fourierdlem1 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem1 46554

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 13374	. . 3 ⊢ ((𝑄‘𝐼)[,](𝑄‘(𝐼 + 1))) ⊆ ℝ^*
2		fourierdlem1.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((𝑄‘𝐼)[,](𝑄‘(𝐼 + 1))))
3	1, 2	sselid 3920	. 2 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
4		fourierdlem1.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
5		iccssxr 13374	. . . 4 ⊢ (𝐴[,]𝐵) ⊆ ℝ^*
6		fourierdlem1.q	. . . . 5 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
7		fourierdlem1.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (0..^𝑀))
8		elfzofz 13621	. . . . . 6 ⊢ (𝐼 ∈ (0..^𝑀) → 𝐼 ∈ (0...𝑀))
9	7, 8	syl 17	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (0...𝑀))
10	6, 9	ffvelcdmd 7031	. . . 4 ⊢ (𝜑 → (𝑄‘𝐼) ∈ (𝐴[,]𝐵))
11	5, 10	sselid 3920	. . 3 ⊢ (𝜑 → (𝑄‘𝐼) ∈ ℝ^*)
12		fourierdlem1.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
13		iccgelb 13346	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝑄‘𝐼) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (𝑄‘𝐼))
14	4, 12, 10, 13	syl3anc 1374	. . 3 ⊢ (𝜑 → 𝐴 ≤ (𝑄‘𝐼))
15		fzofzp1 13710	. . . . . . . . 9 ⊢ (𝐼 ∈ (0..^𝑀) → (𝐼 + 1) ∈ (0...𝑀))
16	7, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐼 + 1) ∈ (0...𝑀))
17	6, 16	ffvelcdmd 7031	. . . . . . 7 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ (𝐴[,]𝐵))
18	5, 17	sselid 3920	. . . . . 6 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ ℝ^*)
19		elicc4 13357	. . . . . 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 1374	. . . . 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 13104	. 2 ⊢ (𝜑 → 𝐴 ≤ 𝑋)
24		iccleub 13345	. . . 4 ⊢ (((𝑄‘𝐼) ∈ ℝ^* ∧ (𝑄‘(𝐼 + 1)) ∈ ℝ^* ∧ 𝑋 ∈ ((𝑄‘𝐼)[,](𝑄‘(𝐼 + 1)))) → 𝑋 ≤ (𝑄‘(𝐼 + 1)))
25	11, 18, 2, 24	syl3anc 1374	. . 3 ⊢ (𝜑 → 𝑋 ≤ (𝑄‘(𝐼 + 1)))
26		elicc4 13357	. . . . . 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 1374	. . . . 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 13104	. 2 ⊢ (𝜑 → 𝑋 ≤ 𝐵)
31		elicc1 13333	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 ∈ ℝ^ ∧ 𝐴 ≤ 𝑋 ∧ 𝑋 ≤ 𝐵)))
32	4, 12, 31	syl2anc 585	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 ∈ ℝ^* ∧ 𝐴 ≤ 𝑋 ∧ 𝑋 ≤ 𝐵)))
33	3, 23, 30, 32	mpbir3and 1344	1 ⊢ (𝜑 → 𝑋 ∈ (𝐴[,]𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2114 class class class wbr 5086 ⟶wf 6488 ‘cfv 6492 (class class class)co 7360 0cc0 11029 1c1 11030 + caddc 11032 ℝ^*cxr 11169 ≤ cle 11171 [,]cicc 13292 ...cfz 13452 ..^cfzo 13599

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-n0 12429 df-z 12516 df-uz 12780 df-icc 13296 df-fz 13453 df-fzo 13600

This theorem is referenced by: fourierdlem8 46561 fourierdlem73 46625 fourierdlem81 46633 fourierdlem92 46644 fourierdlem93 46645

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