℝ βŠ† β„‚
31, 2sstri 3984 . . . . . . 7 (0[,)+∞) βŠ† β„‚
43sseli 3971 . . . . . 6 (π‘₯ ∈ (0[,)+∞) β†’ π‘₯ ∈ β„‚)
5 simpll 764 . . . . . . . . . . 11 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ β„‚) ∧ Β¬ π‘₯ = 0) β†’ 𝐴 ∈ β„‚)
6 1cnd 11208 . . . . . . . . . . 11 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ β„‚) ∧ Β¬ π‘₯ = 0) β†’ 1 ∈ β„‚)
7 simplr 766 . . . . . . . . . . 11 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ β„‚) ∧ Β¬ π‘₯ = 0) β†’ π‘₯ ∈ β„‚)
8 ax-1ne0 11176 . . . . . . . . . . . 12 1 β‰  0
98a1i 11 . . . . . . . . . . 11 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ β„‚) ∧ Β¬ π‘₯ = 0) β†’ 1 β‰  0)
10 simpr 484 . . . . . . . . . . . 12 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ β„‚) ∧ Β¬ π‘₯ = 0) β†’ Β¬ π‘₯ = 0)
1110neqned 2939 . . . . . . . . . . 11 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ β„‚) ∧ Β¬ π‘₯ = 0) β†’ π‘₯ β‰  0)
125, 6, 7, 9, 11divdiv2d 12021 . . . . . . . . . 10 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ β„‚) ∧ Β¬ π‘₯ = 0) β†’ (𝐴 / (1 / π‘₯)) = ((𝐴 Β· π‘₯) / 1))
13 mulcl 11191 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ β„‚) β†’ (𝐴 Β· π‘₯) ∈ β„‚)
1413adantr 480 . . . . . . . . . . 11 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ β„‚) ∧ Β¬ π‘₯ = 0) β†’ (𝐴 Β· π‘₯) ∈ β„‚)
1514div1d 11981 . . . . . . . . . 10 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ β„‚) ∧ Β¬ π‘₯ = 0) β†’ ((𝐴 Β· π‘₯) / 1) = (𝐴 Β· π‘₯))
1612, 15eqtrd 2764 . . . . . . . . 9 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ β„‚) ∧ Β¬ π‘₯ = 0) β†’ (𝐴 / (1 / π‘₯)) = (𝐴 Β· π‘₯))
1716oveq2d 7418 . . . . . . . 8 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ β„‚) ∧ Β¬ π‘₯ = 0) β†’ (1 + (𝐴 / (1 / π‘₯))) = (1 + (𝐴 Β· π‘₯)))
1817oveq1d 7417 . . . . . . 7 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ β„‚) ∧ Β¬ π‘₯ = 0) β†’ ((1 + (𝐴 / (1 / π‘₯)))↑𝑐(1 / π‘₯)) = ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯)))
1918ifeq2da 4553 . . . . . 6 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ β„‚) β†’ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 / (1 / π‘₯)))↑𝑐(1 / π‘₯))) = if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯))))
204, 19sylan2 592 . . . . 5 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ (0[,)+∞)) β†’ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 / (1 / π‘₯)))↑𝑐(1 / π‘₯))) = if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯))))
2120mpteq2dva 5239 . . . 4 (𝐴 ∈ β„‚ β†’ (π‘₯ ∈ (0[,)+∞) ↦ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 / (1 / π‘₯)))↑𝑐(1 / π‘₯)))) = (π‘₯ ∈ (0[,)+∞) ↦ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯)))))
22 resmpt 6028 . . . . 5 ((0[,)+∞) βŠ† β„‚ β†’ ((π‘₯ ∈ β„‚ ↦ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯)))) β†Ύ (0[,)+∞)) = (π‘₯ ∈ (0[,)+∞) ↦ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯)))))
233, 22ax-mp 5 . . . 4 ((π‘₯ ∈ β„‚ ↦ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯)))) β†Ύ (0[,)+∞)) = (π‘₯ ∈ (0[,)+∞) ↦ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯))))
2421, 23eqtr4di 2782 . . 3 (𝐴 ∈ β„‚ β†’ (π‘₯ ∈ (0[,)+∞) ↦ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 / (1 / π‘₯)))↑𝑐(1 / π‘₯)))) = ((π‘₯ ∈ β„‚ ↦ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯)))) β†Ύ (0[,)+∞)))
25 0e0icopnf 13436 . . . . 5 0 ∈ (0[,)+∞)
2625a1i 11 . . . 4 (𝐴 ∈ β„‚ β†’ 0 ∈ (0[,)+∞))
27 eqeq2 2736 . . . . . . . . 9 ((expβ€˜(𝐴 Β· 1)) = if((𝐴 Β· π‘₯) = 0, (expβ€˜(𝐴 Β· 1)), (expβ€˜(𝐴 Β· ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))))) β†’ (if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯))) = (expβ€˜(𝐴 Β· 1)) ↔ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯))) = if((𝐴 Β· π‘₯) = 0, (expβ€˜(𝐴 Β· 1)), (expβ€˜(𝐴 Β· ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯)))))))
28 eqeq2 2736 . . . . . . . . 9 ((expβ€˜(𝐴 Β· ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯)))) = if((𝐴 Β· π‘₯) = 0, (expβ€˜(𝐴 Β· 1)), (expβ€˜(𝐴 Β· ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))))) β†’ (if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯))) = (expβ€˜(𝐴 Β· ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯)))) ↔ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯))) = if((𝐴 Β· π‘₯) = 0, (expβ€˜(𝐴 Β· 1)), (expβ€˜(𝐴 Β· ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯)))))))
29 efrlimOLD.1 . . . . . . . . . . . . . 14 𝑆 = (0(ballβ€˜(abs ∘ βˆ’ ))(1 / ((absβ€˜π΄) + 1)))
30 cnxmet 24633 . . . . . . . . . . . . . . 15 (abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚)
31 0cnd 11206 . . . . . . . . . . . . . . 15 (𝐴 ∈ β„‚ β†’ 0 ∈ β„‚)
32 abscl 15227 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ β„‚ β†’ (absβ€˜π΄) ∈ ℝ)
33 peano2re 11386 . . . . . . . . . . . . . . . . . . 19 ((absβ€˜π΄) ∈ ℝ β†’ ((absβ€˜π΄) + 1) ∈ ℝ)
3432, 33syl 17 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ β„‚ β†’ ((absβ€˜π΄) + 1) ∈ ℝ)
35 0red 11216 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ β„‚ β†’ 0 ∈ ℝ)
36 absge0 15236 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ β„‚ β†’ 0 ≀ (absβ€˜π΄))
3732ltp1d 12143 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ β„‚ β†’ (absβ€˜π΄) < ((absβ€˜π΄) + 1))
3835, 32, 34, 36, 37lelttrd 11371 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ β„‚ β†’ 0 < ((absβ€˜π΄) + 1))
3934, 38elrpd 13014 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ β„‚ β†’ ((absβ€˜π΄) + 1) ∈ ℝ+)
4039rpreccld 13027 . . . . . . . . . . . . . . . 16 (𝐴 ∈ β„‚ β†’ (1 / ((absβ€˜π΄) + 1)) ∈ ℝ+)
4140rpxrd 13018 . . . . . . . . . . . . . . 15 (𝐴 ∈ β„‚ β†’ (1 / ((absβ€˜π΄) + 1)) ∈ ℝ*)
42 blssm 24268 . . . . . . . . . . . . . . 15 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 0 ∈ β„‚ ∧ (1 / ((absβ€˜π΄) + 1)) ∈ ℝ*) β†’ (0(ballβ€˜(abs ∘ βˆ’ ))(1 / ((absβ€˜π΄) + 1))) βŠ† β„‚)
4330, 31, 41, 42mp3an2i 1462 . . . . . . . . . . . . . 14 (𝐴 ∈ β„‚ β†’ (0(ballβ€˜(abs ∘ βˆ’ ))(1 / ((absβ€˜π΄) + 1))) βŠ† β„‚)
4429, 43eqsstrid 4023 . . . . . . . . . . . . 13 (𝐴 ∈ β„‚ β†’ 𝑆 βŠ† β„‚)
4544sselda 3975 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ π‘₯ ∈ β„‚)
46 mul0or 11853 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ β„‚) β†’ ((𝐴 Β· π‘₯) = 0 ↔ (𝐴 = 0 ∨ π‘₯ = 0)))
4745, 46syldan 590 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ ((𝐴 Β· π‘₯) = 0 ↔ (𝐴 = 0 ∨ π‘₯ = 0)))
4847biimpa 476 . . . . . . . . . 10 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 Β· π‘₯) = 0) β†’ (𝐴 = 0 ∨ π‘₯ = 0))
497, 11reccld 11982 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ β„‚) ∧ Β¬ π‘₯ = 0) β†’ (1 / π‘₯) ∈ β„‚)
5045, 49syldanl 601 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ Β¬ π‘₯ = 0) β†’ (1 / π‘₯) ∈ β„‚)
5150adantlr 712 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ 𝐴 = 0) ∧ Β¬ π‘₯ = 0) β†’ (1 / π‘₯) ∈ β„‚)
52511cxpd 26582 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ 𝐴 = 0) ∧ Β¬ π‘₯ = 0) β†’ (1↑𝑐(1 / π‘₯)) = 1)
53 simplr 766 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ 𝐴 = 0) ∧ Β¬ π‘₯ = 0) β†’ 𝐴 = 0)
