Step | Hyp | Ref
| Expression |
1 | | dvco.bg |
. . . 4
⊢ (𝜑 → 𝐶(𝑇 D 𝐺)𝐿) |
2 | | eqid 2622 |
. . . . 5
⊢ (𝐽 ↾t 𝑇) = (𝐽 ↾t 𝑇) |
3 | | dvco.j |
. . . . 5
⊢ 𝐽 =
(TopOpen‘ℂfld) |
4 | | eqid 2622 |
. . . . 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 3613 |
. . . . . 6
⊢ (𝜑 → 𝑋 ⊆ ℂ) |
10 | 6, 9 | fssd 6057 |
. . . . 5
⊢ (𝜑 → 𝐺:𝑌⟶ℂ) |
11 | | dvco.y |
. . . . 5
⊢ (𝜑 → 𝑌 ⊆ 𝑇) |
12 | 2, 3, 4, 5, 10, 11 | eldv 23662 |
. . . 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 222 |
. . 3
⊢ (𝜑 → (𝐶 ∈ ((int‘(𝐽 ↾t 𝑇))‘𝑌) ∧ 𝐿 ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶))) |
14 | 13 | simpld 475 |
. 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 23663 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐺‘𝐶)(𝑆 D 𝐹)𝐾) → 𝐾 ∈ ℂ) |
18 | 15, 17 | mpdan 702 |
. . . . . 6
⊢ (𝜑 → 𝐾 ∈ ℂ) |
19 | 18 | ad2antrr 762 |
. . . . 5
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → 𝐾 ∈ ℂ) |
20 | 16 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝐹:𝑋⟶ℂ) |
21 | | eldifi 3732 |
. . . . . . . . . 10
⊢ (𝑧 ∈ (𝑌 ∖ {𝐶}) → 𝑧 ∈ 𝑌) |
22 | | ffvelrn 6357 |
. . . . . . . . . 10
⊢ ((𝐺:𝑌⟶𝑋 ∧ 𝑧 ∈ 𝑌) → (𝐺‘𝑧) ∈ 𝑋) |
23 | 6, 21, 22 | syl2an 494 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐺‘𝑧) ∈ 𝑋) |
24 | 20, 23 | ffvelrnd 6360 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐹‘(𝐺‘𝑧)) ∈ ℂ) |
25 | 24 | adantr 481 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐹‘(𝐺‘𝑧)) ∈ ℂ) |
26 | 6 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝐺:𝑌⟶𝑋) |
27 | 5, 10, 11 | dvbss 23665 |
. . . . . . . . . . . 12
⊢ (𝜑 → dom (𝑇 D 𝐺) ⊆ 𝑌) |
28 | | reldv 23634 |
. . . . . . . . . . . . 13
⊢ Rel
(𝑇 D 𝐺) |
29 | | releldm 5358 |
. . . . . . . . . . . . 13
⊢ ((Rel
(𝑇 D 𝐺) ∧ 𝐶(𝑇 D 𝐺)𝐿) → 𝐶 ∈ dom (𝑇 D 𝐺)) |
30 | 28, 1, 29 | sylancr 695 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐶 ∈ dom (𝑇 D 𝐺)) |
31 | 27, 30 | sseldd 3604 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐶 ∈ 𝑌) |
32 | 31 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝐶 ∈ 𝑌) |
33 | 26, 32 | ffvelrnd 6360 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐺‘𝐶) ∈ 𝑋) |
34 | 20, 33 | ffvelrnd 6360 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐹‘(𝐺‘𝐶)) ∈ ℂ) |
35 | 34 | adantr 481 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐹‘(𝐺‘𝐶)) ∈ ℂ) |
36 | 25, 35 | subcld 10392 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) ∈ ℂ) |
37 | 10 | ad2antrr 762 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → 𝐺:𝑌⟶ℂ) |
38 | 21 | ad2antlr 763 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → 𝑧 ∈ 𝑌) |
