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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppcnlem1 | Structured version Visualization version GIF version |
Description: Lemma for knoppcn 31664. (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 6557 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → (((2 · 𝑁)↑𝑛) · 𝑦) = (((2 · 𝑁)↑𝑛) · 𝐴)) | |
4 | 3 | fveq2d 6107 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)) = (𝑇‘(((2 · 𝑁)↑𝑛) · 𝐴))) |
5 | 4 | oveq2d 6565 | . . . . 5 ⊢ (𝑦 = 𝐴 → ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))) = ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝐴)))) |
6 | 5 | mpteq2dv 4673 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))) = (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝐴))))) |
7 | 6 | adantl 481 | . . 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 11175 | . . . . 5 ⊢ ℕ0 ∈ V | |
10 | 9 | mptex 6390 | . . . 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 6197 | . 2 ⊢ (𝜑 → (𝐹‘𝐴) = (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝐴))))) |
13 | oveq2 6557 | . . . 4 ⊢ (𝑛 = 𝑀 → (𝐶↑𝑛) = (𝐶↑𝑀)) | |
14 | oveq2 6557 | . . . . . 6 ⊢ (𝑛 = 𝑀 → ((2 · 𝑁)↑𝑛) = ((2 · 𝑁)↑𝑀)) | |
15 | 14 | oveq1d 6564 | . . . . 5 ⊢ (𝑛 = 𝑀 → (((2 · 𝑁)↑𝑛) · 𝐴) = (((2 · 𝑁)↑𝑀) · 𝐴)) |
16 | 15 | fveq2d 6107 | . . . 4 ⊢ (𝑛 = 𝑀 → (𝑇‘(((2 · 𝑁)↑𝑛) · 𝐴)) = (𝑇‘(((2 · 𝑁)↑𝑀) · 𝐴))) |
17 | 13, 16 | oveq12d 6567 | . . 3 ⊢ (𝑛 = 𝑀 → ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝐴))) = ((𝐶↑𝑀) · (𝑇‘(((2 · 𝑁)↑𝑀) · 𝐴)))) |
18 | 17 | adantl 481 | . 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 | ovex 6577 | . . 3 ⊢ ((𝐶↑𝑀) · (𝑇‘(((2 · 𝑁)↑𝑀) · 𝐴))) ∈ V | |
21 | 20 | a1i 11 | . 2 ⊢ (𝜑 → ((𝐶↑𝑀) · (𝑇‘(((2 · 𝑁)↑𝑀) · 𝐴))) ∈ V) |
22 | 12, 18, 19, 21 | fvmptd 6197 | 1 ⊢ (𝜑 → ((𝐹‘𝐴)‘𝑀) = ((𝐶↑𝑀) · (𝑇‘(((2 · 𝑁)↑𝑀) · 𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 · cmul 9820 2c2 10947 ℕ0cn0 11169 ↑cexp 12722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-nn 10898 df-n0 11170 |
This theorem is referenced by: knoppcnlem3 31655 knoppcnlem4 31656 knoppcnlem10 31662 knoppndvlem6 31678 knoppndvlem7 31679 knoppndvlem11 31683 |
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