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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppcnlem1 | Structured version Visualization version GIF version |
Description: Lemma for knoppcn 32494. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
Ref | Expression |
---|---|
knoppcnlem1.f | ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) |
knoppcnlem1.2 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
knoppcnlem1.3 | ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
Ref | Expression |
---|---|
knoppcnlem1 | ⊢ (𝜑 → ((𝐹‘𝐴)‘𝑀) = ((𝐶↑𝑀) · (𝑇‘(((2 · 𝑁)↑𝑀) · 𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | knoppcnlem1.f | . . . 4 ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))) |
3 | oveq2 6658 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → (((2 · 𝑁)↑𝑛) · 𝑦) = (((2 · 𝑁)↑𝑛) · 𝐴)) | |
4 | 3 | fveq2d 6195 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)) = (𝑇‘(((2 · 𝑁)↑𝑛) · 𝐴))) |
5 | 4 | oveq2d 6666 | . . . . 5 ⊢ (𝑦 = 𝐴 → ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))) = ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝐴)))) |
6 | 5 | mpteq2dv 4745 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))) = (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝐴))))) |
7 | 6 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))) = (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝐴))))) |
8 | knoppcnlem1.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
9 | nn0ex 11298 | . . . . 5 ⊢ ℕ0 ∈ V | |
10 | 9 | mptex 6486 | . . . 4 ⊢ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝐴)))) ∈ V |
11 | 10 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝐴)))) ∈ V) |
12 | 2, 7, 8, 11 | fvmptd 6288 | . 2 ⊢ (𝜑 → (𝐹‘𝐴) = (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝐴))))) |
13 | oveq2 6658 | . . . 4 ⊢ (𝑛 = 𝑀 → (𝐶↑𝑛) = (𝐶↑𝑀)) | |
14 | oveq2 6658 | . . . . . 6 ⊢ (𝑛 = 𝑀 → ((2 · 𝑁)↑𝑛) = ((2 · 𝑁)↑𝑀)) | |
15 | 14 | oveq1d 6665 | . . . . 5 ⊢ (𝑛 = 𝑀 → (((2 · 𝑁)↑𝑛) · 𝐴) = (((2 · 𝑁)↑𝑀) · 𝐴)) |
16 | 15 | fveq2d 6195 | . . . 4 ⊢ (𝑛 = 𝑀 → (𝑇‘(((2 · 𝑁)↑𝑛) · 𝐴)) = (𝑇‘(((2 · 𝑁)↑𝑀) · 𝐴))) |
17 | 13, 16 | oveq12d 6668 | . . 3 ⊢ (𝑛 = 𝑀 → ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝐴))) = ((𝐶↑𝑀) · (𝑇‘(((2 · 𝑁)↑𝑀) · 𝐴)))) |
18 | 17 | adantl 482 | . 2 ⊢ ((𝜑 ∧ 𝑛 = 𝑀) → ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝐴))) = ((𝐶↑𝑀) · (𝑇‘(((2 · 𝑁)↑𝑀) · 𝐴)))) |
19 | knoppcnlem1.3 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℕ0) | |
20 | ovexd 6680 | . 2 ⊢ (𝜑 → ((𝐶↑𝑀) · (𝑇‘(((2 · 𝑁)↑𝑀) · 𝐴))) ∈ V) | |
21 | 12, 18, 19, 20 | fvmptd 6288 | 1 ⊢ (𝜑 → ((𝐹‘𝐴)‘𝑀) = ((𝐶↑𝑀) · (𝑇‘(((2 · 𝑁)↑𝑀) · 𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 · cmul 9941 2c2 11070 ℕ0cn0 11292 ↑cexp 12860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-i2m1 10004 ax-1ne0 10005 ax-rrecex 10008 ax-cnre 10009 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-nn 11021 df-n0 11293 |
This theorem is referenced by: knoppcnlem3 32485 knoppcnlem4 32486 knoppcnlem10 32492 knoppndvlem6 32508 knoppndvlem7 32509 knoppndvlem11 32513 |
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