ltmin - Metamath Proof Explorer

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Theorem ltmin 13242

Description: Two ways of saying a number is less than the minimum of two others. (Contributed by NM, 1-Sep-2006.)

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

**Proof of Theorem ltmin**
Step	Hyp	Ref	Expression
1		rexr 11314	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
2		rexr 11314	. 2 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
3		rexr 11314	. 2 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ^*)
4		xrltmin 13230	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ↔ (𝐴 < 𝐵 ∧ 𝐴 < 𝐶)))
5	1, 2, 3, 4	syl3an 1161	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 206 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2108 ifcif 4534 class class class wbr 5151 ℝcr 11161 ℝ^*cxr 11301 < clt 11302 ≤ cle 11303

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5305 ax-nul 5315 ax-pow 5374 ax-pr 5441 ax-un 7761 ax-cnex 11218 ax-resscn 11219 ax-pre-lttri 11236 ax-pre-lttrn 11237

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3483 df-sbc 3795 df-csb 3912 df-dif 3969 df-un 3971 df-in 3973 df-ss 3983 df-nul 4343 df-if 4535 df-pw 4610 df-sn 4635 df-pr 4637 df-op 4641 df-uni 4916 df-br 5152 df-opab 5214 df-mpt 5235 df-id 5587 df-po 5601 df-so 5602 df-xp 5699 df-rel 5700 df-cnv 5701 df-co 5702 df-dm 5703 df-rn 5704 df-res 5705 df-ima 5706 df-iota 6522 df-fun 6571 df-fn 6572 df-f 6573 df-f1 6574 df-fo 6575 df-f1o 6576 df-fv 6577 df-er 8753 df-en 8994 df-dom 8995 df-sdom 8996 df-pnf 11304 df-mnf 11305 df-xr 11306 df-ltxr 11307 df-le 11308

This theorem is referenced by: addcnlem 24911 dveflem 26043 logcnlem4 26713 cxpcn3lem 26816 hoidmv1lelem2 46576

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