xrmaxeq - Metamath Proof Explorer

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Theorem xrmaxeq 13006

Description: The maximum of two extended reals is equal to the first if the first is bigger. (Contributed by Mario Carneiro, 25-Mar-2015.)

**Assertion**
Ref	Expression
xrmaxeq	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ≤ 𝐴) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) = 𝐴)

**Proof of Theorem xrmaxeq**
Step	Hyp	Ref	Expression
1		xrletri3 12981	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 = 𝐴 ↔ (𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)))
2	1	ancoms 459	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 = 𝐴 ↔ (𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)))
3	2	biimpar 478	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐵 = 𝐴)
4	3	anassrs 468	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 ≤ 𝐴) ∧ 𝐴 ≤ 𝐵) → 𝐵 = 𝐴)
5	4	ifeq1da 4503	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 ≤ 𝐴) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) = if(𝐴 ≤ 𝐵, 𝐴, 𝐴))
6	5	3impa 1109	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ≤ 𝐴) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) = if(𝐴 ≤ 𝐵, 𝐴, 𝐴))
7		ifid 4512	. 2 ⊢ if(𝐴 ≤ 𝐵, 𝐴, 𝐴) = 𝐴
8	6, 7	eqtrdi 2792	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ≤ 𝐴) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ifcif 4472 class class class wbr 5089 ℝ^*cxr 11101 ≤ cle 11103

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-cnex 11020 ax-resscn 11021 ax-pre-lttri 11038 ax-pre-lttrn 11039

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4852 df-br 5090 df-opab 5152 df-mpt 5173 df-id 5512 df-po 5526 df-so 5527 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-er 8561 df-en 8797 df-dom 8798 df-sdom 8799 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108

This theorem is referenced by: max0sub 13023