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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 13453 | . . 3 ⊢ ((𝑄‘𝐼)[,](𝑄‘(𝐼 + 1))) ⊆ ℝ* | |
2 | fourierdlem1.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ ((𝑄‘𝐼)[,](𝑄‘(𝐼 + 1)))) | |
3 | 1, 2 | sselid 3977 | . 2 ⊢ (𝜑 → 𝑋 ∈ ℝ*) |
4 | fourierdlem1.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
5 | iccssxr 13453 | . . . 4 ⊢ (𝐴[,]𝐵) ⊆ ℝ* | |
6 | fourierdlem1.q | . . . . 5 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵)) | |
7 | fourierdlem1.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (0..^𝑀)) | |
8 | elfzofz 13694 | . . . . . 6 ⊢ (𝐼 ∈ (0..^𝑀) → 𝐼 ∈ (0...𝑀)) | |
9 | 7, 8 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (0...𝑀)) |
10 | 6, 9 | ffvelcdmd 7089 | . . . 4 ⊢ (𝜑 → (𝑄‘𝐼) ∈ (𝐴[,]𝐵)) |
11 | 5, 10 | sselid 3977 | . . 3 ⊢ (𝜑 → (𝑄‘𝐼) ∈ ℝ*) |
12 | fourierdlem1.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
13 | iccgelb 13426 | . . . 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 13776 | . . . . . . . . 9 ⊢ (𝐼 ∈ (0..^𝑀) → (𝐼 + 1) ∈ (0...𝑀)) | |
16 | 7, 15 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐼 + 1) ∈ (0...𝑀)) |
17 | 6, 16 | ffvelcdmd 7089 | . . . . . . 7 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ (𝐴[,]𝐵)) |
18 | 5, 17 | sselid 3977 | . . . . . 6 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ ℝ*) |
19 | elicc4 13437 | . . . . . 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 231 | . . . 4 ⊢ (𝜑 → ((𝑄‘𝐼) ≤ 𝑋 ∧ 𝑋 ≤ (𝑄‘(𝐼 + 1)))) |
22 | 21 | simpld 493 | . . 3 ⊢ (𝜑 → (𝑄‘𝐼) ≤ 𝑋) |
23 | 4, 11, 3, 14, 22 | xrletrd 13187 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝑋) |
24 | iccleub 13425 | . . . 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 13437 | . . . . . 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 231 | . . . 4 ⊢ (𝜑 → (𝐴 ≤ (𝑄‘(𝐼 + 1)) ∧ (𝑄‘(𝐼 + 1)) ≤ 𝐵)) |
29 | 28 | simprd 494 | . . 3 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ≤ 𝐵) |
30 | 3, 18, 12, 25, 29 | xrletrd 13187 | . 2 ⊢ (𝜑 → 𝑋 ≤ 𝐵) |
31 | elicc1 13414 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 ∈ ℝ* ∧ 𝐴 ≤ 𝑋 ∧ 𝑋 ≤ 𝐵))) | |
32 | 4, 12, 31 | syl2anc 582 | . 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 205 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2099 class class class wbr 5144 ⟶wf 6540 ‘cfv 6544 (class class class)co 7414 0cc0 11147 1c1 11148 + caddc 11150 ℝ*cxr 11286 ≤ cle 11288 [,]cicc 13373 ...cfz 13530 ..^cfzo 13673 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-nn 12257 df-n0 12517 df-z 12603 df-uz 12867 df-icc 13377 df-fz 13531 df-fzo 13674 |
This theorem is referenced by: fourierdlem8 45770 fourierdlem73 45834 fourierdlem81 45842 fourierdlem92 45853 fourierdlem93 45854 |
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