5453oveq1d 7417 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ 𝐴 = 0) ∧ Β¬ π‘₯ = 0) β†’ (𝐴 Β· π‘₯) = (0 Β· π‘₯))
5545ad2antrr 723 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ 𝐴 = 0) ∧ Β¬ π‘₯ = 0) β†’ π‘₯ ∈ β„‚)
5655mul02d 11411 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ 𝐴 = 0) ∧ Β¬ π‘₯ = 0) β†’ (0 Β· π‘₯) = 0)
5754, 56eqtrd 2764 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ 𝐴 = 0) ∧ Β¬ π‘₯ = 0) β†’ (𝐴 Β· π‘₯) = 0)
5857oveq2d 7418 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ 𝐴 = 0) ∧ Β¬ π‘₯ = 0) β†’ (1 + (𝐴 Β· π‘₯)) = (1 + 0))
59 1p0e1 12335 . . . . . . . . . . . . . . . . 17 (1 + 0) = 1
6058, 59eqtrdi 2780 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ 𝐴 = 0) ∧ Β¬ π‘₯ = 0) β†’ (1 + (𝐴 Β· π‘₯)) = 1)
6160oveq1d 7417 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ 𝐴 = 0) ∧ Β¬ π‘₯ = 0) β†’ ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯)) = (1↑𝑐(1 / π‘₯)))
6253fveq2d 6886 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ 𝐴 = 0) ∧ Β¬ π‘₯ = 0) β†’ (expβ€˜π΄) = (expβ€˜0))
63 ef0 16037 . . . . . . . . . . . . . . . 16 (expβ€˜0) = 1
6462, 63eqtrdi 2780 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ 𝐴 = 0) ∧ Β¬ π‘₯ = 0) β†’ (expβ€˜π΄) = 1)
6552, 61, 643eqtr4d 2774 . . . . . . . . . . . . . 14 ((((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ 𝐴 = 0) ∧ Β¬ π‘₯ = 0) β†’ ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯)) = (expβ€˜π΄))
6665ifeq2da 4553 . . . . . . . . . . . . 13 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ 𝐴 = 0) β†’ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯))) = if(π‘₯ = 0, (expβ€˜π΄), (expβ€˜π΄)))
67 ifid 4561 . . . . . . . . . . . . 13 if(π‘₯ = 0, (expβ€˜π΄), (expβ€˜π΄)) = (expβ€˜π΄)
6866, 67eqtrdi 2780 . . . . . . . . . . . 12 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ 𝐴 = 0) β†’ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯))) = (expβ€˜π΄))
69 iftrue 4527 . . . . . . . . . . . . 13 (π‘₯ = 0 β†’ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯))) = (expβ€˜π΄))
7069adantl 481 . . . . . . . . . . . 12 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ π‘₯ = 0) β†’ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯))) = (expβ€˜π΄))
7168, 70jaodan 954 . . . . . . . . . . 11 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 = 0 ∨ π‘₯ = 0)) β†’ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯))) = (expβ€˜π΄))
72 mulrid 11211 . . . . . . . . . . . . 13 (𝐴 ∈ β„‚ β†’ (𝐴 Β· 1) = 𝐴)
7372ad2antrr 723 . . . . . . . . . . . 12 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 = 0 ∨ π‘₯ = 0)) β†’ (𝐴 Β· 1) = 𝐴)
7473fveq2d 6886 . . . . . . . . . . 11 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 = 0 ∨ π‘₯ = 0)) β†’ (expβ€˜(𝐴 Β· 1)) = (expβ€˜π΄))
7571, 74eqtr4d 2767 . . . . . . . . . 10 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 = 0 ∨ π‘₯ = 0)) β†’ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯))) = (expβ€˜(𝐴 Β· 1)))
7648, 75syldan 590 . . . . . . . . 9 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 Β· π‘₯) = 0) β†’ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯))) = (expβ€˜(𝐴 Β· 1)))
77 mulne0b 11854 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ β„‚) β†’ ((𝐴 β‰  0 ∧ π‘₯ β‰  0) ↔ (𝐴 Β· π‘₯) β‰  0))
7845, 77syldan 590 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ ((𝐴 β‰  0 ∧ π‘₯ β‰  0) ↔ (𝐴 Β· π‘₯) β‰  0))
79 df-ne 2933 . . . . . . . . . . . 12 ((𝐴 Β· π‘₯) β‰  0 ↔ Β¬ (𝐴 Β· π‘₯) = 0)
8078, 79bitrdi 287 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ ((𝐴 β‰  0 ∧ π‘₯ β‰  0) ↔ Β¬ (𝐴 Β· π‘₯) = 0))
81 simprr 770 . . . . . . . . . . . . . . 15 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 β‰  0 ∧ π‘₯ β‰  0)) β†’ π‘₯ β‰  0)
8281neneqd 2937 . . . . . . . . . . . . . 14 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 β‰  0 ∧ π‘₯ β‰  0)) β†’ Β¬ π‘₯ = 0)
8382iffalsed 4532 . . . . . . . . . . . . 13 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 β‰  0 ∧ π‘₯ β‰  0)) β†’ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯))) = ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯)))
84 ax-1cn 11165 . . . . . . . . . . . . . . . 16 1 ∈ β„‚
8545, 13syldan 590 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ (𝐴 Β· π‘₯) ∈ β„‚)
86 addcl 11189 . . . . . . . . . . . . . . . 16 ((1 ∈ β„‚ ∧ (𝐴 Β· π‘₯) ∈ β„‚) β†’ (1 + (𝐴 Β· π‘₯)) ∈ β„‚)
8784, 85, 86sylancr 586 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ (1 + (𝐴 Β· π‘₯)) ∈ β„‚)
8887adantr 480 . . . . . . . . . . . . . 14 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 β‰  0 ∧ π‘₯ β‰  0)) β†’ (1 + (𝐴 Β· π‘₯)) ∈ β„‚)
89 eqid 2724 . . . . . . . . . . . . . . . . . . 19 (1(ballβ€˜(abs ∘ βˆ’ ))1) = (1(ballβ€˜(abs ∘ βˆ’ ))1)
9089dvlog2lem 26527 . . . . . . . . . . . . . . . . . 18 (1(ballβ€˜(abs ∘ βˆ’ ))1) βŠ† (β„‚ βˆ– (-∞(,]0))
91 eqid 2724 . . . . . . . . . . . . . . . . . . 19 (β„‚ βˆ– (-∞(,]0)) = (β„‚ βˆ– (-∞(,]0))
9291logdmss 26517 . . . . . . . . . . . . . . . . . 18 (β„‚ βˆ– (-∞(,]0)) βŠ† (β„‚ βˆ– {0})
9390, 92sstri 3984 . . . . . . . . . . . . . . . . 17 (1(ballβ€˜(abs ∘ βˆ’ ))1) βŠ† (β„‚ βˆ– {0})
94 eqid 2724 . . . . . . . . . . . . . . . . . . . . . 22 (abs ∘ βˆ’ ) = (abs ∘ βˆ’ )
9594cnmetdval 24631 . . . . . . . . . . . . . . . . . . . . 21 (((1 + (𝐴 Β· π‘₯)) ∈ β„‚ ∧ 1 ∈ β„‚) β†’ ((1 + (𝐴 Β· π‘₯))(abs ∘ βˆ’ )1) = (absβ€˜((1 + (𝐴 Β· π‘₯)) βˆ’ 1)))
9687, 84, 95sylancl 585 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ ((1 + (𝐴 Β· π‘₯))(abs ∘ βˆ’ )1) = (absβ€˜((1 + (𝐴 Β· π‘₯)) βˆ’ 1)))
97 pncan2 11466 . . . . . . . . . . . . . . . . . . . . . 22 ((1 ∈ β„‚ ∧ (𝐴 Β· π‘₯) ∈ β„‚) β†’ ((1 + (𝐴 Β· π‘₯)) βˆ’ 1) = (𝐴 Β· π‘₯))
9884, 85, 97sylancr 586 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ ((1 + (𝐴 Β· π‘₯)) βˆ’ 1) = (𝐴 Β· π‘₯))
9998fveq2d 6886 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ (absβ€˜((1 + (𝐴 Β· π‘₯)) βˆ’ 1)) = (absβ€˜(𝐴 Β· π‘₯)))
10096, 99eqtrd 2764 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ ((1 + (𝐴 Β· π‘₯))(abs ∘ βˆ’ )1) = (absβ€˜(𝐴 Β· π‘₯)))
10185abscld 15385 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ (absβ€˜(𝐴 Β· π‘₯)) ∈ ℝ)
10234adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ ((absβ€˜π΄) + 1) ∈ ℝ)
10345abscld 15385 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ (absβ€˜π‘₯) ∈ ℝ)
104102, 103remulcld 11243 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ (((absβ€˜π΄) + 1) Β· (absβ€˜π‘₯)) ∈ ℝ)
105 1red 11214 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ 1 ∈ ℝ)
106 absmul 15243 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ β„‚) β†’ (absβ€˜(𝐴 Β· π‘₯)) = ((absβ€˜π΄) Β· (absβ€˜π‘₯)))
10745, 106syldan 590 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ (absβ€˜(𝐴 Β· π‘₯)) = ((absβ€˜π΄) Β· (absβ€˜π‘₯)))
10832adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ (absβ€˜π΄) ∈ ℝ)
109108, 33syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ ((absβ€˜π΄) + 1) ∈ ℝ)
