Theorem somin1 6122

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

Assertion

Ref	Expression
somin1	⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐴)

Proof of Theorem somin1

Step	Hyp	Ref	Expression
1		iftrue 4488	. . . . 5 ⊢ (𝐴𝑅𝐵 → if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴)
2	1	olcd 885	. . . 4 ⊢ (𝐴𝑅𝐵 → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴))
3	2	adantl 485	. . 3 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ 𝐴𝑅𝐵) → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴))
4		sotric 5587	. . . . . . 7 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵𝑅𝐴)))
5		orcom 881	. . . . . . . . 9 ⊢ ((𝐴 = 𝐵 ∨ 𝐵𝑅𝐴) ↔ (𝐵𝑅𝐴 ∨ 𝐴 = 𝐵))
6		eqcom 2771	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
7	6	orbi2i 923	. . . . . . . . 9 ⊢ ((𝐵𝑅𝐴 ∨ 𝐴 = 𝐵) ↔ (𝐵𝑅𝐴 ∨ 𝐵 = 𝐴))
8	5, 7	bitri 277	. . . . . . . 8 ⊢ ((𝐴 = 𝐵 ∨ 𝐵𝑅𝐴) ↔ (𝐵𝑅𝐴 ∨ 𝐵 = 𝐴))
9	8	notbii 322	. . . . . . 7 ⊢ (¬ (𝐴 = 𝐵 ∨ 𝐵𝑅𝐴) ↔ ¬ (𝐵𝑅𝐴 ∨ 𝐵 = 𝐴))
10	4, 9	bitrdi 289	. . . . . 6 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝑅𝐵 ↔ ¬ (𝐵𝑅𝐴 ∨ 𝐵 = 𝐴)))
11	10	con2bid 356	. . . . 5 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐵𝑅𝐴 ∨ 𝐵 = 𝐴) ↔ ¬ 𝐴𝑅𝐵))
12	11	biimpar 481	. . . 4 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ ¬ 𝐴𝑅𝐵) → (𝐵𝑅𝐴 ∨ 𝐵 = 𝐴))
13		iffalse 4491	. . . . . 6 ⊢ (¬ 𝐴𝑅𝐵 → if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐵)
14		breq1 5105	. . . . . . 7 ⊢ (if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐵 → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ↔ 𝐵𝑅𝐴))
15		eqeq1 2768	. . . . . . 7 ⊢ (if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐵 → (if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴 ↔ 𝐵 = 𝐴))
16	14, 15	orbi12d 929	. . . . . 6 ⊢ (if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐵 → ((if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴) ↔ (𝐵𝑅𝐴 ∨ 𝐵 = 𝐴)))
17	13, 16	syl 17	. . . . 5 ⊢ (¬ 𝐴𝑅𝐵 → ((if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴) ↔ (𝐵𝑅𝐴 ∨ 𝐵 = 𝐴)))
18	17	adantl 485	. . . 4 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ ¬ 𝐴𝑅𝐵) → ((if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴) ↔ (𝐵𝑅𝐴 ∨ 𝐵 = 𝐴)))
19	12, 18	mpbird 259	. . 3 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ ¬ 𝐴𝑅𝐵) → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴))
20	3, 19	pm2.61dan 822	. 2 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴))
21		poleloe 6120	. . 3 ⊢ (𝐴 ∈ 𝑋 → (if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐴 ↔ (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴)))
22	21	ad2antrl 738	. 2 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐴 ↔ (if(𝐴𝑅𝐵, 𝐴, 𝐵)𝑅𝐴 ∨ if(𝐴𝑅𝐵, 𝐴, 𝐵) = 𝐴)))
23	20, 22	mpbird 259	1 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   = wceq 1562   ∈ wcel 2144   ∪ cun 3904  ifcif 4482   class class class wbr 5102   I cid 5543   Or wor 5556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-id 5544  df-po 5557  df-so 5558  df-xp 5655  df-rel 5656

This theorem is referenced by:  somin2  6124  soltmin  6125