fourierdlem1 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem1 46079

Description: A partition interval is a subset of the partitioned interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
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))))

**Assertion**
Ref	Expression
fourierdlem1	⊢ (𝜑 → 𝑋 ∈ (𝐴[,]𝐵))

**Proof of Theorem fourierdlem1**
Step	Hyp	Ref	Expression
1		iccssxr 13367	. . 3 ⊢ ((𝑄‘𝐼)[,](𝑄‘(𝐼 + 1))) ⊆ ℝ^*
2		fourierdlem1.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((𝑄‘𝐼)[,](𝑄‘(𝐼 + 1))))
3	1, 2	sselid 3941	. 2 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
4		fourierdlem1.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
5		iccssxr 13367	. . . 4 ⊢ (𝐴[,]𝐵) ⊆ ℝ^*
6		fourierdlem1.q	. . . . 5 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
7		fourierdlem1.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (0..^𝑀))
8		elfzofz 13612	. . . . . 6 ⊢ (𝐼 ∈ (0..^𝑀) → 𝐼 ∈ (0...𝑀))
9	7, 8	syl 17	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (0...𝑀))
10	6, 9	ffvelcdmd 7039	. . . 4 ⊢ (𝜑 → (𝑄‘𝐼) ∈ (𝐴[,]𝐵))
11	5, 10	sselid 3941	. . 3 ⊢ (𝜑 → (𝑄‘𝐼) ∈ ℝ^*)
12		fourierdlem1.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
13		iccgelb 13339	. . . 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 13701	. . . . . . . . 9 ⊢ (𝐼 ∈ (0..^𝑀) → (𝐼 + 1) ∈ (0...𝑀))
16	7, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐼 + 1) ∈ (0...𝑀))
17	6, 16	ffvelcdmd 7039	. . . . . . 7 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ (𝐴[,]𝐵))
18	5, 17	sselid 3941	. . . . . 6 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ ℝ^*)
19		elicc4 13350	. . . . . 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 13098	. 2 ⊢ (𝜑 → 𝐴 ≤ 𝑋)
24		iccleub 13338	. . . 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 13350	. . . . . 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 13098	. 2 ⊢ (𝜑 → 𝑋 ≤ 𝐵)
31		elicc1 13326	. . 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 1086 ∈ wcel 2109 class class class wbr 5102 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 0cc0 11044 1c1 11045 + caddc 11047 ℝ^*cxr 11183 ≤ cle 11185 [,]cicc 13285 ...cfz 13444 ..^cfzo 13591

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-n0 12419 df-z 12506 df-uz 12770 df-icc 13289 df-fz 13445 df-fzo 13592

This theorem is referenced by: fourierdlem8 46086 fourierdlem73 46150 fourierdlem81 46158 fourierdlem92 46169 fourierdlem93 46170

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