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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fourierdlem1 | Structured version Visualization version GIF version |
Description: A partition interval is a subset of the partitioned interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
fourierdlem1.a | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
fourierdlem1.b | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
fourierdlem1.q | ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵)) |
fourierdlem1.i | ⊢ (𝜑 → 𝐼 ∈ (0..^𝑀)) |
fourierdlem1.x | ⊢ (𝜑 → 𝑋 ∈ ((𝑄‘𝐼)[,](𝑄‘(𝐼 + 1)))) |
Ref | Expression |
---|---|
fourierdlem1 | ⊢ (𝜑 → 𝑋 ∈ (𝐴[,]𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccssxr 12808 | . . 3 ⊢ ((𝑄‘𝐼)[,](𝑄‘(𝐼 + 1))) ⊆ ℝ* | |
2 | fourierdlem1.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ ((𝑄‘𝐼)[,](𝑄‘(𝐼 + 1)))) | |
3 | 1, 2 | sseldi 3913 | . 2 ⊢ (𝜑 → 𝑋 ∈ ℝ*) |
4 | fourierdlem1.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
5 | iccssxr 12808 | . . . 4 ⊢ (𝐴[,]𝐵) ⊆ ℝ* | |
6 | fourierdlem1.q | . . . . 5 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵)) | |
7 | fourierdlem1.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (0..^𝑀)) | |
8 | elfzofz 13048 | . . . . . 6 ⊢ (𝐼 ∈ (0..^𝑀) → 𝐼 ∈ (0...𝑀)) | |
9 | 7, 8 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (0...𝑀)) |
10 | 6, 9 | ffvelrnd 6829 | . . . 4 ⊢ (𝜑 → (𝑄‘𝐼) ∈ (𝐴[,]𝐵)) |
11 | 5, 10 | sseldi 3913 | . . 3 ⊢ (𝜑 → (𝑄‘𝐼) ∈ ℝ*) |
12 | fourierdlem1.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
13 | iccgelb 12781 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ (𝑄‘𝐼) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (𝑄‘𝐼)) | |
14 | 4, 12, 10, 13 | syl3anc 1368 | . . 3 ⊢ (𝜑 → 𝐴 ≤ (𝑄‘𝐼)) |
15 | fzofzp1 13129 | . . . . . . . . 9 ⊢ (𝐼 ∈ (0..^𝑀) → (𝐼 + 1) ∈ (0...𝑀)) | |
16 | 7, 15 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐼 + 1) ∈ (0...𝑀)) |
17 | 6, 16 | ffvelrnd 6829 | . . . . . . 7 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ (𝐴[,]𝐵)) |
18 | 5, 17 | sseldi 3913 | . . . . . 6 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ ℝ*) |
19 | elicc4 12792 | . . . . . 6 ⊢ (((𝑄‘𝐼) ∈ ℝ* ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ* ∧ 𝑋 ∈ ℝ*) → (𝑋 ∈ ((𝑄‘𝐼)[,](𝑄‘(𝐼 + 1))) ↔ ((𝑄‘𝐼) ≤ 𝑋 ∧ 𝑋 ≤ (𝑄‘(𝐼 + 1))))) | |
20 | 11, 18, 3, 19 | syl3anc 1368 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ ((𝑄‘𝐼)[,](𝑄‘(𝐼 + 1))) ↔ ((𝑄‘𝐼) ≤ 𝑋 ∧ 𝑋 ≤ (𝑄‘(𝐼 + 1))))) |
21 | 2, 20 | mpbid 235 | . . . 4 ⊢ (𝜑 → ((𝑄‘𝐼) ≤ 𝑋 ∧ 𝑋 ≤ (𝑄‘(𝐼 + 1)))) |
22 | 21 | simpld 498 | . . 3 ⊢ (𝜑 → (𝑄‘𝐼) ≤ 𝑋) |
23 | 4, 11, 3, 14, 22 | xrletrd 12543 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝑋) |
24 | iccleub 12780 | . . . 4 ⊢ (((𝑄‘𝐼) ∈ ℝ* ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ* ∧ 𝑋 ∈ ((𝑄‘𝐼)[,](𝑄‘(𝐼 + 1)))) → 𝑋 ≤ (𝑄‘(𝐼 + 1))) | |
25 | 11, 18, 2, 24 | syl3anc 1368 | . . 3 ⊢ (𝜑 → 𝑋 ≤ (𝑄‘(𝐼 + 1))) |
26 | elicc4 12792 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ*) → ((𝑄‘(𝐼 + 1)) ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ (𝑄‘(𝐼 + 1)) ∧ (𝑄‘(𝐼 + 1)) ≤ 𝐵))) | |
27 | 4, 12, 18, 26 | syl3anc 1368 | . . . . 5 ⊢ (𝜑 → ((𝑄‘(𝐼 + 1)) ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ (𝑄‘(𝐼 + 1)) ∧ (𝑄‘(𝐼 + 1)) ≤ 𝐵))) |
28 | 17, 27 | mpbid 235 | . . . 4 ⊢ (𝜑 → (𝐴 ≤ (𝑄‘(𝐼 + 1)) ∧ (𝑄‘(𝐼 + 1)) ≤ 𝐵)) |
29 | 28 | simprd 499 | . . 3 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ≤ 𝐵) |
30 | 3, 18, 12, 25, 29 | xrletrd 12543 | . 2 ⊢ (𝜑 → 𝑋 ≤ 𝐵) |
31 | elicc1 12770 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 ∈ ℝ* ∧ 𝐴 ≤ 𝑋 ∧ 𝑋 ≤ 𝐵))) | |
32 | 4, 12, 31 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 ∈ ℝ* ∧ 𝐴 ≤ 𝑋 ∧ 𝑋 ≤ 𝐵))) |
33 | 3, 23, 30, 32 | mpbir3and 1339 | 1 ⊢ (𝜑 → 𝑋 ∈ (𝐴[,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 ∈ wcel 2111 class class class wbr 5030 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 0cc0 10526 1c1 10527 + caddc 10529 ℝ*cxr 10663 ≤ cle 10665 [,]cicc 12729 ...cfz 12885 ..^cfzo 13028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-icc 12733 df-fz 12886 df-fzo 13029 |
This theorem is referenced by: fourierdlem8 42757 fourierdlem73 42821 fourierdlem81 42829 fourierdlem92 42840 fourierdlem93 42841 |
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