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 6674 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → (𝑉‘𝑖) = (𝑉‘𝐼)) |
5 | 4 | oveq1d 7171 | . . 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 42414 | . . . . . . . . 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 8428 | . . . . . 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 6852 | . . . 4 ⊢ (𝜑 → (𝑉‘𝐼) ∈ ℝ) |
17 | fourierdlem13.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
18 | 16, 17 | resubcld 11068 | . . 3 ⊢ (𝜑 → ((𝑉‘𝐼) − 𝑋) ∈ ℝ) |
19 | 2, 5, 6, 18 | fvmptd 6775 | . 2 ⊢ (𝜑 → (𝑄‘𝐼) = ((𝑉‘𝐼) − 𝑋)) |
20 | 19 | oveq2d 7172 | . . 3 ⊢ (𝜑 → (𝑋 + (𝑄‘𝐼)) = (𝑋 + ((𝑉‘𝐼) − 𝑋))) |
21 | 17 | recnd 10669 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
22 | 16 | recnd 10669 | . . . 4 ⊢ (𝜑 → (𝑉‘𝐼) ∈ ℂ) |
23 | 21, 22 | pncan3d 11000 | . . 3 ⊢ (𝜑 → (𝑋 + ((𝑉‘𝐼) − 𝑋)) = (𝑉‘𝐼)) |
24 | 20, 23 | eqtr2d 2857 | . 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 1537 ∈ wcel 2114 ∀wral 3138 {crab 3142 class class class wbr 5066 ↦ cmpt 5146 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ↑m cmap 8406 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 < clt 10675 − cmin 10870 ℕcn 11638 ...cfz 12893 ..^cfzo 13034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-sub 10872 df-neg 10873 |
This theorem is referenced by: fourierdlem72 42483 fourierdlem103 42514 fourierdlem104 42515 |
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