Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fourierdlem27 | Structured version Visualization version GIF version |
Description: A partition open interval is a subset of the partitioned open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
fourierdlem27.a | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
fourierdlem27.b | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
fourierdlem27.q | ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵)) |
fourierdlem27.i | ⊢ (𝜑 → 𝐼 ∈ (0..^𝑀)) |
Ref | Expression |
---|---|
fourierdlem27 | ⊢ (𝜑 → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ (𝐴(,)𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fourierdlem27.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
2 | 1 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝐴 ∈ ℝ*) |
3 | fourierdlem27.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
4 | 3 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝐵 ∈ ℝ*) |
5 | elioore 12205 | . . . . 5 ⊢ (𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) → 𝑥 ∈ ℝ) | |
6 | 5 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 ∈ ℝ) |
7 | iccssxr 12256 | . . . . . . 7 ⊢ (𝐴[,]𝐵) ⊆ ℝ* | |
8 | fourierdlem27.q | . . . . . . . 8 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵)) | |
9 | fourierdlem27.i | . . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (0..^𝑀)) | |
10 | elfzofz 12485 | . . . . . . . . 9 ⊢ (𝐼 ∈ (0..^𝑀) → 𝐼 ∈ (0...𝑀)) | |
11 | 9, 10 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ (0...𝑀)) |
12 | 8, 11 | ffvelrnd 6360 | . . . . . . 7 ⊢ (𝜑 → (𝑄‘𝐼) ∈ (𝐴[,]𝐵)) |
13 | 7, 12 | sseldi 3601 | . . . . . 6 ⊢ (𝜑 → (𝑄‘𝐼) ∈ ℝ*) |
14 | 13 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) ∈ ℝ*) |
15 | 6 | rexrd 10089 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 ∈ ℝ*) |
16 | iccgelb 12230 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ (𝑄‘𝐼) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (𝑄‘𝐼)) | |
17 | 1, 3, 12, 16 | syl3anc 1326 | . . . . . 6 ⊢ (𝜑 → 𝐴 ≤ (𝑄‘𝐼)) |
18 | 17 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝐴 ≤ (𝑄‘𝐼)) |
19 | fzofzp1 12565 | . . . . . . . . . 10 ⊢ (𝐼 ∈ (0..^𝑀) → (𝐼 + 1) ∈ (0...𝑀)) | |
20 | 9, 19 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (𝐼 + 1) ∈ (0...𝑀)) |
21 | 8, 20 | ffvelrnd 6360 | . . . . . . . 8 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ (𝐴[,]𝐵)) |
22 | 7, 21 | sseldi 3601 | . . . . . . 7 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ ℝ*) |
23 | 22 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘(𝐼 + 1)) ∈ ℝ*) |
24 | simpr 477 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) | |
25 | ioogtlb 39717 | . . . . . 6 ⊢ (((𝑄‘𝐼) ∈ ℝ* ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ* ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) < 𝑥) | |
26 | 14, 23, 24, 25 | syl3anc 1326 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) < 𝑥) |
27 | 2, 14, 15, 18, 26 | xrlelttrd 11991 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝐴 < 𝑥) |
28 | iooltub 39735 | . . . . . 6 ⊢ (((𝑄‘𝐼) ∈ ℝ* ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ* ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 < (𝑄‘(𝐼 + 1))) | |
29 | 14, 23, 24, 28 | syl3anc 1326 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 < (𝑄‘(𝐼 + 1))) |
30 | iccleub 12229 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ (𝑄‘(𝐼 + 1)) ∈ (𝐴[,]𝐵)) → (𝑄‘(𝐼 + 1)) ≤ 𝐵) | |
31 | 1, 3, 21, 30 | syl3anc 1326 | . . . . . 6 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ≤ 𝐵) |
32 | 31 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘(𝐼 + 1)) ≤ 𝐵) |
33 | 15, 23, 4, 29, 32 | xrltletrd 11992 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 < 𝐵) |
34 | 2, 4, 6, 27, 33 | eliood 39720 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 ∈ (𝐴(,)𝐵)) |
35 | 34 | ralrimiva 2966 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))𝑥 ∈ (𝐴(,)𝐵)) |
36 | dfss3 3592 | . 2 ⊢ (((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ (𝐴(,)𝐵) ↔ ∀𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))𝑥 ∈ (𝐴(,)𝐵)) | |
37 | 35, 36 | sylibr 224 | 1 ⊢ (𝜑 → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ (𝐴(,)𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 class class class wbr 4653 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 ℝ*cxr 10073 < clt 10074 ≤ cle 10075 (,)cioo 12175 [,]cicc 12178 ...cfz 12326 ..^cfzo 12465 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-ioo 12179 df-icc 12182 df-fz 12327 df-fzo 12466 |
This theorem is referenced by: fourierdlem102 40425 fourierdlem114 40437 |
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