Step | Hyp | Ref
| Expression |
1 | | dvco.bg |
. . . 4
⊢ (𝜑 → 𝐶(𝑇 D 𝐺)𝐿) |
2 | | eqid 2737 |
. . . . 5
⊢ (𝐽 ↾t 𝑇) = (𝐽 ↾t 𝑇) |
3 | | dvco.j |
. . . . 5
⊢ 𝐽 =
(TopOpen‘ℂfld) |
4 | | eqid 2737 |
. . . . 5
⊢ (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) |
5 | | dvcobr.t |
. . . . 5
⊢ (𝜑 → 𝑇 ⊆ ℂ) |
6 | | dvco.g |
. . . . . 6
⊢ (𝜑 → 𝐺:𝑌⟶𝑋) |
7 | | dvco.x |
. . . . . . 7
⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
8 | | dvcobr.s |
. . . . . . 7
⊢ (𝜑 → 𝑆 ⊆ ℂ) |
9 | 7, 8 | sstrd 3911 |
. . . . . 6
⊢ (𝜑 → 𝑋 ⊆ ℂ) |
10 | 6, 9 | fssd 6563 |
. . . . 5
⊢ (𝜑 → 𝐺:𝑌⟶ℂ) |
11 | | dvco.y |
. . . . 5
⊢ (𝜑 → 𝑌 ⊆ 𝑇) |
12 | 2, 3, 4, 5, 10, 11 | eldv 24795 |
. . . 4
⊢ (𝜑 → (𝐶(𝑇 D 𝐺)𝐿 ↔ (𝐶 ∈ ((int‘(𝐽 ↾t 𝑇))‘𝑌) ∧ 𝐿 ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)))) |
13 | 1, 12 | mpbid 235 |
. . 3
⊢ (𝜑 → (𝐶 ∈ ((int‘(𝐽 ↾t 𝑇))‘𝑌) ∧ 𝐿 ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶))) |
14 | 13 | simpld 498 |
. 2
⊢ (𝜑 → 𝐶 ∈ ((int‘(𝐽 ↾t 𝑇))‘𝑌)) |
15 | | dvco.bf |
. . . . . . 7
⊢ (𝜑 → (𝐺‘𝐶)(𝑆 D 𝐹)𝐾) |
16 | | dvco.f |
. . . . . . . 8
⊢ (𝜑 → 𝐹:𝑋⟶ℂ) |
17 | 8, 16, 7 | dvcl 24796 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐺‘𝐶)(𝑆 D 𝐹)𝐾) → 𝐾 ∈ ℂ) |
18 | 15, 17 | mpdan 687 |
. . . . . 6
⊢ (𝜑 → 𝐾 ∈ ℂ) |
19 | 18 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → 𝐾 ∈ ℂ) |
20 | 16 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝐹:𝑋⟶ℂ) |
21 | | eldifi 4041 |
. . . . . . . . . 10
⊢ (𝑧 ∈ (𝑌 ∖ {𝐶}) → 𝑧 ∈ 𝑌) |
22 | | ffvelrn 6902 |
. . . . . . . . . 10
⊢ ((𝐺:𝑌⟶𝑋 ∧ 𝑧 ∈ 𝑌) → (𝐺‘𝑧) ∈ 𝑋) |
23 | 6, 21, 22 | syl2an 599 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐺‘𝑧) ∈ 𝑋) |
24 | 20, 23 | ffvelrnd 6905 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐹‘(𝐺‘𝑧)) ∈ ℂ) |
25 | 24 | adantr 484 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐹‘(𝐺‘𝑧)) ∈ ℂ) |
26 | 6 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝐺:𝑌⟶𝑋) |
27 | 5, 10, 11 | dvbss 24798 |
. . . . . . . . . . . 12
⊢ (𝜑 → dom (𝑇 D 𝐺) ⊆ 𝑌) |
28 | | reldv 24767 |
. . . . . . . . . . . . 13
⊢ Rel
(𝑇 D 𝐺) |
29 | | releldm 5813 |
. . . . . . . . . . . . 13
⊢ ((Rel
(𝑇 D 𝐺) ∧ 𝐶(𝑇 D 𝐺)𝐿) → 𝐶 ∈ dom (𝑇 D 𝐺)) |
30 | 28, 1, 29 | sylancr 590 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐶 ∈ dom (𝑇 D 𝐺)) |
31 | 27, 30 | sseldd 3902 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐶 ∈ 𝑌) |
32 | 31 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝐶 ∈ 𝑌) |
33 | 26, 32 | ffvelrnd 6905 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐺‘𝐶) ∈ 𝑋) |
34 | 20, 33 | ffvelrnd 6905 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐹‘(𝐺‘𝐶)) ∈ ℂ) |
35 | 34 | adantr 484 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐹‘(𝐺‘𝐶)) ∈ ℂ) |
36 | 25, 35 | subcld 11189 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) ∈ ℂ) |
37 | 10 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → 𝐺:𝑌⟶ℂ) |
38 | 21 | ad2antlr 727 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → 𝑧 ∈ 𝑌) |
