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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 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝐴 ∈ ℝ*) |
3 | fourierdlem27.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
4 | 3 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝐵 ∈ ℝ*) |
5 | elioore 12076 | . . . . 5 ⊢ (𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) → 𝑥 ∈ ℝ) | |
6 | 5 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 ∈ ℝ) |
7 | iccssxr 12127 | . . . . . . 7 ⊢ (𝐴[,]𝐵) ⊆ ℝ* | |
8 | fourierdlem27.q | . . . . . . . 8 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵)) | |
9 | fourierdlem27.i | . . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (0..^𝑀)) | |
10 | elfzofz 12354 | . . . . . . . . 9 ⊢ (𝐼 ∈ (0..^𝑀) → 𝐼 ∈ (0...𝑀)) | |
11 | 9, 10 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ (0...𝑀)) |
12 | 8, 11 | ffvelrnd 6268 | . . . . . . 7 ⊢ (𝜑 → (𝑄‘𝐼) ∈ (𝐴[,]𝐵)) |
13 | 7, 12 | sseldi 3566 | . . . . . 6 ⊢ (𝜑 → (𝑄‘𝐼) ∈ ℝ*) |
14 | 13 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) ∈ ℝ*) |
15 | 6 | rexrd 9968 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 ∈ ℝ*) |
16 | iccgelb 12101 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ (𝑄‘𝐼) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (𝑄‘𝐼)) | |
17 | 1, 3, 12, 16 | syl3anc 1318 | . . . . . 6 ⊢ (𝜑 → 𝐴 ≤ (𝑄‘𝐼)) |
18 | 17 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝐴 ≤ (𝑄‘𝐼)) |
19 | fzofzp1 12431 | . . . . . . . . . 10 ⊢ (𝐼 ∈ (0..^𝑀) → (𝐼 + 1) ∈ (0...𝑀)) | |
20 | 9, 19 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (𝐼 + 1) ∈ (0...𝑀)) |
21 | 8, 20 | ffvelrnd 6268 | . . . . . . . 8 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ (𝐴[,]𝐵)) |
22 | 7, 21 | sseldi 3566 | . . . . . . 7 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ ℝ*) |
23 | 22 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘(𝐼 + 1)) ∈ ℝ*) |
24 | simpr 476 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) | |
25 | ioogtlb 38564 | . . . . . 6 ⊢ (((𝑄‘𝐼) ∈ ℝ* ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ* ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) < 𝑥) | |
26 | 14, 23, 24, 25 | syl3anc 1318 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) < 𝑥) |
27 | 2, 14, 15, 18, 26 | xrlelttrd 11867 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝐴 < 𝑥) |
28 | iooltub 38582 | . . . . . 6 ⊢ (((𝑄‘𝐼) ∈ ℝ* ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ* ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 < (𝑄‘(𝐼 + 1))) | |
29 | 14, 23, 24, 28 | syl3anc 1318 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 < (𝑄‘(𝐼 + 1))) |
30 | iccleub 12100 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ (𝑄‘(𝐼 + 1)) ∈ (𝐴[,]𝐵)) → (𝑄‘(𝐼 + 1)) ≤ 𝐵) | |
31 | 1, 3, 21, 30 | syl3anc 1318 | . . . . . 6 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ≤ 𝐵) |
32 | 31 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘(𝐼 + 1)) ≤ 𝐵) |
33 | 15, 23, 4, 29, 32 | xrltletrd 11868 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 < 𝐵) |
34 | 2, 4, 6, 27, 33 | eliood 38567 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 ∈ (𝐴(,)𝐵)) |
35 | 34 | ralrimiva 2949 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))𝑥 ∈ (𝐴(,)𝐵)) |
36 | dfss3 3558 | . 2 ⊢ (((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ (𝐴(,)𝐵) ↔ ∀𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))𝑥 ∈ (𝐴(,)𝐵)) | |
37 | 35, 36 | sylibr 223 | 1 ⊢ (𝜑 → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ (𝐴(,)𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 class class class wbr 4583 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 (,)cioo 12046 [,]cicc 12049 ...cfz 12197 ..^cfzo 12334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-ioo 12050 df-icc 12053 df-fz 12198 df-fzo 12335 |
This theorem is referenced by: fourierdlem102 39101 fourierdlem114 39113 |
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