11045absge0d 15393 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ 0 ≀ (absβ€˜π‘₯))
111108lep1d 12144 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ (absβ€˜π΄) ≀ ((absβ€˜π΄) + 1))
112108, 109, 103, 110, 111lemul1ad 12152 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ ((absβ€˜π΄) Β· (absβ€˜π‘₯)) ≀ (((absβ€˜π΄) + 1) Β· (absβ€˜π‘₯)))
113107, 112eqbrtrd 5161 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ (absβ€˜(𝐴 Β· π‘₯)) ≀ (((absβ€˜π΄) + 1) Β· (absβ€˜π‘₯)))
114 0cn 11205 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ β„‚
11594cnmetdval 24631 . . . . . . . . . . . . . . . . . . . . . . . 24 ((π‘₯ ∈ β„‚ ∧ 0 ∈ β„‚) β†’ (π‘₯(abs ∘ βˆ’ )0) = (absβ€˜(π‘₯ βˆ’ 0)))
11645, 114, 115sylancl 585 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ (π‘₯(abs ∘ βˆ’ )0) = (absβ€˜(π‘₯ βˆ’ 0)))
11745subid1d 11559 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ (π‘₯ βˆ’ 0) = π‘₯)
118117fveq2d 6886 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ (absβ€˜(π‘₯ βˆ’ 0)) = (absβ€˜π‘₯))
119116, 118eqtrd 2764 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ (π‘₯(abs ∘ βˆ’ )0) = (absβ€˜π‘₯))
120 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ π‘₯ ∈ 𝑆)
121120, 29eleqtrdi 2835 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ π‘₯ ∈ (0(ballβ€˜(abs ∘ βˆ’ ))(1 / ((absβ€˜π΄) + 1))))
12230a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ (abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚))
12341adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ (1 / ((absβ€˜π΄) + 1)) ∈ ℝ*)
124 0cnd 11206 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ 0 ∈ β„‚)
125 elbl3 24242 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ (1 / ((absβ€˜π΄) + 1)) ∈ ℝ*) ∧ (0 ∈ β„‚ ∧ π‘₯ ∈ β„‚)) β†’ (π‘₯ ∈ (0(ballβ€˜(abs ∘ βˆ’ ))(1 / ((absβ€˜π΄) + 1))) ↔ (π‘₯(abs ∘ βˆ’ )0) < (1 / ((absβ€˜π΄) + 1))))
126122, 123, 124, 45, 125syl22anc 836 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ (π‘₯ ∈ (0(ballβ€˜(abs ∘ βˆ’ ))(1 / ((absβ€˜π΄) + 1))) ↔ (π‘₯(abs ∘ βˆ’ )0) < (1 / ((absβ€˜π΄) + 1))))
127121, 126mpbid 231 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ (π‘₯(abs ∘ βˆ’ )0) < (1 / ((absβ€˜π΄) + 1)))
128119, 127eqbrtrrd 5163 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ (absβ€˜π‘₯) < (1 / ((absβ€˜π΄) + 1)))
12938adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ 0 < ((absβ€˜π΄) + 1))
130 ltmuldiv2 12087 . . . . . . . . . . . . . . . . . . . . . 22 (((absβ€˜π‘₯) ∈ ℝ ∧ 1 ∈ ℝ ∧ (((absβ€˜π΄) + 1) ∈ ℝ ∧ 0 < ((absβ€˜π΄) + 1))) β†’ ((((absβ€˜π΄) + 1) Β· (absβ€˜π‘₯)) < 1 ↔ (absβ€˜π‘₯) < (1 / ((absβ€˜π΄) + 1))))
131103, 105, 109, 129, 130syl112anc 1371 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ ((((absβ€˜π΄) + 1) Β· (absβ€˜π‘₯)) < 1 ↔ (absβ€˜π‘₯) < (1 / ((absβ€˜π΄) + 1))))
132128, 131mpbird 257 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ (((absβ€˜π΄) + 1) Β· (absβ€˜π‘₯)) < 1)
133101, 104, 105, 113, 132lelttrd 11371 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ (absβ€˜(𝐴 Β· π‘₯)) < 1)
134100, 133eqbrtrd 5161 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ ((1 + (𝐴 Β· π‘₯))(abs ∘ βˆ’ )1) < 1)
135 1rp 12979 . . . . . . . . . . . . . . . . . . . 20 1 ∈ ℝ+
136 rpxr 12984 . . . . . . . . . . . . . . . . . . . 20 (1 ∈ ℝ+ β†’ 1 ∈ ℝ*)
137135, 136mp1i 13 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ 1 ∈ ℝ*)
138 1cnd 11208 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ 1 ∈ β„‚)
139 elbl3 24242 . . . . . . . . . . . . . . . . . . 19 ((((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 1 ∈ ℝ*) ∧ (1 ∈ β„‚ ∧ (1 + (𝐴 Β· π‘₯)) ∈ β„‚)) β†’ ((1 + (𝐴 Β· π‘₯)) ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) ↔ ((1 + (𝐴 Β· π‘₯))(abs ∘ βˆ’ )1) < 1))
140122, 137, 138, 87, 139syl22anc 836 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ ((1 + (𝐴 Β· π‘₯)) ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) ↔ ((1 + (𝐴 Β· π‘₯))(abs ∘ βˆ’ )1) < 1))
141134, 140mpbird 257 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ (1 + (𝐴 Β· π‘₯)) ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1))
14293, 141sselid 3973 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ (1 + (𝐴 Β· π‘₯)) ∈ (β„‚ βˆ– {0}))
143 eldifsni 4786 . . . . . . . . . . . . . . . 16 ((1 + (𝐴 Β· π‘₯)) ∈ (β„‚ βˆ– {0}) β†’ (1 + (𝐴 Β· π‘₯)) β‰  0)
144142, 143syl 17 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ (1 + (𝐴 Β· π‘₯)) β‰  0)
145144adantr 480 . . . . . . . . . . . . . 14 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 β‰  0 ∧ π‘₯ β‰  0)) β†’ (1 + (𝐴 Β· π‘₯)) β‰  0)
14645adantr 480 . . . . . . . . . . . . . . 15 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 β‰  0 ∧ π‘₯ β‰  0)) β†’ π‘₯ ∈ β„‚)
147146, 81reccld 11982 . . . . . . . . . . . . . 14 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 β‰  0 ∧ π‘₯ β‰  0)) β†’ (1 / π‘₯) ∈ β„‚)
14888, 145, 147cxpefd 26587 . . . . . . . . . . . . 13 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 β‰  0 ∧ π‘₯ β‰  0)) β†’ ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯)) = (expβ€˜((1 / π‘₯) Β· (logβ€˜(1 + (𝐴 Β· π‘₯))))))
14987, 144logcld 26445 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ (logβ€˜(1 + (𝐴 Β· π‘₯))) ∈ β„‚)
150149adantr 480 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 β‰  0 ∧ π‘₯ β‰  0)) β†’ (logβ€˜(1 + (𝐴 Β· π‘₯))) ∈ β„‚)
151147, 150mulcomd 11234 . . . . . . . . . . . . . . 15 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 β‰  0 ∧ π‘₯ β‰  0)) β†’ ((1 / π‘₯) Β· (logβ€˜(1 + (𝐴 Β· π‘₯)))) = ((logβ€˜(1 + (𝐴 Β· π‘₯))) Β· (1 / π‘₯)))
152 simpll 764 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 β‰  0 ∧ π‘₯ β‰  0)) β†’ 𝐴 ∈ β„‚)
153 simprl 768 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 β‰  0 ∧ π‘₯ β‰  0)) β†’ 𝐴 β‰  0)
154152, 153dividd 11987 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 β‰  0 ∧ π‘₯ β‰  0)) β†’ (𝐴 / 𝐴) = 1)
155154oveq1d 7417 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 β‰  0 ∧ π‘₯ β‰  0)) β†’ ((𝐴 / 𝐴) / π‘₯) = (1 / π‘₯))
156152, 152, 146, 153, 81divdiv1d 12020 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 β‰  0 ∧ π‘₯ β‰  0)) β†’ ((𝐴 / 𝐴) / π‘₯) = (𝐴 / (𝐴 Β· π‘₯)))
157155, 156eqtr3d 2766 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 β‰  0 ∧ π‘₯ β‰  0)) β†’ (1 / π‘₯) = (𝐴 / (𝐴 Β· π‘₯)))
158157oveq2d 7418 . . . . . . . . . . . . . . 15 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 β‰  0 ∧ π‘₯ β‰  0)) β†’ ((logβ€˜(1 + (𝐴 Β· π‘₯))) Β· (1 / π‘₯)) = ((logβ€˜(1 + (𝐴 Β· π‘₯))) Β· (𝐴 / (𝐴 Β· π‘₯))))
15985adantr 480 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 β‰  0 ∧ π‘₯ β‰  0)) β†’ (𝐴 Β· π‘₯) ∈ β„‚)
16078biimpa 476 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 β‰  0 ∧ π‘₯ β‰  0)) β†’ (𝐴 Β· π‘₯) β‰  0)
161150, 152, 159, 160div12d 12025 . . . . . . . . . . . . . . 15 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 β‰  0 ∧ π‘₯ β‰  0)) β†’ ((logβ€˜(1 + (𝐴 Β· π‘₯))) Β· (𝐴 / (𝐴 Β· π‘₯))) = (𝐴 Β· ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))))
