Theorem soltmin 6135

Description: Being less than a minimum, for a general total order. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Assertion

Ref	Expression
soltmin	⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑅𝐶)))

Proof of Theorem soltmin

Step	Hyp	Ref	Expression
1		sopo 5581	. . . . . 6 ⊢ (𝑅 Or 𝑋 → 𝑅 Po 𝑋)
2	1	ad2antrr 739	. . . . 5 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝑅 Po 𝑋)
3		simplr1 1234	. . . . . 6 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴 ∈ 𝑋)
4		simplr2 1235	. . . . . . 7 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐵 ∈ 𝑋)
5		simplr3 1236	. . . . . . 7 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐶 ∈ 𝑋)
6	4, 5	ifcld 4531	. . . . . 6 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋)
7	3, 6, 4	3jca 1146	. . . . 5 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → (𝐴 ∈ 𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
8		simpr 490	. . . . 5 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
9		simpll 779	. . . . . 6 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝑅 Or 𝑋)
10		somin1 6132	. . . . . 6 ⊢ ((𝑅 Or 𝑋 ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐵)
11	9, 4, 5, 10	syl12anc 850	. . . . 5 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐵)
12		poltletr 6131	. . . . . 6 ⊢ ((𝑅 Po 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐵) → 𝐴𝑅𝐵))
13	12	imp 412	. . . . 5 ⊢ (((𝑅 Po 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ (𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐵)) → 𝐴𝑅𝐵)
14	2, 7, 8, 11, 13	syl22anc 852	. . . 4 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴𝑅𝐵)
15	3, 6, 5	3jca 1146	. . . . 5 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → (𝐴 ∈ 𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋))
16		somin2 6134	. . . . . 6 ⊢ ((𝑅 Or 𝑋 ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐶)
17	9, 4, 5, 16	syl12anc 850	. . . . 5 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐶)
18		poltletr 6131	. . . . . 6 ⊢ ((𝑅 Po 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐶) → 𝐴𝑅𝐶))
19	18	imp 412	. . . . 5 ⊢ (((𝑅 Po 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐶)) → 𝐴𝑅𝐶)
20	2, 15, 8, 17, 19	syl22anc 852	. . . 4 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴𝑅𝐶)
21	14, 20	jca 521	. . 3 ⊢ (((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → (𝐴𝑅𝐵 ∧ 𝐴𝑅𝐶))
22	21	ex 418	. 2 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐴𝑅𝐵 ∧ 𝐴𝑅𝐶)))
23		breq2 5109	. . 3 ⊢ (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐴𝑅𝐵 ↔ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
24		breq2 5109	. . 3 ⊢ (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐴𝑅𝐶 ↔ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
25	23, 24	ifboth 4524	. 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐴𝑅𝐶) → 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
26	22, 25	impbid1 228	1 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑅𝐶)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2146   ∪ cun 4151  ifcif 4484   class class class wbr 5105   I cid 5548   Po wpo 5560   Or wor 5561

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-opab 5170  df-id 5549  df-po 5562  df-so 5563  df-xp 5660  df-rel 5661

This theorem is used by:  wemaplem2  9551