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Theorem xrmaxlt 13156

Description: Two ways of saying the maximum of two extended reals is less than a third. (Contributed by NM, 7-Feb-2007.)

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

**Proof of Theorem xrmaxlt**
Step	Hyp	Ref	Expression
1		xrmax1 13150	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
2	1	3adant3 1133	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
3		ifcl 4572	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ∈ ℝ^)
4	3	ancoms 460	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ∈ ℝ^)
5	4	3adant3 1133	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ∈ ℝ^)
6		xrlelttr 13131	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ∧ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) < 𝐶) → 𝐴 < 𝐶))
7	5, 6	syld3an2 1412	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ∧ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) < 𝐶) → 𝐴 < 𝐶))
8	2, 7	mpand 694	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (if(𝐴 ≤ 𝐵, 𝐵, 𝐴) < 𝐶 → 𝐴 < 𝐶))
9		xrmax2 13151	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
10	9	3adant3 1133	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
11		simp2 1138	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐵 ∈ ℝ^)
12		simp3 1139	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐶 ∈ ℝ^)
13		xrlelttr 13131	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ∧ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) < 𝐶) → 𝐵 < 𝐶))
14	11, 5, 12, 13	syl3anc 1372	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ∧ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) < 𝐶) → 𝐵 < 𝐶))
15	10, 14	mpand 694	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (if(𝐴 ≤ 𝐵, 𝐵, 𝐴) < 𝐶 → 𝐵 < 𝐶))
16	8, 15	jcad 514	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (if(𝐴 ≤ 𝐵, 𝐵, 𝐴) < 𝐶 → (𝐴 < 𝐶 ∧ 𝐵 < 𝐶)))
17		breq1 5150	. . . 4 ⊢ (𝐵 = if(𝐴 ≤ 𝐵, 𝐵, 𝐴) → (𝐵 < 𝐶 ↔ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) < 𝐶))
18		breq1 5150	. . . 4 ⊢ (𝐴 = if(𝐴 ≤ 𝐵, 𝐵, 𝐴) → (𝐴 < 𝐶 ↔ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) < 𝐶))
19	17, 18	ifboth 4566	. . 3 ⊢ ((𝐵 < 𝐶 ∧ 𝐴 < 𝐶) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) < 𝐶)
20	19	ancoms 460	. 2 ⊢ ((𝐴 < 𝐶 ∧ 𝐵 < 𝐶) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) < 𝐶)
21	16, 20	impbid1 224	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (if(𝐴 ≤ 𝐵, 𝐵, 𝐴) < 𝐶 ↔ (𝐴 < 𝐶 ∧ 𝐵 < 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 ∈ wcel 2107 ifcif 4527 class class class wbr 5147 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-pre-lttri 11180 ax-pre-lttrn 11181

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250

This theorem is referenced by: maxlt 13168 iooin 13354 txmetcnp 24038 mbfmax 25148 dvlip2 25494 ply1divmo 25635

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