39 | 37, 38 | ffvelrnd 6360 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐺‘𝑧) ∈ ℂ) |
40 | 31 | ad2antrr 762 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → 𝐶 ∈ 𝑌) |
41 | 37, 40 | ffvelrnd 6360 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐺‘𝐶) ∈ ℂ) |
42 | 39, 41 | subcld 10392 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → ((𝐺‘𝑧) − (𝐺‘𝐶)) ∈ ℂ) |
43 | | simpr 477 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) |
44 | 39, 41 | subeq0ad 10402 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐺‘𝑧) − (𝐺‘𝐶)) = 0 ↔ (𝐺‘𝑧) = (𝐺‘𝐶))) |
45 | 44 | necon3abid 2830 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐺‘𝑧) − (𝐺‘𝐶)) ≠ 0 ↔ ¬ (𝐺‘𝑧) = (𝐺‘𝐶))) |
46 | 43, 45 | mpbird 247 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → ((𝐺‘𝑧) − (𝐺‘𝐶)) ≠ 0) |
47 | 36, 42, 46 | divcld 10801 |
. . . . 5
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))) ∈ ℂ) |
48 | 19, 47 | ifclda 4120 |
. . . 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 3613 |
. . . . 5
⊢ (𝜑 → 𝑌 ⊆ ℂ) |
50 | 10, 49, 31 | dvlem 23660 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) ∈ ℂ) |
51 | | ssid 3624 |
. . . . 5
⊢ ℂ
⊆ ℂ |
52 | 51 | a1i 11 |
. . . 4
⊢ (𝜑 → ℂ ⊆
ℂ) |
53 | 3 | cnfldtopon 22586 |
. . . . . . 7
⊢ 𝐽 ∈
(TopOn‘ℂ) |
54 | | txtopon 21394 |
. . . . . . 7
⊢ ((𝐽 ∈ (TopOn‘ℂ)
∧ 𝐽 ∈
(TopOn‘ℂ)) → (𝐽 ×t 𝐽) ∈ (TopOn‘(ℂ ×
ℂ))) |
55 | 53, 53, 54 | mp2an 708 |
. . . . . 6
⊢ (𝐽 ×t 𝐽) ∈ (TopOn‘(ℂ
× ℂ)) |
56 | 55 | toponunii 20721 |
. . . . . . 7
⊢ (ℂ
× ℂ) = ∪ (𝐽 ×t 𝐽) |
57 | 56 | restid 16094 |
. . . . . 6
⊢ ((𝐽 ×t 𝐽) ∈ (TopOn‘(ℂ
× ℂ)) → ((𝐽 ×t 𝐽) ↾t (ℂ ×
ℂ)) = (𝐽
×t 𝐽)) |
58 | 55, 57 | ax-mp 5 |
. . . . 5
⊢ ((𝐽 ×t 𝐽) ↾t (ℂ
× ℂ)) = (𝐽
×t 𝐽) |
59 | 58 | eqcomi 2631 |
. . . 4
⊢ (𝐽 ×t 𝐽) = ((𝐽 ×t 𝐽) ↾t (ℂ ×
ℂ)) |
60 | 23 | anim1i 592 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) ≠ (𝐺‘𝐶)) → ((𝐺‘𝑧) ∈ 𝑋 ∧ (𝐺‘𝑧) ≠ (𝐺‘𝐶))) |
61 | | eldifsn 4317 |
. . . . . . 7
⊢ ((𝐺‘𝑧) ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↔ ((𝐺‘𝑧) ∈ 𝑋 ∧ (𝐺‘𝑧) ≠ (𝐺‘𝐶))) |
62 | 60, 61 | sylibr 224 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) ≠ (𝐺‘𝐶)) → (𝐺‘𝑧) ∈ (𝑋 ∖ {(𝐺‘𝐶)})) |
63 | 62 | anasss 679 |
. . . . 5
⊢ ((𝜑 ∧ (𝑧 ∈ (𝑌 ∖ {𝐶}) ∧ (𝐺‘𝑧) ≠ (𝐺‘𝐶))) → (𝐺‘𝑧) ∈ (𝑋 ∖ {(𝐺‘𝐶)})) |
64 | | eldifsni 4320 |
. . . . . . . 8
⊢ (𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) → 𝑦 ≠ (𝐺‘𝐶)) |
65 | | ifnefalse 4098 |
. . . . . . . 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)))) |
66 | 64, 65 | 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)))) |
67 | 66 | adantl 482 |
. . . . . 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)))) |
68 | 6, 31 | ffvelrnd 6360 |
