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 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → 𝑖 = 𝐼) | |
4 | 3 | fveq2d 6667 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → (𝑉‘𝑖) = (𝑉‘𝐼)) |
5 | 4 | oveq1d 7163 | . . 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 42385 | . . . . . . . . 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 234 | . . . . . . 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 497 | . . . . . 6 ⊢ (𝜑 → 𝑉 ∈ (ℝ ↑m (0...𝑀))) |
14 | elmapi 8420 | . . . . . 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 6845 | . . . 4 ⊢ (𝜑 → (𝑉‘𝐼) ∈ ℝ) |
17 | fourierdlem13.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
18 | 16, 17 | resubcld 11060 | . . 3 ⊢ (𝜑 → ((𝑉‘𝐼) − 𝑋) ∈ ℝ) |
19 | 2, 5, 6, 18 | fvmptd 6768 | . 2 ⊢ (𝜑 → (𝑄‘𝐼) = ((𝑉‘𝐼) − 𝑋)) |
20 | 19 | oveq2d 7164 | . . 3 ⊢ (𝜑 → (𝑋 + (𝑄‘𝐼)) = (𝑋 + ((𝑉‘𝐼) − 𝑋))) |
21 | 17 | recnd 10661 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
22 | 16 | recnd 10661 | . . . 4 ⊢ (𝜑 → (𝑉‘𝐼) ∈ ℂ) |
23 | 21, 22 | pncan3d 10992 | . . 3 ⊢ (𝜑 → (𝑋 + ((𝑉‘𝐼) − 𝑋)) = (𝑉‘𝐼)) |
24 | 20, 23 | eqtr2d 2855 | . 2 ⊢ (𝜑 → (𝑉‘𝐼) = (𝑋 + (𝑄‘𝐼))) |
25 | 19, 24 | jca 514 | 1 ⊢ (𝜑 → ((𝑄‘𝐼) = ((𝑉‘𝐼) − 𝑋) ∧ (𝑉‘𝐼) = (𝑋 + (𝑄‘𝐼)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ∀wral 3136 {crab 3140 class class class wbr 5057 ↦ cmpt 5137 ⟶wf 6344 ‘cfv 6348 (class class class)co 7148 ↑m cmap 8398 ℝcr 10528 0cc0 10529 1c1 10530 + caddc 10532 < clt 10667 − cmin 10862 ℕcn 11630 ...cfz 12884 ..^cfzo 13025 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-1st 7681 df-2nd 7682 df-er 8281 df-map 8400 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-ltxr 10672 df-sub 10864 df-neg 10865 |
This theorem is referenced by: fourierdlem72 42454 fourierdlem103 42485 fourierdlem104 42486 |
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