39 | 37, 38 | ffvelrnd 6905 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐺‘𝑧) ∈ ℂ) |
40 | 31 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → 𝐶 ∈ 𝑌) |
41 | 37, 40 | ffvelrnd 6905 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐺‘𝐶) ∈ ℂ) |
42 | 39, 41 | subcld 11189 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → ((𝐺‘𝑧) − (𝐺‘𝐶)) ∈ ℂ) |
43 | | simpr 488 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) |
44 | 39, 41 | subeq0ad 11199 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐺‘𝑧) − (𝐺‘𝐶)) = 0 ↔ (𝐺‘𝑧) = (𝐺‘𝐶))) |
45 | 44 | necon3abid 2977 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐺‘𝑧) − (𝐺‘𝐶)) ≠ 0 ↔ ¬ (𝐺‘𝑧) = (𝐺‘𝐶))) |
46 | 43, 45 | mpbird 260 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → ((𝐺‘𝑧) − (𝐺‘𝐶)) ≠ 0) |
47 | 36, 42, 46 | divcld 11608 |
. . . . 5
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))) ∈ ℂ) |
48 | 19, 47 | ifclda 4474 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) ∈ ℂ) |
49 | 11, 5 | sstrd 3911 |
. . . . 5
⊢ (𝜑 → 𝑌 ⊆ ℂ) |
50 | 10, 49, 31 | dvlem 24793 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) ∈ ℂ) |
51 | | ssidd 3924 |
. . . 4
⊢ (𝜑 → ℂ ⊆
ℂ) |
52 | 3 | cnfldtopon 23680 |
. . . . . 6
⊢ 𝐽 ∈
(TopOn‘ℂ) |
53 | | txtopon 22488 |
. . . . . 6
⊢ ((𝐽 ∈ (TopOn‘ℂ)
∧ 𝐽 ∈
(TopOn‘ℂ)) → (𝐽 ×t 𝐽) ∈ (TopOn‘(ℂ ×
ℂ))) |
54 | 52, 52, 53 | mp2an 692 |
. . . . 5
⊢ (𝐽 ×t 𝐽) ∈ (TopOn‘(ℂ
× ℂ)) |
55 | 54 | toponrestid 21818 |
. . . 4
⊢ (𝐽 ×t 𝐽) = ((𝐽 ×t 𝐽) ↾t (ℂ ×
ℂ)) |
56 | 23 | anim1i 618 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) ≠ (𝐺‘𝐶)) → ((𝐺‘𝑧) ∈ 𝑋 ∧ (𝐺‘𝑧) ≠ (𝐺‘𝐶))) |
57 | | eldifsn 4700 |
. . . . . . 7
⊢ ((𝐺‘𝑧) ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↔ ((𝐺‘𝑧) ∈ 𝑋 ∧ (𝐺‘𝑧) ≠ (𝐺‘𝐶))) |
58 | 56, 57 | sylibr 237 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) ≠ (𝐺‘𝐶)) → (𝐺‘𝑧) ∈ (𝑋 ∖ {(𝐺‘𝐶)})) |
59 | 58 | anasss 470 |
. . . . 5
⊢ ((𝜑 ∧ (𝑧 ∈ (𝑌 ∖ {𝐶}) ∧ (𝐺‘𝑧) ≠ (𝐺‘𝐶))) → (𝐺‘𝑧) ∈ (𝑋 ∖ {(𝐺‘𝐶)})) |
60 | | eldifsni 4703 |
. . . . . . . 8
⊢ (𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) → 𝑦 ≠ (𝐺‘𝐶)) |
61 | | ifnefalse 4451 |
. . . . . . . 8
⊢ (𝑦 ≠ (𝐺‘𝐶) → if(𝑦 = (𝐺‘𝐶), 𝐾, (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) = (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) |
62 | 60, 61 | syl 17 |
. . . . . . 7
⊢ (𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) → if(𝑦 = (𝐺‘𝐶), 𝐾, (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) = (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) |
63 | 62 | adantl 485 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)})) → if(𝑦 = (𝐺‘𝐶), 𝐾, (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) = (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) |
64 | 6, 31 | ffvelrnd 6905 |
. . . . . . 7
⊢ (𝜑 → (𝐺‘𝐶) ∈ 𝑋) |
65 | 16, 9, 64 | dvlem 24793 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)})) → (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶))) ∈ ℂ) |
