Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fourierdlem13 | Structured version Visualization version GIF version |
Description: Value of 𝑉 in terms of value of 𝑄. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
fourierdlem13.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
fourierdlem13.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
fourierdlem13.x | ⊢ (𝜑 → 𝑋 ∈ ℝ) |
fourierdlem13.p | ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 + 𝑋) ∧ (𝑝‘𝑚) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) |
fourierdlem13.m | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
fourierdlem13.v | ⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀)) |
fourierdlem13.i | ⊢ (𝜑 → 𝐼 ∈ (0...𝑀)) |
fourierdlem13.q | ⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)) |
Ref | Expression |
---|---|
fourierdlem13 | ⊢ (𝜑 → ((𝑄‘𝐼) = ((𝑉‘𝐼) − 𝑋) ∧ (𝑉‘𝐼) = (𝑋 + (𝑄‘𝐼)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fourierdlem13.q | . . . 4 ⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)) | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋))) |
3 | simpr 476 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → 𝑖 = 𝐼) | |
4 | 3 | fveq2d 6107 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → (𝑉‘𝑖) = (𝑉‘𝐼)) |
5 | 4 | oveq1d 6564 | . . 3 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → ((𝑉‘𝑖) − 𝑋) = ((𝑉‘𝐼) − 𝑋)) |
6 | fourierdlem13.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ (0...𝑀)) | |
7 | fourierdlem13.v | . . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀)) | |
8 | fourierdlem13.m | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
9 | fourierdlem13.p | . . . . . . . . . 10 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 + 𝑋) ∧ (𝑝‘𝑚) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) | |
10 | 9 | fourierdlem2 39002 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → (𝑉 ∈ (𝑃‘𝑀) ↔ (𝑉 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑉‘0) = (𝐴 + 𝑋) ∧ (𝑉‘𝑀) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉‘𝑖) < (𝑉‘(𝑖 + 1)))))) |
11 | 8, 10 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝑉 ∈ (𝑃‘𝑀) ↔ (𝑉 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑉‘0) = (𝐴 + 𝑋) ∧ (𝑉‘𝑀) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉‘𝑖) < (𝑉‘(𝑖 + 1)))))) |
12 | 7, 11 | mpbid 221 | . . . . . . 7 ⊢ (𝜑 → (𝑉 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑉‘0) = (𝐴 + 𝑋) ∧ (𝑉‘𝑀) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉‘𝑖) < (𝑉‘(𝑖 + 1))))) |
13 | 12 | simpld 474 | . . . . . 6 ⊢ (𝜑 → 𝑉 ∈ (ℝ ↑𝑚 (0...𝑀))) |
14 | elmapi 7765 | . . . . . 6 ⊢ (𝑉 ∈ (ℝ ↑𝑚 (0...𝑀)) → 𝑉:(0...𝑀)⟶ℝ) | |
15 | 13, 14 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑉:(0...𝑀)⟶ℝ) |
16 | 15, 6 | ffvelrnd 6268 | . . . 4 ⊢ (𝜑 → (𝑉‘𝐼) ∈ ℝ) |
17 | fourierdlem13.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
18 | 16, 17 | resubcld 10337 | . . 3 ⊢ (𝜑 → ((𝑉‘𝐼) − 𝑋) ∈ ℝ) |
19 | 2, 5, 6, 18 | fvmptd 6197 | . 2 ⊢ (𝜑 → (𝑄‘𝐼) = ((𝑉‘𝐼) − 𝑋)) |
20 | 19 | oveq2d 6565 | . . 3 ⊢ (𝜑 → (𝑋 + (𝑄‘𝐼)) = (𝑋 + ((𝑉‘𝐼) − 𝑋))) |
21 | 17 | recnd 9947 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
22 | 16 | recnd 9947 | . . . 4 ⊢ (𝜑 → (𝑉‘𝐼) ∈ ℂ) |
23 | 21, 22 | pncan3d 10274 | . . 3 ⊢ (𝜑 → (𝑋 + ((𝑉‘𝐼) − 𝑋)) = (𝑉‘𝐼)) |
24 | 20, 23 | eqtr2d 2645 | . 2 ⊢ (𝜑 → (𝑉‘𝐼) = (𝑋 + (𝑄‘𝐼))) |
25 | 19, 24 | jca 553 | 1 ⊢ (𝜑 → ((𝑄‘𝐼) = ((𝑉‘𝐼) − 𝑋) ∧ (𝑉‘𝐼) = (𝑋 + (𝑄‘𝐼)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 class class class wbr 4583 ↦ cmpt 4643 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 − cmin 10145 ℕcn 10897 ...cfz 12197 ..^cfzo 12334 |
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-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
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-nel 2783 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-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 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-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-sub 10147 df-neg 10148 |
This theorem is referenced by: fourierdlem72 39071 fourierdlem103 39102 fourierdlem104 39103 |
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