162151, 158, 1613eqtrd 2768 . . . . . . . . . . . . . 14 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 β‰  0 ∧ π‘₯ β‰  0)) β†’ ((1 / π‘₯) Β· (logβ€˜(1 + (𝐴 Β· π‘₯)))) = (𝐴 Β· ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))))
163162fveq2d 6886 . . . . . . . . . . . . 13 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 β‰  0 ∧ π‘₯ β‰  0)) β†’ (expβ€˜((1 / π‘₯) Β· (logβ€˜(1 + (𝐴 Β· π‘₯))))) = (expβ€˜(𝐴 Β· ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯)))))
16483, 148, 1633eqtrd 2768 . . . . . . . . . . . 12 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 β‰  0 ∧ π‘₯ β‰  0)) β†’ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯))) = (expβ€˜(𝐴 Β· ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯)))))
165164ex 412 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ ((𝐴 β‰  0 ∧ π‘₯ β‰  0) β†’ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯))) = (expβ€˜(𝐴 Β· ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))))))
16680, 165sylbird 260 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ (Β¬ (𝐴 Β· π‘₯) = 0 β†’ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯))) = (expβ€˜(𝐴 Β· ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))))))
167166imp 406 . . . . . . . . 9 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ Β¬ (𝐴 Β· π‘₯) = 0) β†’ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯))) = (expβ€˜(𝐴 Β· ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯)))))
16827, 28, 76, 167ifbothda 4559 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯))) = if((𝐴 Β· π‘₯) = 0, (expβ€˜(𝐴 Β· 1)), (expβ€˜(𝐴 Β· ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))))))
169168mpteq2dva 5239 . . . . . . 7 (𝐴 ∈ β„‚ β†’ (π‘₯ ∈ 𝑆 ↦ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯)))) = (π‘₯ ∈ 𝑆 ↦ if((𝐴 Β· π‘₯) = 0, (expβ€˜(𝐴 Β· 1)), (expβ€˜(𝐴 Β· ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯)))))))
17044resmptd 6031 . . . . . . 7 (𝐴 ∈ β„‚ β†’ ((π‘₯ ∈ β„‚ ↦ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯)))) β†Ύ 𝑆) = (π‘₯ ∈ 𝑆 ↦ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯)))))
171 1cnd 11208 . . . . . . . . 9 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ (𝐴 Β· π‘₯) = 0) β†’ 1 ∈ β„‚)
172149adantr 480 . . . . . . . . . 10 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ Β¬ (𝐴 Β· π‘₯) = 0) β†’ (logβ€˜(1 + (𝐴 Β· π‘₯))) ∈ β„‚)
17385adantr 480 . . . . . . . . . 10 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ Β¬ (𝐴 Β· π‘₯) = 0) β†’ (𝐴 Β· π‘₯) ∈ β„‚)
174 simpr 484 . . . . . . . . . . 11 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ Β¬ (𝐴 Β· π‘₯) = 0) β†’ Β¬ (𝐴 Β· π‘₯) = 0)
175174neqned 2939 . . . . . . . . . 10 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ Β¬ (𝐴 Β· π‘₯) = 0) β†’ (𝐴 Β· π‘₯) β‰  0)
176172, 173, 175divcld 11989 . . . . . . . . 9 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) ∧ Β¬ (𝐴 Β· π‘₯) = 0) β†’ ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯)) ∈ β„‚)
177171, 176ifclda 4556 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))) ∈ β„‚)
178 eqidd 2725 . . . . . . . 8 (𝐴 ∈ β„‚ β†’ (π‘₯ ∈ 𝑆 ↦ if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯)))) = (π‘₯ ∈ 𝑆 ↦ if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯)))))
179 eqidd 2725 . . . . . . . 8 (𝐴 ∈ β„‚ β†’ (𝑦 ∈ β„‚ ↦ (expβ€˜(𝐴 Β· 𝑦))) = (𝑦 ∈ β„‚ ↦ (expβ€˜(𝐴 Β· 𝑦))))
180 oveq2 7410 . . . . . . . . . 10 (𝑦 = if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))) β†’ (𝐴 Β· 𝑦) = (𝐴 Β· if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯)))))
181180fveq2d 6886 . . . . . . . . 9 (𝑦 = if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))) β†’ (expβ€˜(𝐴 Β· 𝑦)) = (expβ€˜(𝐴 Β· if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))))))
182 oveq2 7410 . . . . . . . . . . 11 (if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))) = 1 β†’ (𝐴 Β· if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯)))) = (𝐴 Β· 1))
183182fveq2d 6886 . . . . . . . . . 10 (if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))) = 1 β†’ (expβ€˜(𝐴 Β· if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))))) = (expβ€˜(𝐴 Β· 1)))
184 oveq2 7410 . . . . . . . . . . 11 (if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))) = ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯)) β†’ (𝐴 Β· if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯)))) = (𝐴 Β· ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))))
185184fveq2d 6886 . . . . . . . . . 10 (if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))) = ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯)) β†’ (expβ€˜(𝐴 Β· if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))))) = (expβ€˜(𝐴 Β· ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯)))))
186183, 185ifsb 4534 . . . . . . . . 9 (expβ€˜(𝐴 Β· if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))))) = if((𝐴 Β· π‘₯) = 0, (expβ€˜(𝐴 Β· 1)), (expβ€˜(𝐴 Β· ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯)))))
187181, 186eqtrdi 2780 . . . . . . . 8 (𝑦 = if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))) β†’ (expβ€˜(𝐴 Β· 𝑦)) = if((𝐴 Β· π‘₯) = 0, (expβ€˜(𝐴 Β· 1)), (expβ€˜(𝐴 Β· ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))))))
188177, 178, 179, 187fmptco 7120 . . . . . . 7 (𝐴 ∈ β„‚ β†’ ((𝑦 ∈ β„‚ ↦ (expβ€˜(𝐴 Β· 𝑦))) ∘ (π‘₯ ∈ 𝑆 ↦ if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))))) = (π‘₯ ∈ 𝑆 ↦ if((𝐴 Β· π‘₯) = 0, (expβ€˜(𝐴 Β· 1)), (expβ€˜(𝐴 Β· ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯)))))))
189169, 170, 1883eqtr4d 2774 . . . . . 6 (𝐴 ∈ β„‚ β†’ ((π‘₯ ∈ β„‚ ↦ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯)))) β†Ύ 𝑆) = ((𝑦 ∈ β„‚ ↦ (expβ€˜(𝐴 Β· 𝑦))) ∘ (π‘₯ ∈ 𝑆 ↦ if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))))))
190 eqidd 2725 . . . . . . . . . 10 (𝐴 ∈ β„‚ β†’ (π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯))) = (π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯))))
191 eqidd 2725 . . . . . . . . . 10 (𝐴 ∈ β„‚ β†’ (𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) ↦ if(𝑦 = 1, 1, ((logβ€˜π‘¦) / (𝑦 βˆ’ 1)))) = (𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) ↦ if(𝑦 = 1, 1, ((logβ€˜π‘¦) / (𝑦 βˆ’ 1)))))
192 eqeq1 2728 . . . . . . . . . . 11 (𝑦 = (1 + (𝐴 Β· π‘₯)) β†’ (𝑦 = 1 ↔ (1 + (𝐴 Β· π‘₯)) = 1))
193 fveq2 6882 . . . . . . . . . . . 12 (𝑦 = (1 + (𝐴 Β· π‘₯)) β†’ (logβ€˜π‘¦) = (logβ€˜(1 + (𝐴 Β· π‘₯))))
194 oveq1 7409 . . . . . . . . . . . 12 (𝑦 = (1 + (𝐴 Β· π‘₯)) β†’ (𝑦 βˆ’ 1) = ((1 + (𝐴 Β· π‘₯)) βˆ’ 1))
195193, 194oveq12d 7420 . . . . . . . . . . 11 (𝑦 = (1 + (𝐴 Β· π‘₯)) β†’ ((logβ€˜π‘¦) / (𝑦 βˆ’ 1)) = ((logβ€˜(1 + (𝐴 Β· π‘₯))) / ((1 + (𝐴 Β· π‘₯)) βˆ’ 1)))
196192, 195ifbieq2d 4547 . . . . . . . . . 10 (𝑦 = (1 + (𝐴 Β· π‘₯)) β†’ if(𝑦 = 1, 1, ((logβ€˜π‘¦) / (𝑦 βˆ’ 1))) = if((1 + (𝐴 Β· π‘₯)) = 1, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / ((1 + (𝐴 Β· π‘₯)) βˆ’ 1))))
