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Theorem xrltmin 13148

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 13143	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐵)
2	1	3adant1 1130	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐵)
3		simp1 1136	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
4		ifcl 4536	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∈ ℝ^)
5	4	3adant1 1130	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∈ ℝ^)
6		simp2 1137	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐵 ∈ ℝ^)
7		xrltletr 13123	. . . . 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 1373	. . . 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 694	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → 𝐴 < 𝐵))
10		xrmin2 13144	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐶)
11	10	3adant1 1130	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐶)
12		xrltletr 13123	. . . . 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 1413	. . . 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 694	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → 𝐴 < 𝐶))
15	9, 14	jcad 512	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → (𝐴 < 𝐵 ∧ 𝐴 < 𝐶)))
16		breq2 5113	. . 3 ⊢ (𝐵 = if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → (𝐴 < 𝐵 ↔ 𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶)))
17		breq2 5113	. . 3 ⊢ (𝐶 = if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → (𝐴 < 𝐶 ↔ 𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶)))
18	16, 17	ifboth 4530	. 2 ⊢ ((𝐴 < 𝐵 ∧ 𝐴 < 𝐶) → 𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶))
19	15, 18	impbid1 225	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 1086 ∈ wcel 2109 ifcif 4490 class class class wbr 5109 ℝ^*cxr 11213 < clt 11214 ≤ cle 11215

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 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-resscn 11131 ax-pre-lttri 11148 ax-pre-lttrn 11149

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-br 5110 df-opab 5172 df-mpt 5191 df-id 5535 df-po 5548 df-so 5549 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-er 8673 df-en 8921 df-dom 8922 df-sdom 8923 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220

This theorem is referenced by: ltmin 13160 iooin 13346 blin 24315 lhop1 25925 ioondisj1 45485

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