. . . . . . 7
⊢ (𝜑 → (𝐺‘𝐶) ∈ 𝑋) |
69 | 16, 9, 68 | dvlem 23660 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)})) → (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶))) ∈ ℂ) |
70 | 67, 69 | eqeltrd 2701 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)})) → if(𝑦 = (𝐺‘𝐶), 𝐾, (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) ∈ ℂ) |
71 | | limcresi 23649 |
. . . . . . 7
⊢ (𝐺 limℂ 𝐶) ⊆ ((𝐺 ↾ (𝑌 ∖ {𝐶})) limℂ 𝐶) |
72 | 6 | feqmptd 6249 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧))) |
73 | 72 | reseq1d 5395 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺 ↾ (𝑌 ∖ {𝐶})) = ((𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧)) ↾ (𝑌 ∖ {𝐶}))) |
74 | | difss 3737 |
. . . . . . . . . 10
⊢ (𝑌 ∖ {𝐶}) ⊆ 𝑌 |
75 | | resmpt 5449 |
. . . . . . . . . 10
⊢ ((𝑌 ∖ {𝐶}) ⊆ 𝑌 → ((𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧)) ↾ (𝑌 ∖ {𝐶})) = (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧))) |
76 | 74, 75 | ax-mp 5 |
. . . . . . . . 9
⊢ ((𝑧 ∈ 𝑌 ↦ (𝐺‘𝑧)) ↾ (𝑌 ∖ {𝐶})) = (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧)) |
77 | 73, 76 | syl6eq 2672 |
. . . . . . . 8
⊢ (𝜑 → (𝐺 ↾ (𝑌 ∖ {𝐶})) = (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧))) |
78 | 77 | oveq1d 6665 |
. . . . . . 7
⊢ (𝜑 → ((𝐺 ↾ (𝑌 ∖ {𝐶})) limℂ 𝐶) = ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧)) limℂ 𝐶)) |
79 | 71, 78 | syl5sseq 3653 |
. . . . . 6
⊢ (𝜑 → (𝐺 limℂ 𝐶) ⊆ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧)) limℂ 𝐶)) |
80 | | eqid 2622 |
. . . . . . . . . 10
⊢ (𝐽 ↾t 𝑌) = (𝐽 ↾t 𝑌) |
81 | 80, 3 | dvcnp2 23683 |
. . . . . . . . 9
⊢ (((𝑇 ⊆ ℂ ∧ 𝐺:𝑌⟶ℂ ∧ 𝑌 ⊆ 𝑇) ∧ 𝐶 ∈ dom (𝑇 D 𝐺)) → 𝐺 ∈ (((𝐽 ↾t 𝑌) CnP 𝐽)‘𝐶)) |
82 | 5, 10, 11, 30, 81 | syl31anc 1329 |
. . . . . . . 8
⊢ (𝜑 → 𝐺 ∈ (((𝐽 ↾t 𝑌) CnP 𝐽)‘𝐶)) |
83 | 3, 80 | cnplimc 23651 |
. . . . . . . . 9
⊢ ((𝑌 ⊆ ℂ ∧ 𝐶 ∈ 𝑌) → (𝐺 ∈ (((𝐽 ↾t 𝑌) CnP 𝐽)‘𝐶) ↔ (𝐺:𝑌⟶ℂ ∧ (𝐺‘𝐶) ∈ (𝐺 limℂ 𝐶)))) |
84 | 49, 31, 83 | syl2anc 693 |
. . . . . . . 8
⊢ (𝜑 → (𝐺 ∈ (((𝐽 ↾t 𝑌) CnP 𝐽)‘𝐶) ↔ (𝐺:𝑌⟶ℂ ∧ (𝐺‘𝐶) ∈ (𝐺 limℂ 𝐶)))) |
85 | 82, 84 | mpbid 222 |
. . . . . . 7
⊢ (𝜑 → (𝐺:𝑌⟶ℂ ∧ (𝐺‘𝐶) ∈ (𝐺 limℂ 𝐶))) |
86 | 85 | simprd 479 |
. . . . . 6
⊢ (𝜑 → (𝐺‘𝐶) ∈ (𝐺 limℂ 𝐶)) |
87 | 79, 86 | sseldd 3604 |
. . . . 5
⊢ (𝜑 → (𝐺‘𝐶) ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (𝐺‘𝑧)) limℂ 𝐶)) |
88 | | eqid 2622 |
. . . . . . . . 9
⊢ (𝐽 ↾t 𝑆) = (𝐽 ↾t 𝑆) |
89 | | eqid 2622 |
. . . . . . . . 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)))) |
90 | 88, 3, 89, 8, 16, 7 | eldv 23662 |
. . . . . . . 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))))) |
91 | 15, 90 | mpbid 222 |