66 | 63, 65 | eqeltrd 2838 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)})) → if(𝑦 = (𝐺‘𝐶), 𝐾, (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) ∈ ℂ) |
67 | | limcresi 24782 |
. . . . . . 7
⊢ (𝐺 limℂ 𝐶) ⊆ ((𝐺 ↾ (𝑌 ∖ {𝐶})) limℂ 𝐶) |
68 | 6 | feqmptd 6780 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧))) |
69 | 68 | reseq1d 5850 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺 ↾ (𝑌 ∖ {𝐶})) = ((𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧)) ↾ (𝑌 ∖ {𝐶}))) |
70 | | difss 4046 |
. . . . . . . . . 10
⊢ (𝑌 ∖ {𝐶}) ⊆ 𝑌 |
71 | | resmpt 5905 |
. . . . . . . . . 10
⊢ ((𝑌 ∖ {𝐶}) ⊆ 𝑌 → ((𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧)) ↾ (𝑌 ∖ {𝐶})) = (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧))) |
72 | 70, 71 | ax-mp 5 |
. . . . . . . . 9
⊢ ((𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧)) ↾ (𝑌 ∖ {𝐶})) = (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧)) |
73 | 69, 72 | eqtrdi 2794 |
. . . . . . . 8
⊢ (𝜑 → (𝐺 ↾ (𝑌 ∖ {𝐶})) = (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧))) |
74 | 73 | oveq1d 7228 |
. . . . . . 7
⊢ (𝜑 → ((𝐺 ↾ (𝑌 ∖ {𝐶})) limℂ 𝐶) = ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧)) limℂ 𝐶)) |
75 | 67, 74 | sseqtrid 3953 |
. . . . . 6
⊢ (𝜑 → (𝐺 limℂ 𝐶) ⊆ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧)) limℂ 𝐶)) |
76 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝐽 ↾t 𝑌) = (𝐽 ↾t 𝑌) |
77 | 76, 3 | dvcnp2 24817 |
. . . . . . . . 9
⊢ (((𝑇 ⊆ ℂ ∧ 𝐺:𝑌⟶ℂ ∧ 𝑌 ⊆ 𝑇) ∧ 𝐶 ∈ dom (𝑇 D 𝐺)) → 𝐺 ∈ (((𝐽 ↾t 𝑌) CnP 𝐽)‘𝐶)) |
78 | 5, 10, 11, 30, 77 | syl31anc 1375 |
. . . . . . . 8
⊢ (𝜑 → 𝐺 ∈ (((𝐽 ↾t 𝑌) CnP 𝐽)‘𝐶)) |
79 | 3, 76 | cnplimc 24784 |
. . . . . . . . 9
⊢ ((𝑌 ⊆ ℂ ∧ 𝐶 ∈ 𝑌) → (𝐺 ∈ (((𝐽 ↾t 𝑌) CnP 𝐽)‘𝐶) ↔ (𝐺:𝑌⟶ℂ ∧ (𝐺‘𝐶) ∈ (𝐺 limℂ 𝐶)))) |
80 | 49, 31, 79 | syl2anc 587 |
. . . . . . . 8
⊢ (𝜑 → (𝐺 ∈ (((𝐽 ↾t 𝑌) CnP 𝐽)‘𝐶) ↔ (𝐺:𝑌⟶ℂ ∧ (𝐺‘𝐶) ∈ (𝐺 limℂ 𝐶)))) |
81 | 78, 80 | mpbid 235 |
. . . . . . 7
⊢ (𝜑 → (𝐺:𝑌⟶ℂ ∧ (𝐺‘𝐶) ∈ (𝐺 limℂ 𝐶))) |
82 | 81 | simprd 499 |
. . . . . 6
⊢ (𝜑 → (𝐺‘𝐶) ∈ (𝐺 limℂ 𝐶)) |
83 | 75, 82 | sseldd 3902 |
. . . . 5
⊢ (𝜑 → (𝐺‘𝐶) ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧)) limℂ 𝐶)) |
84 | | eqid 2737 |
. . . . . . . . 9
⊢ (𝐽 ↾t 𝑆) = (𝐽 ↾t 𝑆) |
85 | | eqid 2737 |
. . . . . . . . 9
⊢ (𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↦ (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) = (𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↦ (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) |
86 | 84, 3, 85, 8, 16, 7 | eldv 24795 |
. . . . . . . 8
⊢ (𝜑 → ((𝐺‘𝐶)(𝑆 D 𝐹)𝐾 ↔ ((𝐺‘𝐶) ∈ ((int‘(𝐽 ↾t 𝑆))‘𝑋) ∧ 𝐾 ∈ ((𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↦ (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) limℂ (𝐺‘𝐶))))) |
87 | 15, 86 | mpbid 235 |
. . . . . . 7
⊢ (𝜑 → ((𝐺‘𝐶) ∈ ((int‘(𝐽 ↾t 𝑆))‘𝑋) ∧ 𝐾 ∈ ((𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↦ (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) limℂ (𝐺‘𝐶)))) |
88 | 87 | simprd 499 |
