Theorem xrlemin 13203

Description: Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by Mario Carneiro, 18-Jun-2014.)

Assertion

Ref	Expression
xrlemin	⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ↔ (𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶)))

Proof of Theorem xrlemin

Step	Hyp	Ref	Expression
1		xrmin1 13196	. . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐵)
2	1	3adant1 1127	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐵)
3		simp1 1133	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → 𝐴 ∈ ℝ*)
4		ifcl 4577	. . . . . 6 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∈ ℝ*)
5	4	3adant1 1127	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ∈ ℝ*)
6		simp2 1134	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → 𝐵 ∈ ℝ*)
7		xrletr 13177	. . . . 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 1368	. . . 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 692	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → 𝐴 ≤ 𝐵))
10		xrmin2 13197	. . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐶)
11	10	3adant1 1127	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ≤ 𝐶)
12		xrletr 13177	. . . . 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 1408	. . . 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 692	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → 𝐴 ≤ 𝐶))
15	9, 14	jcad 511	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → (𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶)))
16		breq2 5156	. . 3 ⊢ (𝐵 = if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → (𝐴 ≤ 𝐵 ↔ 𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶)))
17		breq2 5156	. . 3 ⊢ (𝐶 = if(𝐵 ≤ 𝐶, 𝐵, 𝐶) → (𝐴 ≤ 𝐶 ↔ 𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶)))
18	16, 17	ifboth 4571	. 2 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶) → 𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶))
19	15, 18	impbid1 224	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ↔ (𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   ∈ wcel 2098  ifcif 4532   class class class wbr 5152  ℝ^*cxr 11285   ≤ cle 11287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-pre-lttri 11220  ax-pre-lttrn 11221

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-po 5594  df-so 5595  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292

This theorem is referenced by:  lemin  13211  stdbdxmet  24444  stdbdbl  24446  itgspliticc  25786  cvmliftlem10  34937  iccin  47993