. . . . . . 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)))) |
92 | 91 | simprd 479 |
. . . . . 6
⊢ (𝜑 → 𝐾 ∈ ((𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↦ (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶)))) limℂ (𝐺‘𝐶))) |
93 | 66 | mpteq2ia 4740 |
. . . . . . 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)))) |
94 | 93 | oveq1i 6660 |
. . . . . 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)) |
95 | 92, 94 | syl6eleqr 2712 |
. . . . 5
⊢ (𝜑 → 𝐾 ∈ ((𝑦 ∈ (𝑋 ∖ {(𝐺‘𝐶)}) ↦ if(𝑦 = (𝐺‘𝐶), 𝐾, (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶))))) limℂ (𝐺‘𝐶))) |
96 | | eqeq1 2626 |
. . . . . 6
⊢ (𝑦 = (𝐺‘𝑧) → (𝑦 = (𝐺‘𝐶) ↔ (𝐺‘𝑧) = (𝐺‘𝐶))) |
97 | | fveq2 6191 |
. . . . . . . 8
⊢ (𝑦 = (𝐺‘𝑧) → (𝐹‘𝑦) = (𝐹‘(𝐺‘𝑧))) |
98 | 97 | oveq1d 6665 |
. . . . . . 7
⊢ (𝑦 = (𝐺‘𝑧) → ((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) = ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶)))) |
99 | | oveq1 6657 |
. . . . . . 7
⊢ (𝑦 = (𝐺‘𝑧) → (𝑦 − (𝐺‘𝐶)) = ((𝐺‘𝑧) − (𝐺‘𝐶))) |
100 | 98, 99 | oveq12d 6668 |
. . . . . 6
⊢ (𝑦 = (𝐺‘𝑧) → (((𝐹‘𝑦) − (𝐹‘(𝐺‘𝐶))) / (𝑦 − (𝐺‘𝐶))) = (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) |
101 | 96, 100 | ifbieq2d 4111 |
. . . . 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))))) |
102 | | iftrue 4092 |
. . . . . 6
⊢ ((𝐺‘𝑧) = (𝐺‘𝐶) → if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶)))) = 𝐾) |
103 | 102 | ad2antll 765 |
. . . . 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) |
104 | 63, 70, 87, 95, 101, 103 | limcco 23657 |
. . . 4
⊢ (𝜑 → 𝐾 ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ if((𝐺‘𝑧) = (𝐺‘𝐶), 𝐾, (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / ((𝐺‘𝑧) − (𝐺‘𝐶))))) limℂ 𝐶)) |
105 | 13 | simprd 479 |
. . . 4
⊢ (𝜑 → 𝐿 ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)) |
106 | 3 | mulcn 22670 |
. . . . 5
⊢ ·
∈ ((𝐽
×t 𝐽) Cn
𝐽) |
107 | 5, 10, 11 | dvcl 23663 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐶(𝑇 D 𝐺)𝐿) → 𝐿 ∈ ℂ) |
108 | 1, 107 | mpdan 702 |
. . . . . 6
⊢ (𝜑 → 𝐿 ∈ ℂ) |
109 | | opelxpi 5148 |
. . . . . 6
⊢ ((𝐾 ∈ ℂ ∧ 𝐿 ∈ ℂ) →
〈𝐾, 𝐿〉 ∈ (ℂ ×
ℂ)) |
110 | 18, 108, 109 | syl2anc 693 |
. . . . 5
⊢ (𝜑 → 〈𝐾, 𝐿〉 ∈ (ℂ ×
ℂ)) |
111 | 56 | cncnpi 21082 |
. . . . 5
⊢ ((
· ∈ ((𝐽
×t 𝐽) Cn
𝐽) ∧ 〈𝐾, 𝐿〉 ∈ (ℂ × ℂ))
→ · ∈ (((𝐽
×t 𝐽) CnP
𝐽)‘〈𝐾, 𝐿〉)) |
112 | 106, 110,
111 | sylancr 695 |
. . . 4
⊢ (𝜑 → · ∈ (((𝐽 ×t 𝐽) CnP 𝐽)‘〈𝐾, 𝐿〉)) |
113 | 48, 50, 52, 52, 3, 59, 104, 105, 112 | limccnp2 23656 |
. . 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)) |
114 | | oveq1 6657 |
. . . . . . . 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)))) |
