Theorem somincom 5961

Description: Commutativity of minimum in a total order. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Assertion

Ref	Expression
somincom	⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵) = if(𝐵𝑅𝐴, 𝐵, 𝐴))

Proof of Theorem somincom

Step	Hyp	Ref	Expression
1		so2nr 5463	. . . . 5 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ¬ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴))
2		nan 828	. . . . 5 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ¬ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴)) ↔ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ 𝐴𝑅𝐵) → ¬ 𝐵𝑅𝐴))
3	1, 2	mpbi 233	. . . 4 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ 𝐴𝑅𝐵) → ¬ 𝐵𝑅𝐴)
4	3	iffalsed 4436	. . 3 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ 𝐴𝑅𝐵) → if(𝐵𝑅𝐴, 𝐵, 𝐴) = 𝐴)
5	4	eqcomd 2804	. 2 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ 𝐴𝑅𝐵) → 𝐴 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
6		sotric 5465	. . . . 5 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵𝑅𝐴)))
7	6	con2bid 358	. . . 4 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴 = 𝐵 ∨ 𝐵𝑅𝐴) ↔ ¬ 𝐴𝑅𝐵))
8		ifeq2 4430	. . . . . 6 ⊢ (𝐴 = 𝐵 → if(𝐵𝑅𝐴, 𝐵, 𝐴) = if(𝐵𝑅𝐴, 𝐵, 𝐵))
9		ifid 4464	. . . . . 6 ⊢ if(𝐵𝑅𝐴, 𝐵, 𝐵) = 𝐵
10	8, 9	eqtr2di 2850	. . . . 5 ⊢ (𝐴 = 𝐵 → 𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
11		iftrue 4431	. . . . . 6 ⊢ (𝐵𝑅𝐴 → if(𝐵𝑅𝐴, 𝐵, 𝐴) = 𝐵)
12	11	eqcomd 2804	. . . . 5 ⊢ (𝐵𝑅𝐴 → 𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
13	10, 12	jaoi 854	. . . 4 ⊢ ((𝐴 = 𝐵 ∨ 𝐵𝑅𝐴) → 𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
14	7, 13	syl6bir 257	. . 3 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (¬ 𝐴𝑅𝐵 → 𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴)))
15	14	imp 410	. 2 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ ¬ 𝐴𝑅𝐵) → 𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
16	5, 15	ifeqda 4460	1 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵) = if(𝐵𝑅𝐴, 𝐵, 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  ifcif 4425   class class class wbr 5030   Or wor 5437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rab 3115  df-v 3443  df-un 3886  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-po 5438  df-so 5439

This theorem is referenced by:  somin2  5962