197141, 190, 191, 196fmptco 7120 . . . . . . . . 9 (𝐴 ∈ β„‚ β†’ ((𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) ↦ if(𝑦 = 1, 1, ((logβ€˜π‘¦) / (𝑦 βˆ’ 1)))) ∘ (π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯)))) = (π‘₯ ∈ 𝑆 ↦ if((1 + (𝐴 Β· π‘₯)) = 1, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / ((1 + (𝐴 Β· π‘₯)) βˆ’ 1)))))
19859eqeq2i 2737 . . . . . . . . . . . 12 ((1 + (𝐴 Β· π‘₯)) = (1 + 0) ↔ (1 + (𝐴 Β· π‘₯)) = 1)
199138, 85, 124addcand 11416 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ ((1 + (𝐴 Β· π‘₯)) = (1 + 0) ↔ (𝐴 Β· π‘₯) = 0))
200198, 199bitr3id 285 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ ((1 + (𝐴 Β· π‘₯)) = 1 ↔ (𝐴 Β· π‘₯) = 0))
20198oveq2d 7418 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ ((logβ€˜(1 + (𝐴 Β· π‘₯))) / ((1 + (𝐴 Β· π‘₯)) βˆ’ 1)) = ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯)))
202200, 201ifbieq2d 4547 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ 𝑆) β†’ if((1 + (𝐴 Β· π‘₯)) = 1, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / ((1 + (𝐴 Β· π‘₯)) βˆ’ 1))) = if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))))
203202mpteq2dva 5239 . . . . . . . . 9 (𝐴 ∈ β„‚ β†’ (π‘₯ ∈ 𝑆 ↦ if((1 + (𝐴 Β· π‘₯)) = 1, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / ((1 + (𝐴 Β· π‘₯)) βˆ’ 1)))) = (π‘₯ ∈ 𝑆 ↦ if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯)))))
204197, 203eqtrd 2764 . . . . . . . 8 (𝐴 ∈ β„‚ β†’ ((𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) ↦ if(𝑦 = 1, 1, ((logβ€˜π‘¦) / (𝑦 βˆ’ 1)))) ∘ (π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯)))) = (π‘₯ ∈ 𝑆 ↦ if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯)))))
205 eqid 2724 . . . . . . . . . . . 12 ((TopOpenβ€˜β„‚fld) β†Ύt 𝑆) = ((TopOpenβ€˜β„‚fld) β†Ύt 𝑆)
206 eqid 2724 . . . . . . . . . . . . . 14 (TopOpenβ€˜β„‚fld) = (TopOpenβ€˜β„‚fld)
207206cnfldtopon 24643 . . . . . . . . . . . . 13 (TopOpenβ€˜β„‚fld) ∈ (TopOnβ€˜β„‚)
208207a1i 11 . . . . . . . . . . . 12 (𝐴 ∈ β„‚ β†’ (TopOpenβ€˜β„‚fld) ∈ (TopOnβ€˜β„‚))
209 1cnd 11208 . . . . . . . . . . . . . 14 (𝐴 ∈ β„‚ β†’ 1 ∈ β„‚)
210208, 208, 209cnmptc 23510 . . . . . . . . . . . . 13 (𝐴 ∈ β„‚ β†’ (π‘₯ ∈ β„‚ ↦ 1) ∈ ((TopOpenβ€˜β„‚fld) Cn (TopOpenβ€˜β„‚fld)))
211 id 22 . . . . . . . . . . . . . . 15 (𝐴 ∈ β„‚ β†’ 𝐴 ∈ β„‚)
212208, 208, 211cnmptc 23510 . . . . . . . . . . . . . 14 (𝐴 ∈ β„‚ β†’ (π‘₯ ∈ β„‚ ↦ 𝐴) ∈ ((TopOpenβ€˜β„‚fld) Cn (TopOpenβ€˜β„‚fld)))
213208cnmptid 23509 . . . . . . . . . . . . . 14 (𝐴 ∈ β„‚ β†’ (π‘₯ ∈ β„‚ ↦ π‘₯) ∈ ((TopOpenβ€˜β„‚fld) Cn (TopOpenβ€˜β„‚fld)))
214206mulcn 24727 . . . . . . . . . . . . . . 15 Β· ∈ (((TopOpenβ€˜β„‚fld) Γ—t (TopOpenβ€˜β„‚fld)) Cn (TopOpenβ€˜β„‚fld))
215214a1i 11 . . . . . . . . . . . . . 14 (𝐴 ∈ β„‚ β†’ Β· ∈ (((TopOpenβ€˜β„‚fld) Γ—t (TopOpenβ€˜β„‚fld)) Cn (TopOpenβ€˜β„‚fld)))
216208, 212, 213, 215cnmpt12f 23514 . . . . . . . . . . . . 13 (𝐴 ∈ β„‚ β†’ (π‘₯ ∈ β„‚ ↦ (𝐴 Β· π‘₯)) ∈ ((TopOpenβ€˜β„‚fld) Cn (TopOpenβ€˜β„‚fld)))
217206addcn 24725 . . . . . . . . . . . . . 14 + ∈ (((TopOpenβ€˜β„‚fld) Γ—t (TopOpenβ€˜β„‚fld)) Cn (TopOpenβ€˜β„‚fld))
218217a1i 11 . . . . . . . . . . . . 13 (𝐴 ∈ β„‚ β†’ + ∈ (((TopOpenβ€˜β„‚fld) Γ—t (TopOpenβ€˜β„‚fld)) Cn (TopOpenβ€˜β„‚fld)))
219208, 210, 216, 218cnmpt12f 23514 . . . . . . . . . . . 12 (𝐴 ∈ β„‚ β†’ (π‘₯ ∈ β„‚ ↦ (1 + (𝐴 Β· π‘₯))) ∈ ((TopOpenβ€˜β„‚fld) Cn (TopOpenβ€˜β„‚fld)))
220205, 208, 44, 219cnmpt1res 23524 . . . . . . . . . . 11 (𝐴 ∈ β„‚ β†’ (π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯))) ∈ (((TopOpenβ€˜β„‚fld) β†Ύt 𝑆) Cn (TopOpenβ€˜β„‚fld)))
221141fmpttd 7107 . . . . . . . . . . . . 13 (𝐴 ∈ β„‚ β†’ (π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯))):π‘†βŸΆ(1(ballβ€˜(abs ∘ βˆ’ ))1))
222221frnd 6716 . . . . . . . . . . . 12 (𝐴 ∈ β„‚ β†’ ran (π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯))) βŠ† (1(ballβ€˜(abs ∘ βˆ’ ))1))
223 difss 4124 . . . . . . . . . . . . . 14 (β„‚ βˆ– {0}) βŠ† β„‚
22493, 223sstri 3984 . . . . . . . . . . . . 13 (1(ballβ€˜(abs ∘ βˆ’ ))1) βŠ† β„‚
225224a1i 11 . . . . . . . . . . . 12 (𝐴 ∈ β„‚ β†’ (1(ballβ€˜(abs ∘ βˆ’ ))1) βŠ† β„‚)
226 cnrest2 23134 . . . . . . . . . . . 12 (((TopOpenβ€˜β„‚fld) ∈ (TopOnβ€˜β„‚) ∧ ran (π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯))) βŠ† (1(ballβ€˜(abs ∘ βˆ’ ))1) ∧ (1(ballβ€˜(abs ∘ βˆ’ ))1) βŠ† β„‚) β†’ ((π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯))) ∈ (((TopOpenβ€˜β„‚fld) β†Ύt 𝑆) Cn (TopOpenβ€˜β„‚fld)) ↔ (π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯))) ∈ (((TopOpenβ€˜β„‚fld) β†Ύt 𝑆) Cn ((TopOpenβ€˜β„‚fld) β†Ύt (1(ballβ€˜(abs ∘ βˆ’ ))1)))))
227207, 222, 225, 226mp3an2i 1462 . . . . . . . . . . 11 (𝐴 ∈ β„‚ β†’ ((π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯))) ∈ (((TopOpenβ€˜β„‚fld) β†Ύt 𝑆) Cn (TopOpenβ€˜β„‚fld)) ↔ (π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯))) ∈ (((TopOpenβ€˜β„‚fld) β†Ύt 𝑆) Cn ((TopOpenβ€˜β„‚fld) β†Ύt (1(ballβ€˜(abs ∘ βˆ’ ))1)))))
228220, 227mpbid 231 . . . . . . . . . 10 (𝐴 ∈ β„‚ β†’ (π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯))) ∈ (((TopOpenβ€˜β„‚fld) β†Ύt 𝑆) Cn ((TopOpenβ€˜β„‚fld) β†Ύt (1(ballβ€˜(abs ∘ βˆ’ ))1))))
229 blcntr 24263 . . . . . . . . . . . . 13 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 0 ∈ β„‚ ∧ (1 / ((absβ€˜π΄) + 1)) ∈ ℝ+) β†’ 0 ∈ (0(ballβ€˜(abs ∘ βˆ’ ))(1 / ((absβ€˜π΄) + 1))))
23030, 31, 40, 229mp3an2i 1462 . . . . . . . . . . . 12 (𝐴 ∈ β„‚ β†’ 0 ∈ (0(ballβ€˜(abs ∘ βˆ’ ))(1 / ((absβ€˜π΄) + 1))))
231230, 29eleqtrrdi 2836 . . . . . . . . . . 11 (𝐴 ∈ β„‚ β†’ 0 ∈ 𝑆)
232 resttopon 23009 . . . . . . . . . . . . 13 (((TopOpenβ€˜β„‚fld) ∈ (TopOnβ€˜β„‚) ∧ 𝑆 βŠ† β„‚) β†’ ((TopOpenβ€˜β„‚fld) β†Ύt 𝑆) ∈ (TopOnβ€˜π‘†))
233207, 44, 232sylancr 586 . . . . . . . . . . . 12 (𝐴 ∈ β„‚ β†’ ((TopOpenβ€˜β„‚fld) β†Ύt 𝑆) ∈ (TopOnβ€˜π‘†))
234 toponuni 22760 . . . . . . . . . . . 12 (((TopOpenβ€˜β„‚fld) β†Ύt 𝑆) ∈ (TopOnβ€˜π‘†) β†’ 𝑆 = βˆͺ ((TopOpenβ€˜β„‚fld) β†Ύt 𝑆))
235233, 234syl 17 . . . . . . . . . . 11 (𝐴 ∈ β„‚ β†’ 𝑆 = βˆͺ ((TopOpenβ€˜β„‚fld) β†Ύt 𝑆))
236231, 235eleqtrd 2827 . . . . . . . . . 10 (𝐴 ∈ β„‚ β†’ 0 ∈ βˆͺ ((TopOpenβ€˜β„‚fld) β†Ύt 𝑆))
237 eqid 2724 . . . . . . . . . . 11 βˆͺ ((TopOpenβ€˜β„‚fld) β†Ύt 𝑆) = βˆͺ ((TopOpenβ€˜β„‚fld) β†Ύt 𝑆)
238237cncnpi 23126 . . . . . . . . . 10 (((π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯))) ∈ (((TopOpenβ€˜β„‚fld) β†Ύt 𝑆) Cn ((TopOpenβ€˜β„‚fld) β†Ύt (1(ballβ€˜(abs ∘ βˆ’ ))1))) ∧ 0 ∈ βˆͺ ((TopOpenβ€˜β„‚fld) β†Ύt 𝑆)) β†’ (π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯))) ∈ ((((TopOpenβ€˜β„‚fld) β†Ύt 𝑆) CnP ((TopOpenβ€˜β„‚fld) β†Ύt (1(ballβ€˜(abs ∘ βˆ’ ))1)))β€˜0))
239228, 236, 238syl2anc 583 . . . . . . . . 9 (𝐴 ∈ β„‚ β†’ (π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯))) ∈ ((((TopOpenβ€˜β„‚fld) β†Ύt 𝑆) CnP ((TopOpenβ€˜β„‚fld) β†Ύt (1(ballβ€˜(abs ∘ βˆ’ ))1)))β€˜0))
240 cnelprrecn 11200 . . . . . . . . . . 11 β„‚ ∈ {ℝ, β„‚}
241 logf1o 26439 . . . . . . . . . . . . . 14 log:(β„‚ βˆ– {0})–1-1-ontoβ†’ran log
242 f1of 6824 . . . . . . . . . . . . . 14 (log:(β„‚ βˆ– {0})–1-1-ontoβ†’ran log β†’ log:(β„‚ βˆ– {0})⟢ran log)
243241, 242ax-mp 5 . . . . . . . . . . . . 13 log:(β„‚ βˆ– {0})⟢ran log
