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 12820 | . . 3 ⊢ ((𝑄‘𝐼)[,](𝑄‘(𝐼 + 1))) ⊆ ℝ* | |
2 | fourierdlem1.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ ((𝑄‘𝐼)[,](𝑄‘(𝐼 + 1)))) | |
3 | 1, 2 | sseldi 3965 | . 2 ⊢ (𝜑 → 𝑋 ∈ ℝ*) |
4 | fourierdlem1.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
5 | iccssxr 12820 | . . . 4 ⊢ (𝐴[,]𝐵) ⊆ ℝ* | |
6 | fourierdlem1.q | . . . . 5 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵)) | |
7 | fourierdlem1.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (0..^𝑀)) | |
8 | elfzofz 13054 | . . . . . 6 ⊢ (𝐼 ∈ (0..^𝑀) → 𝐼 ∈ (0...𝑀)) | |
9 | 7, 8 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (0...𝑀)) |
10 | 6, 9 | ffvelrnd 6852 | . . . 4 ⊢ (𝜑 → (𝑄‘𝐼) ∈ (𝐴[,]𝐵)) |
11 | 5, 10 | sseldi 3965 | . . 3 ⊢ (𝜑 → (𝑄‘𝐼) ∈ ℝ*) |
12 | fourierdlem1.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
13 | iccgelb 12794 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ (𝑄‘𝐼) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (𝑄‘𝐼)) | |
14 | 4, 12, 10, 13 | syl3anc 1367 | . . 3 ⊢ (𝜑 → 𝐴 ≤ (𝑄‘𝐼)) |
15 | fzofzp1 13135 | . . . . . . . . 9 ⊢ (𝐼 ∈ (0..^𝑀) → (𝐼 + 1) ∈ (0...𝑀)) | |
16 | 7, 15 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐼 + 1) ∈ (0...𝑀)) |
17 | 6, 16 | ffvelrnd 6852 | . . . . . . 7 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ (𝐴[,]𝐵)) |
18 | 5, 17 | sseldi 3965 | . . . . . 6 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ ℝ*) |
19 | elicc4 12804 | . . . . . 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 1367 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ ((𝑄‘𝐼)[,](𝑄‘(𝐼 + 1))) ↔ ((𝑄‘𝐼) ≤ 𝑋 ∧ 𝑋 ≤ (𝑄‘(𝐼 + 1))))) |
21 | 2, 20 | mpbid 234 | . . . 4 ⊢ (𝜑 → ((𝑄‘𝐼) ≤ 𝑋 ∧ 𝑋 ≤ (𝑄‘(𝐼 + 1)))) |
22 | 21 | simpld 497 | . . 3 ⊢ (𝜑 → (𝑄‘𝐼) ≤ 𝑋) |
23 | 4, 11, 3, 14, 22 | xrletrd 12556 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝑋) |
24 | iccleub 12793 | . . . 4 ⊢ (((𝑄‘𝐼) ∈ ℝ* ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ* ∧ 𝑋 ∈ ((𝑄‘𝐼)[,](𝑄‘(𝐼 + 1)))) → 𝑋 ≤ (𝑄‘(𝐼 + 1))) | |
25 | 11, 18, 2, 24 | syl3anc 1367 | . . 3 ⊢ (𝜑 → 𝑋 ≤ (𝑄‘(𝐼 + 1))) |
26 | elicc4 12804 | . . . . . 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 1367 | . . . . 5 ⊢ (𝜑 → ((𝑄‘(𝐼 + 1)) ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ (𝑄‘(𝐼 + 1)) ∧ (𝑄‘(𝐼 + 1)) ≤ 𝐵))) |
28 | 17, 27 | mpbid 234 | . . . 4 ⊢ (𝜑 → (𝐴 ≤ (𝑄‘(𝐼 + 1)) ∧ (𝑄‘(𝐼 + 1)) ≤ 𝐵)) |
29 | 28 | simprd 498 | . . 3 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ≤ 𝐵) |
30 | 3, 18, 12, 25, 29 | xrletrd 12556 | . 2 ⊢ (𝜑 → 𝑋 ≤ 𝐵) |
31 | elicc1 12783 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 ∈ ℝ* ∧ 𝐴 ≤ 𝑋 ∧ 𝑋 ≤ 𝐵))) | |
32 | 4, 12, 31 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 ∈ ℝ* ∧ 𝐴 ≤ 𝑋 ∧ 𝑋 ≤ 𝐵))) |
33 | 3, 23, 30, 32 | mpbir3and 1338 | 1 ⊢ (𝜑 → 𝑋 ∈ (𝐴[,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2114 class class class wbr 5066 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 0cc0 10537 1c1 10538 + caddc 10540 ℝ*cxr 10674 ≤ cle 10676 [,]cicc 12742 ...cfz 12893 ..^cfzo 13034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-icc 12746 df-fz 12894 df-fzo 13035 |
This theorem is referenced by: fourierdlem8 42420 fourierdlem73 42484 fourierdlem81 42492 fourierdlem92 42503 fourierdlem93 42504 |
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