Theorem somincom 6122

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 5583	. . . . 5 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ¬ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴))
2		nan 843	. . . . 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 4492	. . 3 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ 𝐴𝑅𝐵) → if(𝐵𝑅𝐴, 𝐵, 𝐴) = 𝐴)
5	4	eqcomd 2766	. 2 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ 𝐴𝑅𝐵) → 𝐴 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
6		sotric 5585	. . . . 5 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵𝑅𝐴)))
7	6	con2bid 357	. . . 4 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴 = 𝐵 ∨ 𝐵𝑅𝐴) ↔ ¬ 𝐴𝑅𝐵))
8		ifeq2 4486	. . . . . 6 ⊢ (𝐴 = 𝐵 → if(𝐵𝑅𝐴, 𝐵, 𝐴) = if(𝐵𝑅𝐴, 𝐵, 𝐵))
9		ifid 4522	. . . . . 6 ⊢ if(𝐵𝑅𝐴, 𝐵, 𝐵) = 𝐵
10	8, 9	eqtr2di 2812	. . . . 5 ⊢ (𝐴 = 𝐵 → 𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
11		iftrue 4487	. . . . . 6 ⊢ (𝐵𝑅𝐴 → if(𝐵𝑅𝐴, 𝐵, 𝐴) = 𝐵)
12	11	eqcomd 2766	. . . . 5 ⊢ (𝐵𝑅𝐴 → 𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
13	10, 12	jaoi 871	. . . 4 ⊢ ((𝐴 = 𝐵 ∨ 𝐵𝑅𝐴) → 𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
14	7, 13	biimtrrdi 257	. . 3 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (¬ 𝐴𝑅𝐵 → 𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴)))
15	14	imp 412	. 2 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ ¬ 𝐴𝑅𝐵) → 𝐵 = if(𝐵𝑅𝐴, 𝐵, 𝐴))
16	5, 15	ifeqda 4518	1 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵) = if(𝐵𝑅𝐴, 𝐵, 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ifcif 4481   class class class wbr 5102   Or wor 5554

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-po 5555  df-so 5556

This theorem is used by:  somin2  6123