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Theorem xrltmin 13199

Description: Two ways of saying an extended real is less than the minimum of two others. (Contributed by NM, 7-Feb-2007.)

**Assertion**
Ref	Expression
xrltmin	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ↔ (𝐴 < 𝐵 ∧ 𝐴 < 𝐶)))

**Proof of Theorem xrltmin**
Step	Hyp	Ref	Expression
1		xrmin1 13194	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐵)
2	1	3adant1 1146	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐵)
3		simp1 1152	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
4		ifcl 4529	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∈ ℝ^)
5	4	3adant1 1146	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∈ ℝ^)
6		simp2 1153	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐵 ∈ ℝ^)
7		xrltletr 13173	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∧ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐵) → 𝐴 < 𝐵))
8	3, 5, 6, 7	syl3anc 1394	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∧ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐵) → 𝐴 < 𝐵))
9	2, 8	mpan2d 706	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → 𝐴 < 𝐵))
10		xrmin2 13195	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐶)
11	10	3adant1 1146	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐶)
12		xrltletr 13173	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∧ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐶) → 𝐴 < 𝐶))
13	5, 12	syld3an2 1434	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∧ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐶) → 𝐴 < 𝐶))
14	11, 13	mpan2d 706	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → 𝐴 < 𝐶))
15	9, 14	jcad 521	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → (𝐴 < 𝐵 ∧ 𝐴 < 𝐶)))
16		breq2 5109	. . 3 ⊢ (𝐵 = if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → (𝐴 < 𝐵 ↔ 𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶)))
17		breq2 5109	. . 3 ⊢ (𝐶 = if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → (𝐴 < 𝐶 ↔ 𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶)))
18	16, 17	ifboth 4523	. 2 ⊢ ((𝐴 < 𝐵 ∧ 𝐴 < 𝐶) → 𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶))
19	15, 18	impbid1 228	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ↔ (𝐴 < 𝐵 ∧ 𝐴 < 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 ∈ wcel 2145 ifcif 4483 class class class wbr 5105 ℝ^*cxr 11230 < clt 11231 ≤ cle 11232

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-pre-lttri 11162 ax-pre-lttrn 11163

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-br 5106 df-opab 5168 df-mpt 5187 df-id 5547 df-po 5560 df-so 5561 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-er 8682 df-en 8932 df-dom 8933 df-sdom 8934 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237

This theorem is referenced by: ltmin 13211 iooin 13397 blin 24539 lhop1 26134 ioondisj1 46068

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