fourierdlem1 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem1 46268

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 13337	. . 3 ⊢ ((𝑄‘𝐼)[,](𝑄‘(𝐼 + 1))) ⊆ ℝ^*
2		fourierdlem1.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((𝑄‘𝐼)[,](𝑄‘(𝐼 + 1))))
3	1, 2	sselid 3928	. 2 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
4		fourierdlem1.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
5		iccssxr 13337	. . . 4 ⊢ (𝐴[,]𝐵) ⊆ ℝ^*
6		fourierdlem1.q	. . . . 5 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
7		fourierdlem1.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (0..^𝑀))
8		elfzofz 13582	. . . . . 6 ⊢ (𝐼 ∈ (0..^𝑀) → 𝐼 ∈ (0...𝑀))
9	7, 8	syl 17	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (0...𝑀))
10	6, 9	ffvelcdmd 7027	. . . 4 ⊢ (𝜑 → (𝑄‘𝐼) ∈ (𝐴[,]𝐵))
11	5, 10	sselid 3928	. . 3 ⊢ (𝜑 → (𝑄‘𝐼) ∈ ℝ^*)
12		fourierdlem1.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
13		iccgelb 13309	. . . 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 13671	. . . . . . . . 9 ⊢ (𝐼 ∈ (0..^𝑀) → (𝐼 + 1) ∈ (0...𝑀))
16	7, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐼 + 1) ∈ (0...𝑀))
17	6, 16	ffvelcdmd 7027	. . . . . . 7 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ (𝐴[,]𝐵))
18	5, 17	sselid 3928	. . . . . 6 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ ℝ^*)
19		elicc4 13320	. . . . . 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 13067	. 2 ⊢ (𝜑 → 𝐴 ≤ 𝑋)
24		iccleub 13308	. . . 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 13320	. . . . . 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 13067	. 2 ⊢ (𝜑 → 𝑋 ≤ 𝐵)
31		elicc1 13296	. . 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 2113 class class class wbr 5095 ⟶wf 6485 ‘cfv 6489 (class class class)co 7355 0cc0 11017 1c1 11018 + caddc 11020 ℝ^*cxr 11156 ≤ cle 11158 [,]cicc 13255 ...cfz 13414 ..^cfzo 13561

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-om 7806 df-1st 7930 df-2nd 7931 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-er 8631 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-nn 12137 df-n0 12393 df-z 12480 df-uz 12743 df-icc 13259 df-fz 13415 df-fzo 13562

This theorem is referenced by: fourierdlem8 46275 fourierdlem73 46339 fourierdlem81 46347 fourierdlem92 46358 fourierdlem93 46359

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