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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 13470 | . . 3 ⊢ ((𝑄‘𝐼)[,](𝑄‘(𝐼 + 1))) ⊆ ℝ* | |
| 2 | fourierdlem1.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ ((𝑄‘𝐼)[,](𝑄‘(𝐼 + 1)))) | |
| 3 | 1, 2 | sselid 3981 | . 2 ⊢ (𝜑 → 𝑋 ∈ ℝ*) | 
| 4 | fourierdlem1.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
| 5 | iccssxr 13470 | . . . 4 ⊢ (𝐴[,]𝐵) ⊆ ℝ* | |
| 6 | fourierdlem1.q | . . . . 5 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵)) | |
| 7 | fourierdlem1.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (0..^𝑀)) | |
| 8 | elfzofz 13715 | . . . . . 6 ⊢ (𝐼 ∈ (0..^𝑀) → 𝐼 ∈ (0...𝑀)) | |
| 9 | 7, 8 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (0...𝑀)) | 
| 10 | 6, 9 | ffvelcdmd 7105 | . . . 4 ⊢ (𝜑 → (𝑄‘𝐼) ∈ (𝐴[,]𝐵)) | 
| 11 | 5, 10 | sselid 3981 | . . 3 ⊢ (𝜑 → (𝑄‘𝐼) ∈ ℝ*) | 
| 12 | fourierdlem1.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
| 13 | iccgelb 13443 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ (𝑄‘𝐼) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (𝑄‘𝐼)) | |
| 14 | 4, 12, 10, 13 | syl3anc 1373 | . . 3 ⊢ (𝜑 → 𝐴 ≤ (𝑄‘𝐼)) | 
| 15 | fzofzp1 13803 | . . . . . . . . 9 ⊢ (𝐼 ∈ (0..^𝑀) → (𝐼 + 1) ∈ (0...𝑀)) | |
| 16 | 7, 15 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐼 + 1) ∈ (0...𝑀)) | 
| 17 | 6, 16 | ffvelcdmd 7105 | . . . . . . 7 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ (𝐴[,]𝐵)) | 
| 18 | 5, 17 | sselid 3981 | . . . . . 6 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ ℝ*) | 
| 19 | elicc4 13454 | . . . . . 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 1373 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ ((𝑄‘𝐼)[,](𝑄‘(𝐼 + 1))) ↔ ((𝑄‘𝐼) ≤ 𝑋 ∧ 𝑋 ≤ (𝑄‘(𝐼 + 1))))) | 
| 21 | 2, 20 | mpbid 232 | . . . 4 ⊢ (𝜑 → ((𝑄‘𝐼) ≤ 𝑋 ∧ 𝑋 ≤ (𝑄‘(𝐼 + 1)))) | 
| 22 | 21 | simpld 494 | . . 3 ⊢ (𝜑 → (𝑄‘𝐼) ≤ 𝑋) | 
| 23 | 4, 11, 3, 14, 22 | xrletrd 13204 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝑋) | 
| 24 | iccleub 13442 | . . . 4 ⊢ (((𝑄‘𝐼) ∈ ℝ* ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ* ∧ 𝑋 ∈ ((𝑄‘𝐼)[,](𝑄‘(𝐼 + 1)))) → 𝑋 ≤ (𝑄‘(𝐼 + 1))) | |
| 25 | 11, 18, 2, 24 | syl3anc 1373 | . . 3 ⊢ (𝜑 → 𝑋 ≤ (𝑄‘(𝐼 + 1))) | 
| 26 | elicc4 13454 | . . . . . 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 1373 | . . . . 5 ⊢ (𝜑 → ((𝑄‘(𝐼 + 1)) ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ (𝑄‘(𝐼 + 1)) ∧ (𝑄‘(𝐼 + 1)) ≤ 𝐵))) | 
| 28 | 17, 27 | mpbid 232 | . . . 4 ⊢ (𝜑 → (𝐴 ≤ (𝑄‘(𝐼 + 1)) ∧ (𝑄‘(𝐼 + 1)) ≤ 𝐵)) | 
| 29 | 28 | simprd 495 | . . 3 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ≤ 𝐵) | 
| 30 | 3, 18, 12, 25, 29 | xrletrd 13204 | . 2 ⊢ (𝜑 → 𝑋 ≤ 𝐵) | 
| 31 | elicc1 13431 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 ∈ ℝ* ∧ 𝐴 ≤ 𝑋 ∧ 𝑋 ≤ 𝐵))) | |
| 32 | 4, 12, 31 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 ∈ ℝ* ∧ 𝐴 ≤ 𝑋 ∧ 𝑋 ≤ 𝐵))) | 
| 33 | 3, 23, 30, 32 | mpbir3and 1343 | 1 ⊢ (𝜑 → 𝑋 ∈ (𝐴[,]𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2108 class class class wbr 5143 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 ℝ*cxr 11294 ≤ cle 11296 [,]cicc 13390 ...cfz 13547 ..^cfzo 13694 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-icc 13394 df-fz 13548 df-fzo 13695 | 
| This theorem is referenced by: fourierdlem8 46130 fourierdlem73 46194 fourierdlem81 46202 fourierdlem92 46213 fourierdlem93 46214 | 
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