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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 13414 | . . . . 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 13467 | . . . . . . 7 ⊢ (𝐴[,]𝐵) ⊆ ℝ* | |
8 | fourierdlem27.q | . . . . . . . 8 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵)) | |
9 | fourierdlem27.i | . . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (0..^𝑀)) | |
10 | elfzofz 13712 | . . . . . . . . 9 ⊢ (𝐼 ∈ (0..^𝑀) → 𝐼 ∈ (0...𝑀)) | |
11 | 9, 10 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ (0...𝑀)) |
12 | 8, 11 | ffvelcdmd 7105 | . . . . . . 7 ⊢ (𝜑 → (𝑄‘𝐼) ∈ (𝐴[,]𝐵)) |
13 | 7, 12 | sselid 3993 | . . . . . 6 ⊢ (𝜑 → (𝑄‘𝐼) ∈ ℝ*) |
14 | 13 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) ∈ ℝ*) |
15 | 6 | rexrd 11309 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 ∈ ℝ*) |
16 | iccgelb 13440 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ (𝑄‘𝐼) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (𝑄‘𝐼)) | |
17 | 1, 3, 12, 16 | syl3anc 1370 | . . . . . 6 ⊢ (𝜑 → 𝐴 ≤ (𝑄‘𝐼)) |
18 | 17 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝐴 ≤ (𝑄‘𝐼)) |
19 | fzofzp1 13800 | . . . . . . . . . 10 ⊢ (𝐼 ∈ (0..^𝑀) → (𝐼 + 1) ∈ (0...𝑀)) | |
20 | 9, 19 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (𝐼 + 1) ∈ (0...𝑀)) |
21 | 8, 20 | ffvelcdmd 7105 | . . . . . . . 8 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ (𝐴[,]𝐵)) |
22 | 7, 21 | sselid 3993 | . . . . . . 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 45448 | . . . . . 6 ⊢ (((𝑄‘𝐼) ∈ ℝ* ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ* ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) < 𝑥) | |
26 | 14, 23, 24, 25 | syl3anc 1370 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → (𝑄‘𝐼) < 𝑥) |
27 | 2, 14, 15, 18, 26 | xrlelttrd 13199 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝐴 < 𝑥) |
28 | iooltub 45463 | . . . . . 6 ⊢ (((𝑄‘𝐼) ∈ ℝ* ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ* ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 < (𝑄‘(𝐼 + 1))) | |
29 | 14, 23, 24, 28 | syl3anc 1370 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 < (𝑄‘(𝐼 + 1))) |
30 | iccleub 13439 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ (𝑄‘(𝐼 + 1)) ∈ (𝐴[,]𝐵)) → (𝑄‘(𝐼 + 1)) ≤ 𝐵) | |
31 | 1, 3, 21, 30 | syl3anc 1370 | . . . . . 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 13200 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 < 𝐵) |
34 | 2, 4, 6, 27, 33 | eliood 45451 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → 𝑥 ∈ (𝐴(,)𝐵)) |
35 | 34 | ralrimiva 3144 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))𝑥 ∈ (𝐴(,)𝐵)) |
36 | dfss3 3984 | . 2 ⊢ (((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ (𝐴(,)𝐵) ↔ ∀𝑥 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))𝑥 ∈ (𝐴(,)𝐵)) | |
37 | 35, 36 | sylibr 234 | 1 ⊢ (𝜑 → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ (𝐴(,)𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 ∀wral 3059 ⊆ wss 3963 class class class wbr 5148 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 (,)cioo 13384 [,]cicc 13387 ...cfz 13544 ..^cfzo 13691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-ioo 13388 df-icc 13391 df-fz 13545 df-fzo 13692 |
This theorem is referenced by: fourierdlem102 46164 fourierdlem114 46176 |
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