115 | 114 | eqeq1d 2624 |
. . . . . . 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)))) |
116 | | oveq1 6657 |
. . . . . . . 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)))) |
117 | 116 | eqeq1d 2624 |
. . . . . . 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)))) |
118 | 19 | mul01d 10235 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐾 · 0) = 0) |
119 | 9 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝑋 ⊆ ℂ) |
120 | 119, 23 | sseldd 3604 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐺‘𝑧) ∈ ℂ) |
121 | 119, 33 | sseldd 3604 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝐺‘𝐶) ∈ ℂ) |
122 | 120, 121 | subeq0ad 10402 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (((𝐺‘𝑧) − (𝐺‘𝐶)) = 0 ↔ (𝐺‘𝑧) = (𝐺‘𝐶))) |
123 | 122 | biimpar 502 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → ((𝐺‘𝑧) − (𝐺‘𝐶)) = 0) |
124 | 123 | oveq1d 6665 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) = (0 / (𝑧 − 𝐶))) |
125 | 49 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝑌 ⊆ ℂ) |
126 | 21 | adantl 482 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝑧 ∈ 𝑌) |
127 | 125, 126 | sseldd 3604 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝑧 ∈ ℂ) |
128 | 125, 32 | sseldd 3604 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝐶 ∈ ℂ) |
129 | 127, 128 | subcld 10392 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝑧 − 𝐶) ∈ ℂ) |
130 | | eldifsni 4320 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ (𝑌 ∖ {𝐶}) → 𝑧 ≠ 𝐶) |
131 | 130 | adantl 482 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → 𝑧 ≠ 𝐶) |
132 | 127, 128,
131 | subne0d 10401 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (𝑧 − 𝐶) ≠ 0) |
133 | 129, 132 | div0d 10800 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (0 / (𝑧 − 𝐶)) = 0) |
134 | 133 | adantr 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (0 / (𝑧 − 𝐶)) = 0) |
135 | 124, 134 | eqtrd 2656 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) = 0) |
136 | 135 | oveq2d 6666 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐾 · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (𝐾 · 0)) |
137 | | fveq2 6191 |
. . . . . . . . . . . 12
⊢ ((𝐺‘𝑧) = (𝐺‘𝐶) → (𝐹‘(𝐺‘𝑧)) = (𝐹‘(𝐺‘𝐶))) |
138 | 24, 34 | subeq0ad 10402 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) = 0 ↔ (𝐹‘(𝐺‘𝑧)) = (𝐹‘(𝐺‘𝐶)))) |
139 | 137, 138 | syl5ibr 236 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → ((𝐺‘𝑧) = (𝐺‘𝐶) → ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) = 0)) |
140 | 139 | imp 445 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) = 0) |