. . . . . 6
⊢ (𝜑 → 𝐾 ∈ ((𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↦ (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) limℂ (𝐺‘𝐶))) |
89 | 62 | mpteq2ia 5146 |
. . . . . . 7
⊢ (𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↦ if(𝑦 = (𝐺‘𝐶), 𝐾, (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶))))) = (𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↦ (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) |
90 | 89 | oveq1i 7223 |
. . . . . 6
⊢ ((𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↦ if(𝑦 = (𝐺‘𝐶), 𝐾, (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶))))) limℂ (𝐺‘𝐶)) = ((𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↦ (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) limℂ (𝐺‘𝐶)) |
91 | 88, 90 | eleqtrrdi 2849 |
. . . . 5
⊢ (𝜑 → 𝐾 ∈ ((𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↦ if(𝑦 = (𝐺‘𝐶), 𝐾, (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶))))) limℂ (𝐺‘𝐶))) |
92 | | eqeq1 2741 |
. . . . . 6
⊢ (𝑦 = (𝐺‘𝑧) → (𝑦 = (𝐺‘𝐶) ↔ (𝐺‘𝑧) = (𝐺‘𝐶))) |
93 | | fveq2 6717 |
. . . . . . . 8
⊢ (𝑦 = (𝐺‘𝑧) → (𝐹‘𝑦) = (𝐹‘(𝐺‘𝑧))) |
94 | 93 | oveq1d 7228 |
. . . . . . 7
⊢ (𝑦 = (𝐺‘𝑧) → ((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) = ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶)))) |
95 | | oveq1 7220 |
. . . . . . 7
⊢ (𝑦 = (𝐺‘𝑧) → (𝑦 − (𝐺‘𝐶)) = ((𝐺‘𝑧) − (𝐺‘𝐶))) |
96 | 94, 95 | oveq12d 7231 |
. . . . . 6
⊢ (𝑦 = (𝐺‘𝑧) → (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶))) = (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) |
97 | 92, 96 | ifbieq2d 4465 |
. . . . 5
⊢ (𝑦 = (𝐺‘𝑧) → if(𝑦 = (𝐺‘𝐶), 𝐾, (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) = if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))))) |
98 | | iftrue 4445 |
. . . . . 6
⊢ ((𝐺‘𝑧) = (𝐺‘𝐶) → if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) = 𝐾) |
99 | 98 | ad2antll 729 |
. . . . 5
⊢ ((𝜑 ∧ (𝑧 ∈ (𝑌 ∖ {𝐶}) ∧ (𝐺‘𝑧) = (𝐺‘𝐶))) → if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) = 𝐾) |
100 | 59, 66, 83, 91, 97, 99 | limcco 24790 |
. . . 4
⊢ (𝜑 → 𝐾 ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))))) limℂ 𝐶)) |
101 | 13 | simprd 499 |
. . . 4
⊢ (𝜑 → 𝐿 ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)) |
102 | 3 | mulcn 23764 |
. . . . 5
⊢ ·
∈ ((𝐽
×t 𝐽) Cn
𝐽) |
103 | 5, 10, 11 | dvcl 24796 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐶(𝑇 D 𝐺)𝐿) → 𝐿 ∈ ℂ) |
104 | 1, 103 | mpdan 687 |
. . . . . 6
⊢ (𝜑 → 𝐿 ∈ ℂ) |
105 | 18, 104 | opelxpd 5589 |
. . . . 5
⊢ (𝜑 → 〈𝐾, 𝐿〉 ∈ (ℂ ×
ℂ)) |
106 | 54 | toponunii 21813 |
. . . . . 6
⊢ (ℂ
× ℂ) = ∪ (𝐽 ×t 𝐽) |
107 | 106 | cncnpi 22175 |
. . . . 5
⊢ ((
· ∈ ((𝐽
×t 𝐽) Cn
𝐽) ∧ 〈𝐾, 𝐿〉 ∈ (ℂ × ℂ))
→ · ∈ (((𝐽
×t 𝐽) CnP
𝐽)‘〈𝐾, 𝐿〉)) |
108 | 102, 105,
107 | sylancr 590 |
. . . 4
⊢ (𝜑 → · ∈ (((𝐽 ×t 𝐽) CnP 𝐽)‘〈𝐾, 𝐿〉)) |
109 | 48, 50, 51, 51, 3, 55, 100, 101, 108 | limccnp2 24789 |
. . 3
⊢ (𝜑 → (𝐾 · 𝐿) ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))) limℂ 𝐶)) |
