xrmaxeq - Metamath Proof Explorer

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Theorem xrmaxeq 13207

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 13181	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 = 𝐴 ↔ (𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)))
2	1	ancoms 463	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 = 𝐴 ↔ (𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)))
3	2	biimpar 482	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐵 = 𝐴)
4	3	anassrs 472	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 ≤ 𝐴) ∧ 𝐴 ≤ 𝐵) → 𝐵 = 𝐴)
5	4	ifeq1da 4524	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐵 ≤ 𝐴) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) = if(𝐴 ≤ 𝐵, 𝐴, 𝐴))
6	5	3impa 1125	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ≤ 𝐴) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) = if(𝐴 ≤ 𝐵, 𝐴, 𝐴))
7		ifid 4533	. 2 ⊢ if(𝐴 ≤ 𝐵, 𝐴, 𝐴) = 𝐴
8	6, 7	eqtrdi 2820	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ≤ 𝐴) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ifcif 4492 class class class wbr 5113 ℝ^*cxr 11244 ≤ cle 11246

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-nul 5273 ax-pow 5339 ax-pr 5407 ax-un 7735 ax-cnex 11158 ax-resscn 11159 ax-pre-lttri 11176 ax-pre-lttrn 11177

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-br 5114 df-opab 5178 df-mpt 5197 df-id 5559 df-po 5572 df-so 5573 df-xp 5670 df-rel 5671 df-cnv 5672 df-co 5673 df-dm 5674 df-rn 5675 df-res 5676 df-ima 5677 df-iota 6495 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-er 8696 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251

This theorem is referenced by: max0sub 13224