141 | 140 | oveq1d 6665 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶)) = (0 / (𝑧 − 𝐶))) |
142 | 141, 134 | eqtrd 2656 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶)) = 0) |
143 | 118, 136,
142 | 3eqtr4d 2666 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝐾 · (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶))) |
144 | 129 | adantr 481 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝑧 − 𝐶) ∈ ℂ) |
145 | 132 | adantr 481 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) ∧ ¬ (𝐺‘𝑧) = (𝐺‘𝐶)) → (𝑧 − 𝐶) ≠ 0) |
146 | 36, 42, 144, 46, 145 | dmdcan2d 10831 |
. . . . . . 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))) |
147 | 115, 117,
143, 146 | ifbothda 4123 |
. . . . . 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))) |
148 | | fvco3 6275 |
. . . . . . . . 9
⊢ ((𝐺:𝑌⟶𝑋 ∧ 𝑧 ∈ 𝑌) → ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧))) |
149 | 6, 21, 148 | syl2an 494 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → ((𝐹 ∘ 𝐺)‘𝑧) = (𝐹‘(𝐺‘𝑧))) |
150 | | fvco3 6275 |
. . . . . . . . . 10
⊢ ((𝐺:𝑌⟶𝑋 ∧ 𝐶 ∈ 𝑌) → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶))) |
151 | 6, 31, 150 | syl2anc 693 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶))) |
152 | 151 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶))) |
153 | 149, 152 | oveq12d 6668 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → (((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) = ((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶)))) |
154 | 153 | oveq1d 6665 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 ∖ {𝐶})) → ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶)) = (((𝐹‘(𝐺‘𝑧)) − (𝐹‘(𝐺‘𝐶))) / (𝑧 − 𝐶))) |
155 | 147, 154 | eqtr4d 2659 |
. . . . 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))) |
156 | 155 | mpteq2dva 4744 |
. . . 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)))) |
157 | 156 | oveq1d 6665 |
. . 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)) |
158 | 113, 157 | eleqtrd 2703 |
. 2
⊢ (𝜑 → (𝐾 · 𝐿) ∈ ((𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)) |
159 | | eqid 2622 |
. . 3
⊢ (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶))) = (𝑧 ∈ (𝑌 ∖ {𝐶}) ↦ ((((𝐹 ∘ 𝐺)‘𝑧) − ((𝐹 ∘ 𝐺)‘𝐶)) / (𝑧 − 𝐶))) |
160 | | fco 6058 |
. . . 4
⊢ ((𝐹:𝑋⟶ℂ ∧ 𝐺:𝑌⟶𝑋) → (𝐹 ∘ 𝐺):𝑌⟶ℂ) |
161 | 16, 6, 160 | syl2anc 693 |
. . 3
⊢ (𝜑 → (𝐹 ∘ 𝐺):𝑌⟶ℂ) |
162 | 2, 3, 159, 5, 161, 11 | eldv 23662 |
. 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)))) |
163 | 14, 158, 162 | mpbir2and 957 |
1
⊢ (𝜑 → 𝐶(𝑇 D (𝐹 ∘ 𝐺))(𝐾 · 𝐿)) |