110 | | oveq1 7220 |
. . . . . . . 8
⊢ (𝐾 = if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) → (𝐾 · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))) |
111 | 110 | eqeq1d 2739 |
. . . . . . 7
⊢ (𝐾 = if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) → ((𝐾 · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶)) ↔ (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶)))) |
112 | | oveq1 7220 |
. . . . . . . 8
⊢ ((((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))) = if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) → ((((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))) |
113 | 112 | eqeq1d 2739 |
. . . . . . 7
⊢ ((((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))) = if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) → (((((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶)) ↔ (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶)))) |
114 | 19 | mul01d 11031 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐾 · 0) = 0) |
115 | 9 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝑋 ⊆ ℂ) |
116 | 115, 23 | sseldd 3902 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐺‘𝑧) ∈ ℂ) |
117 | 115, 33 | sseldd 3902 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐺‘𝐶) ∈ ℂ) |
118 | 116, 117 | subeq0ad 11199 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (((𝐺‘𝑧) − (𝐺‘𝐶)) = 0 ↔ (𝐺‘𝑧) = (𝐺‘𝐶))) |
119 | 118 | biimpar 481 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → ((𝐺‘𝑧) − (𝐺‘𝐶)) = 0) |
120 | 119 | oveq1d 7228 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) = (0 / (𝑧 − 𝐶))) |
121 | 49 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝑌 ⊆ ℂ) |
122 | 21 | adantl 485 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝑧 ∈ 𝑌) |
123 | 121, 122 | sseldd 3902 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝑧 ∈ ℂ) |
124 | 121, 32 | sseldd 3902 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝐶 ∈ ℂ) |
125 | 123, 124 | subcld 11189 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝑧 − 𝐶) ∈ ℂ) |
126 | | eldifsni 4703 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ (𝑌 ∖ {𝐶}) → 𝑧 ≠ 𝐶) |
127 | 126 | adantl 485 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝑧 ≠ 𝐶) |
128 | 123, 124,
127 | subne0d 11198 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝑧 − 𝐶) ≠ 0) |
129 | 125, 128 | div0d 11607 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (0 / (𝑧 − 𝐶)) = 0) |
130 | 129 | adantr 484 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (0 / (𝑧 − 𝐶)) = 0) |
131 | 120, 130 | eqtrd 2777 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) = 0) |
132 | 131 | oveq2d 7229 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐾 · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (𝐾 · 0)) |
133 | | fveq2 6717 |
. . . . . . . . . . . 12
⊢ ((𝐺‘𝑧) = (𝐺‘𝐶) → (𝐹‘(𝐺‘𝑧)) = (𝐹‘(𝐺‘𝐶))) |
134 | 24, 34 | subeq0ad 11199 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) = 0 ↔ (𝐹‘(𝐺‘𝑧)) = (𝐹‘(𝐺‘𝐶)))) |
