ioossioobi - Mathbox for Glauco Siliprandi

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Theorem ioossioobi 41799

Description: Biconditional form of ioossioo 12832. (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 487	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
2		df-ioo 12745	. . . . . 6 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
3	2	ixxssxr 12753	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ ℝ^*
4		infxrss 12735	. . . . 5 ⊢ (((𝐶(,)𝐷) ⊆ (𝐴(,)𝐵) ∧ (𝐴(,)𝐵) ⊆ ℝ^) → inf((𝐴(,)𝐵), ℝ^, < ) ≤ inf((𝐶(,)𝐷), ℝ^*, < ))
5	1, 3, 4	sylancl 588	. . . 4 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → inf((𝐴(,)𝐵), ℝ^, < ) ≤ inf((𝐶(,)𝐷), ℝ^, < ))
6		ioossioobi.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
7	6	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ^*)
8		ioossioobi.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
9	8	adantr 483	. . . . 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 12767	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^*) → ((𝐶(,)𝐷) ≠ ∅ ↔ 𝐶 < 𝐷))
14	11, 12, 13	syl2anc 586	. . . . . . . 8 ⊢ (𝜑 → ((𝐶(,)𝐷) ≠ ∅ ↔ 𝐶 < 𝐷))
15	10, 14	mpbird 259	. . . . . . 7 ⊢ (𝜑 → (𝐶(,)𝐷) ≠ ∅)
16	15	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → (𝐶(,)𝐷) ≠ ∅)
17		ssn0 4357	. . . . . 6 ⊢ (((𝐶(,)𝐷) ⊆ (𝐴(,)𝐵) ∧ (𝐶(,)𝐷) ≠ ∅) → (𝐴(,)𝐵) ≠ ∅)
18	1, 16, 17	syl2anc 586	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ≠ ∅)
19		idd 24	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤 < 𝐵 → 𝑤 < 𝐵))
20		xrltle 12545	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤 < 𝐵 → 𝑤 ≤ 𝐵))
21		idd 24	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴 < 𝑤 → 𝐴 < 𝑤))
22		xrltle 12545	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴 < 𝑤 → 𝐴 ≤ 𝑤))
23	2, 19, 20, 21, 22	ixxlb 12763	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴(,)𝐵) ≠ ∅) → inf((𝐴(,)𝐵), ℝ^*, < ) = 𝐴)
24	7, 9, 18, 23	syl3anc 1367	. . . 4 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → inf((𝐴(,)𝐵), ℝ^*, < ) = 𝐴)
25	11	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → 𝐶 ∈ ℝ^*)
26	12	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → 𝐷 ∈ ℝ^*)
27		idd 24	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^*) → (𝑤 < 𝐷 → 𝑤 < 𝐷))
28		xrltle 12545	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^*) → (𝑤 < 𝐷 → 𝑤 ≤ 𝐷))
29		idd 24	. . . . . 6 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐶 < 𝑤 → 𝐶 < 𝑤))
30		xrltle 12545	. . . . . 6 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐶 < 𝑤 → 𝐶 ≤ 𝑤))
31	2, 27, 28, 29, 30	ixxlb 12763	. . . . 5 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ (𝐶(,)𝐷) ≠ ∅) → inf((𝐶(,)𝐷), ℝ^*, < ) = 𝐶)
32	25, 26, 16, 31	syl3anc 1367	. . . 4 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → inf((𝐶(,)𝐷), ℝ^*, < ) = 𝐶)
33	5, 24, 32	3brtr3d 5100	. . 3 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → 𝐴 ≤ 𝐶)
34		supxrss 12728	. . . . 5 ⊢ (((𝐶(,)𝐷) ⊆ (𝐴(,)𝐵) ∧ (𝐴(,)𝐵) ⊆ ℝ^) → sup((𝐶(,)𝐷), ℝ^, < ) ≤ sup((𝐴(,)𝐵), ℝ^*, < ))
35	1, 3, 34	sylancl 588	. . . 4 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → sup((𝐶(,)𝐷), ℝ^, < ) ≤ sup((𝐴(,)𝐵), ℝ^, < ))
36	2, 27, 28, 29, 30	ixxub 12762	. . . . 5 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ (𝐶(,)𝐷) ≠ ∅) → sup((𝐶(,)𝐷), ℝ^*, < ) = 𝐷)
37	25, 26, 16, 36	syl3anc 1367	. . . 4 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → sup((𝐶(,)𝐷), ℝ^*, < ) = 𝐷)
38	2, 19, 20, 21, 22	ixxub 12762	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴(,)𝐵) ≠ ∅) → sup((𝐴(,)𝐵), ℝ^*, < ) = 𝐵)
39	7, 9, 18, 38	syl3anc 1367	. . . 4 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → sup((𝐴(,)𝐵), ℝ^*, < ) = 𝐵)
40	35, 37, 39	3brtr3d 5100	. . 3 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → 𝐷 ≤ 𝐵)
41	33, 40	jca 514	. 2 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵))
42	6	adantr 483	. . 3 ⊢ ((𝜑 ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → 𝐴 ∈ ℝ^*)
43	8	adantr 483	. . 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 12832	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
47	42, 43, 44, 45, 46	syl22anc 836	. 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 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ⊆ wss 3939 ∅c0 4294 class class class wbr 5069 (class class class)co 7159 supcsup 8907 infcinf 8908 ℝ^*cxr 10677 < clt 10678 ≤ cle 10679 (,)cioo 12741

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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-sup 8909 df-inf 8910 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-ioo 12745

This theorem is referenced by: fourierdlem50 42448

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