ioossioobi - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > ioossioobi		Structured version Visualization version GIF version

Theorem ioossioobi 44309

Description: Biconditional form of ioossioo 13420. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
ioossioobi.a	⊢ (𝜑 → 𝐴 ∈ ℝ^*)
ioossioobi.b	⊢ (𝜑 → 𝐵 ∈ ℝ^*)
ioossioobi.c	⊢ (𝜑 → 𝐶 ∈ ℝ^*)
ioossioobi.d	⊢ (𝜑 → 𝐷 ∈ ℝ^*)
ioossioobi.cltd	⊢ (𝜑 → 𝐶 < 𝐷)

**Assertion**
Ref	Expression
ioossioobi	⊢ (𝜑 → ((𝐶(,)𝐷) ⊆ (𝐴(,)𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)))

Proof of Theorem ioossioobi Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
2		df-ioo 13330	. . . . . 6 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
3	2	ixxssxr 13338	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ ℝ^*
4		infxrss 13320	. . . . 5 ⊢ (((𝐶(,)𝐷) ⊆ (𝐴(,)𝐵) ∧ (𝐴(,)𝐵) ⊆ ℝ^) → inf((𝐴(,)𝐵), ℝ^, < ) ≤ inf((𝐶(,)𝐷), ℝ^*, < ))
5	1, 3, 4	sylancl 586	. . . 4 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → inf((𝐴(,)𝐵), ℝ^, < ) ≤ inf((𝐶(,)𝐷), ℝ^, < ))
6		ioossioobi.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
7	6	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ^*)
8		ioossioobi.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
9	8	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ^*)
10		ioossioobi.cltd	. . . . . . . 8 ⊢ (𝜑 → 𝐶 < 𝐷)
11		ioossioobi.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
12		ioossioobi.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ ℝ^*)
13		ioon0 13352	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^*) → ((𝐶(,)𝐷) ≠ ∅ ↔ 𝐶 < 𝐷))
14	11, 12, 13	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ((𝐶(,)𝐷) ≠ ∅ ↔ 𝐶 < 𝐷))
15	10, 14	mpbird 256	. . . . . . 7 ⊢ (𝜑 → (𝐶(,)𝐷) ≠ ∅)
16	15	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → (𝐶(,)𝐷) ≠ ∅)
17		ssn0 4400	. . . . . 6 ⊢ (((𝐶(,)𝐷) ⊆ (𝐴(,)𝐵) ∧ (𝐶(,)𝐷) ≠ ∅) → (𝐴(,)𝐵) ≠ ∅)
18	1, 16, 17	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ≠ ∅)
19		idd 24	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤 < 𝐵 → 𝑤 < 𝐵))
20		xrltle 13130	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤 < 𝐵 → 𝑤 ≤ 𝐵))
21		idd 24	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴 < 𝑤 → 𝐴 < 𝑤))
22		xrltle 13130	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴 < 𝑤 → 𝐴 ≤ 𝑤))
23	2, 19, 20, 21, 22	ixxlb 13348	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴(,)𝐵) ≠ ∅) → inf((𝐴(,)𝐵), ℝ^*, < ) = 𝐴)
24	7, 9, 18, 23	syl3anc 1371	. . . 4 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → inf((𝐴(,)𝐵), ℝ^*, < ) = 𝐴)
25	11	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → 𝐶 ∈ ℝ^*)
26	12	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → 𝐷 ∈ ℝ^*)
27		idd 24	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^*) → (𝑤 < 𝐷 → 𝑤 < 𝐷))
28		xrltle 13130	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^*) → (𝑤 < 𝐷 → 𝑤 ≤ 𝐷))
29		idd 24	. . . . . 6 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐶 < 𝑤 → 𝐶 < 𝑤))
30		xrltle 13130	. . . . . 6 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐶 < 𝑤 → 𝐶 ≤ 𝑤))
31	2, 27, 28, 29, 30	ixxlb 13348	. . . . 5 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ (𝐶(,)𝐷) ≠ ∅) → inf((𝐶(,)𝐷), ℝ^*, < ) = 𝐶)
32	25, 26, 16, 31	syl3anc 1371	. . . 4 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → inf((𝐶(,)𝐷), ℝ^*, < ) = 𝐶)
33	5, 24, 32	3brtr3d 5179	. . 3 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → 𝐴 ≤ 𝐶)
34		supxrss 13313	. . . . 5 ⊢ (((𝐶(,)𝐷) ⊆ (𝐴(,)𝐵) ∧ (𝐴(,)𝐵) ⊆ ℝ^) → sup((𝐶(,)𝐷), ℝ^, < ) ≤ sup((𝐴(,)𝐵), ℝ^*, < ))
35	1, 3, 34	sylancl 586	. . . 4 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → sup((𝐶(,)𝐷), ℝ^, < ) ≤ sup((𝐴(,)𝐵), ℝ^, < ))
36	2, 27, 28, 29, 30	ixxub 13347	. . . . 5 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ (𝐶(,)𝐷) ≠ ∅) → sup((𝐶(,)𝐷), ℝ^*, < ) = 𝐷)
37	25, 26, 16, 36	syl3anc 1371	. . . 4 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → sup((𝐶(,)𝐷), ℝ^*, < ) = 𝐷)
38	2, 19, 20, 21, 22	ixxub 13347	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴(,)𝐵) ≠ ∅) → sup((𝐴(,)𝐵), ℝ^*, < ) = 𝐵)
39	7, 9, 18, 38	syl3anc 1371	. . . 4 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → sup((𝐴(,)𝐵), ℝ^*, < ) = 𝐵)
40	35, 37, 39	3brtr3d 5179	. . 3 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → 𝐷 ≤ 𝐵)
41	33, 40	jca 512	. 2 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵))
42	6	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → 𝐴 ∈ ℝ^*)
43	8	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → 𝐵 ∈ ℝ^*)
44		simprl 769	. . 3 ⊢ ((𝜑 ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → 𝐴 ≤ 𝐶)
45		simprr 771	. . 3 ⊢ ((𝜑 ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → 𝐷 ≤ 𝐵)
46		ioossioo 13420	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
47	42, 43, 44, 45, 46	syl22anc 837	. 2 ⊢ ((𝜑 ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
48	41, 47	impbida 799	1 ⊢ (𝜑 → ((𝐶(,)𝐷) ⊆ (𝐴(,)𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ⊆ wss 3948 ∅c0 4322 class class class wbr 5148 (class class class)co 7411 supcsup 9437 infcinf 9438 ℝ^*cxr 11249 < clt 11250 ≤ cle 11251 (,)cioo 13326

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-n0 12475 df-z 12561 df-uz 12825 df-q 12935 df-ioo 13330

This theorem is referenced by: fourierdlem50 44951

W3C validator