fourierdlem1 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem1 46106

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 13391	. . 3 ⊢ ((𝑄‘𝐼)[,](𝑄‘(𝐼 + 1))) ⊆ ℝ^*
2		fourierdlem1.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((𝑄‘𝐼)[,](𝑄‘(𝐼 + 1))))
3	1, 2	sselid 3944	. 2 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
4		fourierdlem1.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
5		iccssxr 13391	. . . 4 ⊢ (𝐴[,]𝐵) ⊆ ℝ^*
6		fourierdlem1.q	. . . . 5 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
7		fourierdlem1.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (0..^𝑀))
8		elfzofz 13636	. . . . . 6 ⊢ (𝐼 ∈ (0..^𝑀) → 𝐼 ∈ (0...𝑀))
9	7, 8	syl 17	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (0...𝑀))
10	6, 9	ffvelcdmd 7057	. . . 4 ⊢ (𝜑 → (𝑄‘𝐼) ∈ (𝐴[,]𝐵))
11	5, 10	sselid 3944	. . 3 ⊢ (𝜑 → (𝑄‘𝐼) ∈ ℝ^*)
12		fourierdlem1.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
13		iccgelb 13363	. . . 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 13725	. . . . . . . . 9 ⊢ (𝐼 ∈ (0..^𝑀) → (𝐼 + 1) ∈ (0...𝑀))
16	7, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐼 + 1) ∈ (0...𝑀))
17	6, 16	ffvelcdmd 7057	. . . . . . 7 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ (𝐴[,]𝐵))
18	5, 17	sselid 3944	. . . . . 6 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ ℝ^*)
19		elicc4 13374	. . . . . 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 13122	. 2 ⊢ (𝜑 → 𝐴 ≤ 𝑋)
24		iccleub 13362	. . . 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 13374	. . . . . 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 13122	. 2 ⊢ (𝜑 → 𝑋 ≤ 𝐵)
31		elicc1 13350	. . 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 5107 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 0cc0 11068 1c1 11069 + caddc 11071 ℝ^*cxr 11207 ≤ cle 11209 [,]cicc 13309 ...cfz 13468 ..^cfzo 13615

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 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

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 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-n0 12443 df-z 12530 df-uz 12794 df-icc 13313 df-fz 13469 df-fzo 13616

This theorem is referenced by: fourierdlem8 46113 fourierdlem73 46177 fourierdlem81 46185 fourierdlem92 46196 fourierdlem93 46197

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