244 logrncn 26437 . . . . . . . . . . . . . 14 (π‘₯ ∈ ran log β†’ π‘₯ ∈ β„‚)
245244ssriv 3979 . . . . . . . . . . . . 13 ran log βŠ† β„‚
246 fss 6725 . . . . . . . . . . . . 13 ((log:(β„‚ βˆ– {0})⟢ran log ∧ ran log βŠ† β„‚) β†’ log:(β„‚ βˆ– {0})βŸΆβ„‚)
247243, 245, 246mp2an 689 . . . . . . . . . . . 12 log:(β„‚ βˆ– {0})βŸΆβ„‚
248 fssres 6748 . . . . . . . . . . . 12 ((log:(β„‚ βˆ– {0})βŸΆβ„‚ ∧ (1(ballβ€˜(abs ∘ βˆ’ ))1) βŠ† (β„‚ βˆ– {0})) β†’ (log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1)):(1(ballβ€˜(abs ∘ βˆ’ ))1)βŸΆβ„‚)
249247, 93, 248mp2an 689 . . . . . . . . . . 11 (log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1)):(1(ballβ€˜(abs ∘ βˆ’ ))1)βŸΆβ„‚
250 blcntr 24263 . . . . . . . . . . . . . 14 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 1 ∈ β„‚ ∧ 1 ∈ ℝ+) β†’ 1 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1))
25130, 84, 135, 250mp3an 1457 . . . . . . . . . . . . 13 1 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1)
252 ovex 7435 . . . . . . . . . . . . . 14 (1 / 𝑦) ∈ V
25389dvlog2 26528 . . . . . . . . . . . . . 14 (β„‚ D (log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1))) = (𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) ↦ (1 / 𝑦))
254252, 253dmmpti 6685 . . . . . . . . . . . . 13 dom (β„‚ D (log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1))) = (1(ballβ€˜(abs ∘ βˆ’ ))1)
255251, 254eleqtrri 2824 . . . . . . . . . . . 12 1 ∈ dom (β„‚ D (log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1)))
256 eqid 2724 . . . . . . . . . . . . 13 ((TopOpenβ€˜β„‚fld) β†Ύt (1(ballβ€˜(abs ∘ βˆ’ ))1)) = ((TopOpenβ€˜β„‚fld) β†Ύt (1(ballβ€˜(abs ∘ βˆ’ ))1))
257253fveq1i 6883 . . . . . . . . . . . . . . . . 17 ((β„‚ D (log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1)))β€˜1) = ((𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) ↦ (1 / 𝑦))β€˜1)
258 oveq2 7410 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 1 β†’ (1 / 𝑦) = (1 / 1))
259 1div1e1 11903 . . . . . . . . . . . . . . . . . . . 20 (1 / 1) = 1
260258, 259eqtrdi 2780 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 1 β†’ (1 / 𝑦) = 1)
261 eqid 2724 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) ↦ (1 / 𝑦)) = (𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) ↦ (1 / 𝑦))
262 1ex 11209 . . . . . . . . . . . . . . . . . . 19 1 ∈ V
263260, 261, 262fvmpt 6989 . . . . . . . . . . . . . . . . . 18 (1 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) β†’ ((𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) ↦ (1 / 𝑦))β€˜1) = 1)
264251, 263ax-mp 5 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) ↦ (1 / 𝑦))β€˜1) = 1
265257, 264eqtr2i 2753 . . . . . . . . . . . . . . . 16 1 = ((β„‚ D (log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1)))β€˜1)
266265a1i 11 . . . . . . . . . . . . . . 15 (𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) β†’ 1 = ((β„‚ D (log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1)))β€˜1))
267 fvres 6901 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) β†’ ((log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1))β€˜π‘¦) = (logβ€˜π‘¦))
268 fvres 6901 . . . . . . . . . . . . . . . . . . . 20 (1 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) β†’ ((log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1))β€˜1) = (logβ€˜1))
269251, 268mp1i 13 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) β†’ ((log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1))β€˜1) = (logβ€˜1))
270 log1 26460 . . . . . . . . . . . . . . . . . . 19 (logβ€˜1) = 0
271269, 270eqtrdi 2780 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) β†’ ((log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1))β€˜1) = 0)
272267, 271oveq12d 7420 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) β†’ (((log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1))β€˜π‘¦) βˆ’ ((log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1))β€˜1)) = ((logβ€˜π‘¦) βˆ’ 0))
27393sseli 3971 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) β†’ 𝑦 ∈ (β„‚ βˆ– {0}))
274 eldifsn 4783 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (β„‚ βˆ– {0}) ↔ (𝑦 ∈ β„‚ ∧ 𝑦 β‰  0))
275273, 274sylib 217 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) β†’ (𝑦 ∈ β„‚ ∧ 𝑦 β‰  0))
276 logcl 26443 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ β„‚ ∧ 𝑦 β‰  0) β†’ (logβ€˜π‘¦) ∈ β„‚)
277275, 276syl 17 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) β†’ (logβ€˜π‘¦) ∈ β„‚)
278277subid1d 11559 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) β†’ ((logβ€˜π‘¦) βˆ’ 0) = (logβ€˜π‘¦))
279272, 278eqtr2d 2765 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) β†’ (logβ€˜π‘¦) = (((log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1))β€˜π‘¦) βˆ’ ((log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1))β€˜1)))
280279oveq1d 7417 . . . . . . . . . . . . . . 15 (𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) β†’ ((logβ€˜π‘¦) / (𝑦 βˆ’ 1)) = ((((log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1))β€˜π‘¦) βˆ’ ((log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1))β€˜1)) / (𝑦 βˆ’ 1)))
281266, 280ifeq12d 4542 . . . . . . . . . . . . . 14 (𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) β†’ if(𝑦 = 1, 1, ((logβ€˜π‘¦) / (𝑦 βˆ’ 1))) = if(𝑦 = 1, ((β„‚ D (log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1)))β€˜1), ((((log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1))β€˜π‘¦) βˆ’ ((log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1))β€˜1)) / (𝑦 βˆ’ 1))))
282281mpteq2ia 5242 . . . . . . . . . . . . 13 (𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) ↦ if(𝑦 = 1, 1, ((logβ€˜π‘¦) / (𝑦 βˆ’ 1)))) = (𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) ↦ if(𝑦 = 1, ((β„‚ D (log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1)))β€˜1), ((((log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1))β€˜π‘¦) βˆ’ ((log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1))β€˜1)) / (𝑦 βˆ’ 1))))
283256, 206, 282dvcnp 25792 . . . . . . . . . . . 12 (((β„‚ ∈ {ℝ, β„‚} ∧ (log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1)):(1(ballβ€˜(abs ∘ βˆ’ ))1)βŸΆβ„‚ ∧ (1(ballβ€˜(abs ∘ βˆ’ ))1) βŠ† β„‚) ∧ 1 ∈ dom (β„‚ D (log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1)))) β†’ (𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) ↦ if(𝑦 = 1, 1, ((logβ€˜π‘¦) / (𝑦 βˆ’ 1)))) ∈ ((((TopOpenβ€˜β„‚fld) β†Ύt (1(ballβ€˜(abs ∘ βˆ’ ))1)) CnP (TopOpenβ€˜β„‚fld))β€˜1))
284255, 283mpan2 688 . . . . . . . . . . 11 ((β„‚ ∈ {ℝ, β„‚} ∧ (log β†Ύ (1(ballβ€˜(abs ∘ βˆ’ ))1)):(1(ballβ€˜(abs ∘ βˆ’ ))1)βŸΆβ„‚ ∧ (1(ballβ€˜(abs ∘ βˆ’ ))1) βŠ† β„‚) β†’ (𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) ↦ if(𝑦 = 1, 1, ((logβ€˜π‘¦) / (𝑦 βˆ’ 1)))) ∈ ((((TopOpenβ€˜β„‚fld) β†Ύt (1(ballβ€˜(abs ∘ βˆ’ ))1)) CnP (TopOpenβ€˜β„‚fld))β€˜1))
