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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 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 13378 | . . . . 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 13431 | . . . . . . 7 ⊢ (𝐴[,]𝐵) ⊆ ℝ* | |
8 | fourierdlem27.q | . . . . . . . 8 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵)) | |
9 | fourierdlem27.i | . . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (0..^𝑀)) | |
10 | elfzofz 13672 | . . . . . . . . 9 ⊢ (𝐼 ∈ (0..^𝑀) → 𝐼 ∈ (0...𝑀)) | |
11 | 9, 10 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ (0...𝑀)) |
12 | 8, 11 | ffvelcdmd 7089 | . . . . . . 7 ⊢ (𝜑 → (𝑄‘𝐼) ∈ (𝐴[,]𝐵)) |
13 | 7, 12 | sselid 3976 | . . . . . 6 ⊢ (𝜑 → (𝑄‘𝐼) ∈ ℝ*) |
14 | 13 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) ∈ ℝ*) |
15 | 6 | rexrd 11286 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 ∈ ℝ*) |
16 | iccgelb 13404 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ (𝑄‘𝐼) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (𝑄‘𝐼)) | |
17 | 1, 3, 12, 16 | syl3anc 1369 | . . . . . 6 ⊢ (𝜑 → 𝐴 ≤ (𝑄‘𝐼)) |
18 | 17 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝐴 ≤ (𝑄‘𝐼)) |
19 | fzofzp1 13753 | . . . . . . . . . 10 ⊢ (𝐼 ∈ (0..^𝑀) → (𝐼 + 1) ∈ (0...𝑀)) | |
20 | 9, 19 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (𝐼 + 1) ∈ (0...𝑀)) |
21 | 8, 20 | ffvelcdmd 7089 | . . . . . . . 8 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ (𝐴[,]𝐵)) |
22 | 7, 21 | sselid 3976 | . . . . . . 7 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ ℝ*) |
23 | 22 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘(𝐼 + 1)) ∈ ℝ*) |
24 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) | |
25 | ioogtlb 44803 | . . . . . 6 ⊢ (((𝑄‘𝐼) ∈ ℝ* ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ* ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) < 𝑥) | |
26 | 14, 23, 24, 25 | syl3anc 1369 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) < 𝑥) |
27 | 2, 14, 15, 18, 26 | xrlelttrd 13163 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝐴 < 𝑥) |
28 | iooltub 44818 | . . . . . 6 ⊢ (((𝑄‘𝐼) ∈ ℝ* ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ* ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 < (𝑄‘(𝐼 + 1))) | |
29 | 14, 23, 24, 28 | syl3anc 1369 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 < (𝑄‘(𝐼 + 1))) |
30 | iccleub 13403 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ (𝑄‘(𝐼 + 1)) ∈ (𝐴[,]𝐵)) → (𝑄‘(𝐼 + 1)) ≤ 𝐵) | |
31 | 1, 3, 21, 30 | syl3anc 1369 | . . . . . 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 13164 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 < 𝐵) |
34 | 2, 4, 6, 27, 33 | eliood 44806 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 ∈ (𝐴(,)𝐵)) |
35 | 34 | ralrimiva 3141 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))𝑥 ∈ (𝐴(,)𝐵)) |
36 | dfss3 3966 | . 2 ⊢ (((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ (𝐴(,)𝐵) ↔ ∀𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))𝑥 ∈ (𝐴(,)𝐵)) | |
37 | 35, 36 | sylibr 233 | 1 ⊢ (𝜑 → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ (𝐴(,)𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2099 ∀wral 3056 ⊆ wss 3944 class class class wbr 5142 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ℝcr 11129 0cc0 11130 1c1 11131 + caddc 11133 ℝ*cxr 11269 < clt 11270 ≤ cle 11271 (,)cioo 13348 [,]cicc 13351 ...cfz 13508 ..^cfzo 13651 |
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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-n0 12495 df-z 12581 df-uz 12845 df-ioo 13352 df-icc 13355 df-fz 13509 df-fzo 13652 |
This theorem is referenced by: fourierdlem102 45519 fourierdlem114 45531 |
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