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Theorem xrltmin 13171

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 13166	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐵)
2	1	3adant1 1139	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐵)
3		simp1 1145	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
4		ifcl 4516	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∈ ℝ^)
5	4	3adant1 1139	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∈ ℝ^)
6		simp2 1146	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐵 ∈ ℝ^)
7		xrltletr 13145	. . . . 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 1382	. . . 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 702	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → 𝐴 < 𝐵))
10		xrmin2 13167	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐶)
11	10	3adant1 1139	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐶)
12		xrltletr 13145	. . . . 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 1422	. . . 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 702	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → 𝐴 < 𝐶))
15	9, 14	jcad 519	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → (𝐴 < 𝐵 ∧ 𝐴 < 𝐶)))
16		breq2 5094	. . 3 ⊢ (𝐵 = if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → (𝐴 < 𝐵 ↔ 𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶)))
17		breq2 5094	. . 3 ⊢ (𝐶 = if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → (𝐴 < 𝐶 ↔ 𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶)))
18	16, 17	ifboth 4510	. 2 ⊢ ((𝐴 < 𝐵 ∧ 𝐴 < 𝐶) → 𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶))
19	15, 18	impbid1 227	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 208 ∧ wa 398 ∧ w3a 1095 ∈ wcel 2132 ifcif 4470 class class class wbr 5090 ℝ^*cxr 11201 < clt 11202 ≤ cle 11203

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-sep 5236 ax-nul 5246 ax-pow 5312 ax-pr 5380 ax-un 7703 ax-cnex 11115 ax-resscn 11116 ax-pre-lttri 11133 ax-pre-lttrn 11134

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-nel 3052 df-ral 3067 df-rex 3077 df-rab 3405 df-v 3446 df-sbc 3736 df-csb 3844 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-nul 4277 df-if 4471 df-pw 4547 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4856 df-br 5091 df-opab 5153 df-mpt 5172 df-id 5531 df-po 5544 df-so 5545 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-res 5648 df-ima 5649 df-iota 6462 df-fun 6508 df-fn 6509 df-f 6510 df-f1 6511 df-fo 6512 df-f1o 6513 df-fv 6514 df-er 8662 df-en 8913 df-dom 8914 df-sdom 8915 df-pnf 11204 df-mnf 11205 df-xr 11206 df-ltxr 11207 df-le 11208

This theorem is referenced by: ltmin 13183 iooin 13369 blin 24450 lhop1 26045 ioondisj1 46008

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