285240, 249, 224, 284mp3an 1457 . . . . . . . . . 10 (𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) ↦ if(𝑦 = 1, 1, ((logβ€˜π‘¦) / (𝑦 βˆ’ 1)))) ∈ ((((TopOpenβ€˜β„‚fld) β†Ύt (1(ballβ€˜(abs ∘ βˆ’ ))1)) CnP (TopOpenβ€˜β„‚fld))β€˜1)
286 oveq2 7410 . . . . . . . . . . . . . . 15 (π‘₯ = 0 β†’ (𝐴 Β· π‘₯) = (𝐴 Β· 0))
287286oveq2d 7418 . . . . . . . . . . . . . 14 (π‘₯ = 0 β†’ (1 + (𝐴 Β· π‘₯)) = (1 + (𝐴 Β· 0)))
288 eqid 2724 . . . . . . . . . . . . . 14 (π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯))) = (π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯)))
289 ovex 7435 . . . . . . . . . . . . . 14 (1 + (𝐴 Β· 0)) ∈ V
290287, 288, 289fvmpt 6989 . . . . . . . . . . . . 13 (0 ∈ 𝑆 β†’ ((π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯)))β€˜0) = (1 + (𝐴 Β· 0)))
291231, 290syl 17 . . . . . . . . . . . 12 (𝐴 ∈ β„‚ β†’ ((π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯)))β€˜0) = (1 + (𝐴 Β· 0)))
292 mul01 11392 . . . . . . . . . . . . . 14 (𝐴 ∈ β„‚ β†’ (𝐴 Β· 0) = 0)
293292oveq2d 7418 . . . . . . . . . . . . 13 (𝐴 ∈ β„‚ β†’ (1 + (𝐴 Β· 0)) = (1 + 0))
294293, 59eqtrdi 2780 . . . . . . . . . . . 12 (𝐴 ∈ β„‚ β†’ (1 + (𝐴 Β· 0)) = 1)
295291, 294eqtrd 2764 . . . . . . . . . . 11 (𝐴 ∈ β„‚ β†’ ((π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯)))β€˜0) = 1)
296295fveq2d 6886 . . . . . . . . . 10 (𝐴 ∈ β„‚ β†’ ((((TopOpenβ€˜β„‚fld) β†Ύt (1(ballβ€˜(abs ∘ βˆ’ ))1)) CnP (TopOpenβ€˜β„‚fld))β€˜((π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯)))β€˜0)) = ((((TopOpenβ€˜β„‚fld) β†Ύt (1(ballβ€˜(abs ∘ βˆ’ ))1)) CnP (TopOpenβ€˜β„‚fld))β€˜1))
297285, 296eleqtrrid 2832 . . . . . . . . 9 (𝐴 ∈ β„‚ β†’ (𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) ↦ if(𝑦 = 1, 1, ((logβ€˜π‘¦) / (𝑦 βˆ’ 1)))) ∈ ((((TopOpenβ€˜β„‚fld) β†Ύt (1(ballβ€˜(abs ∘ βˆ’ ))1)) CnP (TopOpenβ€˜β„‚fld))β€˜((π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯)))β€˜0)))
298 cnpco 23115 . . . . . . . . 9 (((π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯))) ∈ ((((TopOpenβ€˜β„‚fld) β†Ύt 𝑆) CnP ((TopOpenβ€˜β„‚fld) β†Ύt (1(ballβ€˜(abs ∘ βˆ’ ))1)))β€˜0) ∧ (𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) ↦ if(𝑦 = 1, 1, ((logβ€˜π‘¦) / (𝑦 βˆ’ 1)))) ∈ ((((TopOpenβ€˜β„‚fld) β†Ύt (1(ballβ€˜(abs ∘ βˆ’ ))1)) CnP (TopOpenβ€˜β„‚fld))β€˜((π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯)))β€˜0))) β†’ ((𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) ↦ if(𝑦 = 1, 1, ((logβ€˜π‘¦) / (𝑦 βˆ’ 1)))) ∘ (π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯)))) ∈ ((((TopOpenβ€˜β„‚fld) β†Ύt 𝑆) CnP (TopOpenβ€˜β„‚fld))β€˜0))
299239, 297, 298syl2anc 583 . . . . . . . 8 (𝐴 ∈ β„‚ β†’ ((𝑦 ∈ (1(ballβ€˜(abs ∘ βˆ’ ))1) ↦ if(𝑦 = 1, 1, ((logβ€˜π‘¦) / (𝑦 βˆ’ 1)))) ∘ (π‘₯ ∈ 𝑆 ↦ (1 + (𝐴 Β· π‘₯)))) ∈ ((((TopOpenβ€˜β„‚fld) β†Ύt 𝑆) CnP (TopOpenβ€˜β„‚fld))β€˜0))
300204, 299eqeltrrd 2826 . . . . . . 7 (𝐴 ∈ β„‚ β†’ (π‘₯ ∈ 𝑆 ↦ if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯)))) ∈ ((((TopOpenβ€˜β„‚fld) β†Ύt 𝑆) CnP (TopOpenβ€˜β„‚fld))β€˜0))
301208, 208, 211cnmptc 23510 . . . . . . . . . 10 (𝐴 ∈ β„‚ β†’ (𝑦 ∈ β„‚ ↦ 𝐴) ∈ ((TopOpenβ€˜β„‚fld) Cn (TopOpenβ€˜β„‚fld)))
302208cnmptid 23509 . . . . . . . . . 10 (𝐴 ∈ β„‚ β†’ (𝑦 ∈ β„‚ ↦ 𝑦) ∈ ((TopOpenβ€˜β„‚fld) Cn (TopOpenβ€˜β„‚fld)))
303208, 301, 302, 215cnmpt12f 23514 . . . . . . . . 9 (𝐴 ∈ β„‚ β†’ (𝑦 ∈ β„‚ ↦ (𝐴 Β· 𝑦)) ∈ ((TopOpenβ€˜β„‚fld) Cn (TopOpenβ€˜β„‚fld)))
304 efcn 26321 . . . . . . . . . . 11 exp ∈ (ℂ–cnβ†’β„‚)
305206cncfcn1 24775 . . . . . . . . . . 11 (ℂ–cnβ†’β„‚) = ((TopOpenβ€˜β„‚fld) Cn (TopOpenβ€˜β„‚fld))
306304, 305eleqtri 2823 . . . . . . . . . 10 exp ∈ ((TopOpenβ€˜β„‚fld) Cn (TopOpenβ€˜β„‚fld))
307306a1i 11 . . . . . . . . 9 (𝐴 ∈ β„‚ β†’ exp ∈ ((TopOpenβ€˜β„‚fld) Cn (TopOpenβ€˜β„‚fld)))
308208, 303, 307cnmpt11f 23512 . . . . . . . 8 (𝐴 ∈ β„‚ β†’ (𝑦 ∈ β„‚ ↦ (expβ€˜(𝐴 Β· 𝑦))) ∈ ((TopOpenβ€˜β„‚fld) Cn (TopOpenβ€˜β„‚fld)))
309177fmpttd 7107 . . . . . . . . 9 (𝐴 ∈ β„‚ β†’ (π‘₯ ∈ 𝑆 ↦ if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯)))):π‘†βŸΆβ„‚)
310309, 231ffvelcdmd 7078 . . . . . . . 8 (𝐴 ∈ β„‚ β†’ ((π‘₯ ∈ 𝑆 ↦ if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))))β€˜0) ∈ β„‚)
311 unicntop 24646 . . . . . . . . 9 β„‚ = βˆͺ (TopOpenβ€˜β„‚fld)
312311cncnpi 23126 . . . . . . . 8 (((𝑦 ∈ β„‚ ↦ (expβ€˜(𝐴 Β· 𝑦))) ∈ ((TopOpenβ€˜β„‚fld) Cn (TopOpenβ€˜β„‚fld)) ∧ ((π‘₯ ∈ 𝑆 ↦ if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))))β€˜0) ∈ β„‚) β†’ (𝑦 ∈ β„‚ ↦ (expβ€˜(𝐴 Β· 𝑦))) ∈ (((TopOpenβ€˜β„‚fld) CnP (TopOpenβ€˜β„‚fld))β€˜((π‘₯ ∈ 𝑆 ↦ if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))))β€˜0)))
313308, 310, 312syl2anc 583 . . . . . . 7 (𝐴 ∈ β„‚ β†’ (𝑦 ∈ β„‚ ↦ (expβ€˜(𝐴 Β· 𝑦))) ∈ (((TopOpenβ€˜β„‚fld) CnP (TopOpenβ€˜β„‚fld))β€˜((π‘₯ ∈ 𝑆 ↦ if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))))β€˜0)))
314 cnpco 23115 . . . . . . 7 (((π‘₯ ∈ 𝑆 ↦ if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯)))) ∈ ((((TopOpenβ€˜β„‚fld) β†Ύt 𝑆) CnP (TopOpenβ€˜β„‚fld))β€˜0) ∧ (𝑦 ∈ β„‚ ↦ (expβ€˜(𝐴 Β· 𝑦))) ∈ (((TopOpenβ€˜β„‚fld) CnP (TopOpenβ€˜β„‚fld))β€˜((π‘₯ ∈ 𝑆 ↦ if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))))β€˜0))) β†’ ((𝑦 ∈ β„‚ ↦ (expβ€˜(𝐴 Β· 𝑦))) ∘ (π‘₯ ∈ 𝑆 ↦ if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))))) ∈ ((((TopOpenβ€˜β„‚fld) β†Ύt 𝑆) CnP (TopOpenβ€˜β„‚fld))β€˜0))
315300, 313, 314syl2anc 583 . . . . . 6 (𝐴 ∈ β„‚ β†’ ((𝑦 ∈ β„‚ ↦ (expβ€˜(𝐴 Β· 𝑦))) ∘ (π‘₯ ∈ 𝑆 ↦ if((𝐴 Β· π‘₯) = 0, 1, ((logβ€˜(1 + (𝐴 Β· π‘₯))) / (𝐴 Β· π‘₯))))) ∈ ((((TopOpenβ€˜β„‚fld) β†Ύt 𝑆) CnP (TopOpenβ€˜β„‚fld))β€˜0))
316189, 315eqeltrd 2825 . . . . 5 (𝐴 ∈ β„‚ β†’ ((π‘₯ ∈ β„‚ ↦ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯)))) β†Ύ 𝑆) ∈ ((((TopOpenβ€˜β„‚fld) β†Ύt 𝑆) CnP (TopOpenβ€˜β„‚fld))β€˜0))
317206cnfldtop 24644 . . . . . . 7 (TopOpenβ€˜β„‚fld) ∈ Top
318317a1i 11 . . . . . 6 (𝐴 ∈ β„‚ β†’ (TopOpenβ€˜β„‚fld) ∈ Top)
319206cnfldtopn 24642 . . . . . . . . . . 11 (TopOpenβ€˜β„‚fld) = (MetOpenβ€˜(abs ∘ βˆ’ ))
320319blopn 24353 . . . . . . . . . 10 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 0 ∈ β„‚ ∧ (1 / ((absβ€˜π΄) + 1)) ∈ ℝ*) β†’ (0(ballβ€˜(abs ∘ βˆ’ ))(1 / ((absβ€˜π΄) + 1))) ∈ (TopOpenβ€˜β„‚fld))
32130, 31, 41, 320mp3an2i 1462 . . . . . . . . 9 (𝐴 ∈ β„‚ β†’ (0(ballβ€˜(abs ∘ βˆ’ ))(1 / ((absβ€˜π΄) + 1))) ∈ (TopOpenβ€˜β„‚fld))
32229, 321eqeltrid 2829 . . . . . . . 8 (𝐴 ∈ β„‚ β†’ 𝑆 ∈ (TopOpenβ€˜β„‚fld))
323 isopn3i 22930 . . . . . . . 8 (((TopOpenβ€˜β„‚fld) ∈ Top ∧ 𝑆 ∈ (TopOpenβ€˜β„‚fld)) β†’ ((intβ€˜(TopOpenβ€˜β„‚fld))β€˜π‘†) = 𝑆)
324317, 322, 323sylancr 586 . . . . . . 7 (𝐴 ∈ β„‚ β†’ ((intβ€˜(TopOpenβ€˜β„‚fld))β€˜π‘†) = 𝑆)
325231, 324eleqtrrd 2828 . . . . . 6 (𝐴 ∈ β„‚ β†’ 0 ∈ ((intβ€˜(TopOpenβ€˜β„‚fld))β€˜π‘†))
326 efcl 16028 . . . . . . . . 9 (𝐴 ∈ β„‚ β†’ (expβ€˜π΄) ∈ β„‚)
327326ad2antrr 723 . . . . . . . 8 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ β„‚) ∧ π‘₯ = 0) β†’ (expβ€˜π΄) ∈ β„‚)
32884, 14, 86sylancr 586 . . . . . . . . 9 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ β„‚) ∧ Β¬ π‘₯ = 0) β†’ (1 + (𝐴 Β· π‘₯)) ∈ β„‚)
329328, 49cxpcld 26583 . . . . . . . 8 (((𝐴 ∈ β„‚ ∧ π‘₯ ∈ β„‚) ∧ Β¬ π‘₯ = 0) β†’ ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯)) ∈ β„‚)
