ioossioobi - Mathbox for Glauco Siliprandi

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Theorem ioossioobi 45947

Description: Biconditional form of ioossioo 13394. (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 484	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
2		df-ioo 13302	. . . . . 6 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
3	2	ixxssxr 13310	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ ℝ^*
4		infxrss 13292	. . . . 5 ⊢ (((𝐶(,)𝐷) ⊆ (𝐴(,)𝐵) ∧ (𝐴(,)𝐵) ⊆ ℝ^) → inf((𝐴(,)𝐵), ℝ^, < ) ≤ inf((𝐶(,)𝐷), ℝ^*, < ))
5	1, 3, 4	sylancl 587	. . . 4 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → inf((𝐴(,)𝐵), ℝ^, < ) ≤ inf((𝐶(,)𝐷), ℝ^, < ))
6		ioossioobi.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
7	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ^*)
8		ioossioobi.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
9	8	adantr 480	. . . . 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 13324	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^*) → ((𝐶(,)𝐷) ≠ ∅ ↔ 𝐶 < 𝐷))
14	11, 12, 13	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → ((𝐶(,)𝐷) ≠ ∅ ↔ 𝐶 < 𝐷))
15	10, 14	mpbird 257	. . . . . . 7 ⊢ (𝜑 → (𝐶(,)𝐷) ≠ ∅)
16	15	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → (𝐶(,)𝐷) ≠ ∅)
17		ssn0 4345	. . . . . 6 ⊢ (((𝐶(,)𝐷) ⊆ (𝐴(,)𝐵) ∧ (𝐶(,)𝐷) ≠ ∅) → (𝐴(,)𝐵) ≠ ∅)
18	1, 16, 17	syl2anc 585	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ≠ ∅)
19		idd 24	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤 < 𝐵 → 𝑤 < 𝐵))
20		xrltle 13100	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤 < 𝐵 → 𝑤 ≤ 𝐵))
21		idd 24	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴 < 𝑤 → 𝐴 < 𝑤))
22		xrltle 13100	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴 < 𝑤 → 𝐴 ≤ 𝑤))
23	2, 19, 20, 21, 22	ixxlb 13320	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴(,)𝐵) ≠ ∅) → inf((𝐴(,)𝐵), ℝ^*, < ) = 𝐴)
24	7, 9, 18, 23	syl3anc 1374	. . . 4 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → inf((𝐴(,)𝐵), ℝ^*, < ) = 𝐴)
25	11	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → 𝐶 ∈ ℝ^*)
26	12	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → 𝐷 ∈ ℝ^*)
27		idd 24	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^*) → (𝑤 < 𝐷 → 𝑤 < 𝐷))
28		xrltle 13100	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^*) → (𝑤 < 𝐷 → 𝑤 ≤ 𝐷))
29		idd 24	. . . . . 6 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐶 < 𝑤 → 𝐶 < 𝑤))
30		xrltle 13100	. . . . . 6 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐶 < 𝑤 → 𝐶 ≤ 𝑤))
31	2, 27, 28, 29, 30	ixxlb 13320	. . . . 5 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ (𝐶(,)𝐷) ≠ ∅) → inf((𝐶(,)𝐷), ℝ^*, < ) = 𝐶)
32	25, 26, 16, 31	syl3anc 1374	. . . 4 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → inf((𝐶(,)𝐷), ℝ^*, < ) = 𝐶)
33	5, 24, 32	3brtr3d 5117	. . 3 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → 𝐴 ≤ 𝐶)
34		supxrss 13284	. . . . 5 ⊢ (((𝐶(,)𝐷) ⊆ (𝐴(,)𝐵) ∧ (𝐴(,)𝐵) ⊆ ℝ^) → sup((𝐶(,)𝐷), ℝ^, < ) ≤ sup((𝐴(,)𝐵), ℝ^*, < ))
35	1, 3, 34	sylancl 587	. . . 4 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → sup((𝐶(,)𝐷), ℝ^, < ) ≤ sup((𝐴(,)𝐵), ℝ^, < ))
36	2, 27, 28, 29, 30	ixxub 13319	. . . . 5 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ (𝐶(,)𝐷) ≠ ∅) → sup((𝐶(,)𝐷), ℝ^*, < ) = 𝐷)
37	25, 26, 16, 36	syl3anc 1374	. . . 4 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → sup((𝐶(,)𝐷), ℝ^*, < ) = 𝐷)
38	2, 19, 20, 21, 22	ixxub 13319	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴(,)𝐵) ≠ ∅) → sup((𝐴(,)𝐵), ℝ^*, < ) = 𝐵)
39	7, 9, 18, 38	syl3anc 1374	. . . 4 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → sup((𝐴(,)𝐵), ℝ^*, < ) = 𝐵)
40	35, 37, 39	3brtr3d 5117	. . 3 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → 𝐷 ≤ 𝐵)
41	33, 40	jca 511	. 2 ⊢ ((𝜑 ∧ (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) → (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵))
42	6	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → 𝐴 ∈ ℝ^*)
43	8	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → 𝐵 ∈ ℝ^*)
44		simprl 771	. . 3 ⊢ ((𝜑 ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → 𝐴 ≤ 𝐶)
45		simprr 773	. . 3 ⊢ ((𝜑 ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → 𝐷 ≤ 𝐵)
46		ioossioo 13394	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
47	42, 43, 44, 45, 46	syl22anc 839	. 2 ⊢ ((𝜑 ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))
48	41, 47	impbida 801	1 ⊢ (𝜑 → ((𝐶(,)𝐷) ⊆ (𝐴(,)𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ⊆ wss 3890 ∅c0 4274 class class class wbr 5086 (class class class)co 7367 supcsup 9353 infcinf 9354 ℝ^*cxr 11178 < clt 11179 ≤ cle 11180 (,)cioo 13298

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 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

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-rmo 3343 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 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6266 df-ord 6327 df-on 6328 df-lim 6329 df-suc 6330 df-iota 6455 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-sup 9355 df-inf 9356 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-n0 12438 df-z 12525 df-uz 12789 df-q 12899 df-ioo 13302

This theorem is referenced by: fourierdlem50 46584

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