135 | 133, 134 | syl5ibr 249 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → ((𝐺‘𝑧) = (𝐺‘𝐶) → ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) = 0)) |
136 | 135 | imp 410 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) = 0) |
137 | 136 | oveq1d 7228 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶)) = (0 / (𝑧 − 𝐶))) |
138 | 137, 130 | eqtrd 2777 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶)) = 0) |
139 | 114, 132,
138 | 3eqtr4d 2787 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐾 · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶))) |
140 | 125 | adantr 484 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝑧 − 𝐶) ∈ ℂ) |
141 | 128 | adantr 484 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝑧 − 𝐶) ≠ 0) |
142 | 36, 42, 140, 46, 141 | dmdcan2d 11638 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → ((((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶))) |
143 | 111, 113,
139, 142 | ifbothda 4477 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶))) |
144 | | fvco3 6810 |
. . . . . . . . 9
⊢ ((𝐺:𝑌⟶𝑋 ∧ 𝑧 ∈ 𝑌) → ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧))) |
145 | 6, 21, 144 | syl2an 599 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧))) |
146 | | fvco3 6810 |
. . . . . . . . . 10
⊢ ((𝐺:𝑌⟶𝑋 ∧ 𝐶 ∈ 𝑌) → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶))) |
147 | 6, 31, 146 | syl2anc 587 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶))) |
148 | 147 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶))) |
149 | 145, 148 | oveq12d 7231 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) = ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶)))) |
150 | 149 | oveq1d 7228 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶)) = (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶))) |
151 | 143, 150 | eqtr4d 2780 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶))) |
152 | 151 | mpteq2dva 5150 |
. . . 4
⊢ (𝜑 → (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))) = (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶)))) |
153 | 152 | oveq1d 7228 |
. . 3
⊢ (𝜑 → ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))) limℂ 𝐶) = ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)) |
154 | 109, 153 | eleqtrd 2840 |
. 2
⊢ (𝜑 → (𝐾 · 𝐿) ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)) |
155 | | eqid 2737 |
. . 3
⊢ (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶))) = (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶))) |
156 | | fco 6569 |
. . . 4
⊢ ((𝐹:𝑋⟶ℂ ∧ 𝐺:𝑌⟶𝑋) → (𝐹 ∘ 𝐺):𝑌⟶ℂ) |
157 | 16, 6, 156 | syl2anc 587 |
. . 3
⊢ (𝜑 → (𝐹 ∘ 𝐺):𝑌⟶ℂ) |
158 | 2, 3, 155, 5, 157, 11 | eldv 24795 |
. 2
⊢ (𝜑 → (𝐶(𝑇 D (𝐹 ∘ 𝐺))(𝐾 · 𝐿) ↔ (𝐶 ∈ ((int‘(𝐽 ↾t 𝑇))‘𝑌) ∧ (𝐾 · 𝐿) ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)))) |
159 | 14, 154, 158 | mpbir2and 713 |
1
⊢ (𝜑 → 𝐶(𝑇 D (𝐹 ∘ 𝐺))(𝐾 · 𝐿)) |