330327, 329ifclda 4556 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ π‘₯ ∈ β„‚) β†’ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯))) ∈ β„‚)
331330fmpttd 7107 . . . . . 6 (𝐴 ∈ β„‚ β†’ (π‘₯ ∈ β„‚ ↦ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯)))):β„‚βŸΆβ„‚)
332311, 311cnprest 23137 . . . . . 6 ((((TopOpenβ€˜β„‚fld) ∈ Top ∧ 𝑆 βŠ† β„‚) ∧ (0 ∈ ((intβ€˜(TopOpenβ€˜β„‚fld))β€˜π‘†) ∧ (π‘₯ ∈ β„‚ ↦ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯)))):β„‚βŸΆβ„‚)) β†’ ((π‘₯ ∈ β„‚ ↦ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯)))) ∈ (((TopOpenβ€˜β„‚fld) CnP (TopOpenβ€˜β„‚fld))β€˜0) ↔ ((π‘₯ ∈ β„‚ ↦ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯)))) β†Ύ 𝑆) ∈ ((((TopOpenβ€˜β„‚fld) β†Ύt 𝑆) CnP (TopOpenβ€˜β„‚fld))β€˜0)))
333318, 44, 325, 331, 332syl22anc 836 . . . . 5 (𝐴 ∈ β„‚ β†’ ((π‘₯ ∈ β„‚ ↦ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯)))) ∈ (((TopOpenβ€˜β„‚fld) CnP (TopOpenβ€˜β„‚fld))β€˜0) ↔ ((π‘₯ ∈ β„‚ ↦ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯)))) β†Ύ 𝑆) ∈ ((((TopOpenβ€˜β„‚fld) β†Ύt 𝑆) CnP (TopOpenβ€˜β„‚fld))β€˜0)))
334316, 333mpbird 257 . . . 4 (𝐴 ∈ β„‚ β†’ (π‘₯ ∈ β„‚ ↦ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯)))) ∈ (((TopOpenβ€˜β„‚fld) CnP (TopOpenβ€˜β„‚fld))β€˜0))
335311cnpresti 23136 . . . 4 (((0[,)+∞) βŠ† β„‚ ∧ 0 ∈ (0[,)+∞) ∧ (π‘₯ ∈ β„‚ ↦ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯)))) ∈ (((TopOpenβ€˜β„‚fld) CnP (TopOpenβ€˜β„‚fld))β€˜0)) β†’ ((π‘₯ ∈ β„‚ ↦ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯)))) β†Ύ (0[,)+∞)) ∈ ((((TopOpenβ€˜β„‚fld) β†Ύt (0[,)+∞)) CnP (TopOpenβ€˜β„‚fld))β€˜0))
3363, 26, 334, 335mp3an2i 1462 . . 3 (𝐴 ∈ β„‚ β†’ ((π‘₯ ∈ β„‚ ↦ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 Β· π‘₯))↑𝑐(1 / π‘₯)))) β†Ύ (0[,)+∞)) ∈ ((((TopOpenβ€˜β„‚fld) β†Ύt (0[,)+∞)) CnP (TopOpenβ€˜β„‚fld))β€˜0))
33724, 336eqeltrd 2825 . 2 (𝐴 ∈ β„‚ β†’ (π‘₯ ∈ (0[,)+∞) ↦ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 / (1 / π‘₯)))↑𝑐(1 / π‘₯)))) ∈ ((((TopOpenβ€˜β„‚fld) β†Ύt (0[,)+∞)) CnP (TopOpenβ€˜β„‚fld))β€˜0))
338 simpl 482 . . . . . 6 ((𝐴 ∈ β„‚ ∧ π‘˜ ∈ ℝ+) β†’ 𝐴 ∈ β„‚)
339 rpcn 12985 . . . . . . 7 (π‘˜ ∈ ℝ+ β†’ π‘˜ ∈ β„‚)
340339adantl 481 . . . . . 6 ((𝐴 ∈ β„‚ ∧ π‘˜ ∈ ℝ+) β†’ π‘˜ ∈ β„‚)
341 rpne0 12991 . . . . . . 7 (π‘˜ ∈ ℝ+ β†’ π‘˜ β‰  0)
342341adantl 481 . . . . . 6 ((𝐴 ∈ β„‚ ∧ π‘˜ ∈ ℝ+) β†’ π‘˜ β‰  0)
343338, 340, 342divcld 11989 . . . . 5 ((𝐴 ∈ β„‚ ∧ π‘˜ ∈ ℝ+) β†’ (𝐴 / π‘˜) ∈ β„‚)
344 addcl 11189 . . . . 5 ((1 ∈ β„‚ ∧ (𝐴 / π‘˜) ∈ β„‚) β†’ (1 + (𝐴 / π‘˜)) ∈ β„‚)
34584, 343, 344sylancr 586 . . . 4 ((𝐴 ∈ β„‚ ∧ π‘˜ ∈ ℝ+) β†’ (1 + (𝐴 / π‘˜)) ∈ β„‚)
346345, 340cxpcld 26583 . . 3 ((𝐴 ∈ β„‚ ∧ π‘˜ ∈ ℝ+) β†’ ((1 + (𝐴 / π‘˜))β†‘π‘π‘˜) ∈ β„‚)
347 oveq2 7410 . . . . 5 (π‘˜ = (1 / π‘₯) β†’ (𝐴 / π‘˜) = (𝐴 / (1 / π‘₯)))
348347oveq2d 7418 . . . 4 (π‘˜ = (1 / π‘₯) β†’ (1 + (𝐴 / π‘˜)) = (1 + (𝐴 / (1 / π‘₯))))
349 id 22 . . . 4 (π‘˜ = (1 / π‘₯) β†’ π‘˜ = (1 / π‘₯))
350348, 349oveq12d 7420 . . 3 (π‘˜ = (1 / π‘₯) β†’ ((1 + (𝐴 / π‘˜))β†‘π‘π‘˜) = ((1 + (𝐴 / (1 / π‘₯)))↑𝑐(1 / π‘₯)))
351 eqid 2724 . . 3 ((TopOpenβ€˜β„‚fld) β†Ύt (0[,)+∞)) = ((TopOpenβ€˜β„‚fld) β†Ύt (0[,)+∞))
352326, 346, 350, 206, 351rlimcnp3 26840 . 2 (𝐴 ∈ β„‚ β†’ ((π‘˜ ∈ ℝ+ ↦ ((1 + (𝐴 / π‘˜))β†‘π‘π‘˜)) β‡π‘Ÿ (expβ€˜π΄) ↔ (π‘₯ ∈ (0[,)+∞) ↦ if(π‘₯ = 0, (expβ€˜π΄), ((1 + (𝐴 / (1 / π‘₯)))↑𝑐(1 / π‘₯)))) ∈ ((((TopOpenβ€˜β„‚fld) β†Ύt (0[,)+∞)) CnP (TopOpenβ€˜β„‚fld))β€˜0)))
353337, 352mpbird 257 1 (𝐴 ∈ β„‚ β†’ (π‘˜ ∈ ℝ+ ↦ ((1 + (𝐴 / π‘˜))β†‘π‘π‘˜)) β‡π‘Ÿ (expβ€˜π΄))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2932   βˆ– cdif 3938   βŠ† wss 3941  ifcif 4521  {csn 4621  {cpr 4623  βˆͺ cuni 4900   class class class wbr 5139   ↦ cmpt 5222  dom cdm 5667  ran crn 5668   β†Ύ cres 5669   ∘ ccom 5671  βŸΆwf 6530  β€“1-1-ontoβ†’wf1o 6533  β€˜cfv 6534  (class class class)co 7402  β„‚cc 11105  β„cr 11106  0cc0 11107  1c1 11108   + caddc 11110   Β· cmul 11112  +∞cpnf 11244  -∞cmnf 11245  β„*cxr 11246   < clt 11247   ≀ cle 11248   βˆ’ cmin 11443   / cdiv 11870  β„+crp 12975  (,]cioc 13326  [,)cico 13327  abscabs 15183   β‡π‘Ÿ crli 15431  expce 16007   β†Ύt crest 17371  TopOpenctopn 17372  βˆžMetcxmet 21219  ballcbl 21221  β„‚fldccnfld 21234  Topctop 22739  TopOnctopon 22756  intcnt 22865   Cn ccn 23072   CnP ccnp 23073   Γ—t ctx 23408  β€“cnβ†’ccncf 24740   D cdv 25736  logclog 26429  β†‘𝑐ccxp 26430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-inf2 9633  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185  ax-addf 11186  ax-mulf 11187
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-tp 4626  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-iin 4991  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-se 5623  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-isom 6543  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-of 7664  df-om 7850  df-1st 7969  df-2nd 7970  df-supp 8142  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-er 8700  df-map 8819  df-pm 8820  df-ixp 8889  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-fsupp 9359  df-fi 9403  df-sup 9434  df-inf 9435  df-oi 9502  df-card 9931  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-div 11871  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12472  df-z 12558  df-dec 12677  df-uz 12822  df-q 12932  df-rp 12976  df-xneg 13093  df-xadd 13094  df-xmul 13095  df-ioo 13329  df-ioc 13330  df-ico 13331  df-icc 13332  df-fz 13486  df-fzo 13629  df-fl 13758  df-mod 13836  df-seq 13968  df-exp 14029  df-fac 14235  df-bc 14264  df-hash 14292  df-shft 15016  df-cj 15048  df-re 15049  df-im 15050  df-sqrt 15184  df-abs 15185  df-limsup 15417  df-clim 15434  df-rlim 15435  df-sum 15635  df-ef 16013  df-sin 16015  df-cos 16016  df-tan 16017  df-pi 16018  df-struct 17085  df-sets 17102  df-slot 17120  df-ndx 17132  df-base 17150  df-ress 17179  df-plusg 17215  df-mulr 17216  df-starv 17217  df-sca 17218  df-vsca 17219  df-ip 17220  df-tset 17221  df-ple 17222  df-ds 17224  df-unif 17225  df-hom 17226  df-cco 17227  df-rest 17373  df-topn 17374  df-0g 17392  df-gsum 17393  df-topgen 17394  df-pt 17395  df-prds 17398  df-xrs 17453  df-qtop 17458  df-imas 17459  df-xps 17461  df-mre 17535  df-mrc 17536  df-acs 17538  df-mgm 18569  df-sgrp 18648  df-mnd 18664  df-submnd 18710  df-mulg 18992  df-cntz 19229  df-cmn 19698  df-psmet 21226  df-xmet 21227  df-met 21228  df-bl 21229  df-mopn 21230  df-fbas 21231  df-fg 21232  df-cnfld 21235  df-top 22740  df-topon 22757  df-topsp 22779  df-bases 22793  df-cld 22867  df-ntr 22868  df-cls 22869  df-nei 22946  df-lp 22984  df-perf 22985  df-cn 23075  df-cnp 23076  df-haus 23163  df-cmp 23235  df-tx 23410  df-hmeo 23603  df-fil 23694  df-fm 23786  df-flim 23787  df-flf 23788  df-xms 24170  df-ms 24171  df-tms 24172  df-cncf 24742  df-limc 25739  df-dv 25740  df-log 26431  df-cxp 26432
This theorem is referenced by: (None)
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