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Theorem xrmaxlt 12915

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 12909	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
2	1	3adant3 1131	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
3		ifcl 4504	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ∈ ℝ^)
4	3	ancoms 459	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ∈ ℝ^)
5	4	3adant3 1131	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ∈ ℝ^)
6		xrlelttr 12890	. . . . 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 1410	. . . 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 692	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (if(𝐴 ≤ 𝐵, 𝐵, 𝐴) < 𝐶 → 𝐴 < 𝐶))
9		xrmax2 12910	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
10	9	3adant3 1131	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴))
11		simp2 1136	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐵 ∈ ℝ^)
12		simp3 1137	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐶 ∈ ℝ^)
13		xrlelttr 12890	. . . . 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 1370	. . . 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 692	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (if(𝐴 ≤ 𝐵, 𝐵, 𝐴) < 𝐶 → 𝐵 < 𝐶))
16	8, 15	jcad 513	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (if(𝐴 ≤ 𝐵, 𝐵, 𝐴) < 𝐶 → (𝐴 < 𝐶 ∧ 𝐵 < 𝐶)))
17		breq1 5077	. . . 4 ⊢ (𝐵 = if(𝐴 ≤ 𝐵, 𝐵, 𝐴) → (𝐵 < 𝐶 ↔ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) < 𝐶))
18		breq1 5077	. . . 4 ⊢ (𝐴 = if(𝐴 ≤ 𝐵, 𝐵, 𝐴) → (𝐴 < 𝐶 ↔ if(𝐴 ≤ 𝐵, 𝐵, 𝐴) < 𝐶))
19	17, 18	ifboth 4498	. . 3 ⊢ ((𝐵 < 𝐶 ∧ 𝐴 < 𝐶) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) < 𝐶)
20	19	ancoms 459	. 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 396 ∧ w3a 1086 ∈ wcel 2106 ifcif 4459 class class class wbr 5074 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-pre-lttri 10945 ax-pre-lttrn 10946

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015

This theorem is referenced by: maxlt 12927 iooin 13113 txmetcnp 23703 mbfmax 24813 dvlip2 25159 ply1divmo 25300

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