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Theorem soltmin 6136
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 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ (𝐴𝑅𝐵𝐴𝑅𝐶)))

Proof of Theorem soltmin
StepHypRef Expression
1 sopo 5606 . . . . . 6 (𝑅 Or 𝑋𝑅 Po 𝑋)
21ad2antrr 722 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝑅 Po 𝑋)
3 simplr1 1213 . . . . . 6 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴𝑋)
4 simplr2 1214 . . . . . . 7 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐵𝑋)
5 simplr3 1215 . . . . . . 7 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐶𝑋)
64, 5ifcld 4573 . . . . . 6 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋)
73, 6, 43jca 1126 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → (𝐴𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋𝐵𝑋))
8 simpr 483 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
9 simpll 763 . . . . . 6 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝑅 Or 𝑋)
10 somin1 6133 . . . . . 6 ((𝑅 Or 𝑋 ∧ (𝐵𝑋𝐶𝑋)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐵)
119, 4, 5, 10syl12anc 833 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐵)
12 poltletr 6132 . . . . . 6 ((𝑅 Po 𝑋 ∧ (𝐴𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋𝐵𝑋)) → ((𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐵) → 𝐴𝑅𝐵))
1312imp 405 . . . . 5 (((𝑅 Po 𝑋 ∧ (𝐴𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋𝐵𝑋)) ∧ (𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐵)) → 𝐴𝑅𝐵)
142, 7, 8, 11, 13syl22anc 835 . . . 4 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴𝑅𝐵)
153, 6, 53jca 1126 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → (𝐴𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋𝐶𝑋))
16 somin2 6135 . . . . . 6 ((𝑅 Or 𝑋 ∧ (𝐵𝑋𝐶𝑋)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐶)
179, 4, 5, 16syl12anc 833 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐶)
18 poltletr 6132 . . . . . 6 ((𝑅 Po 𝑋 ∧ (𝐴𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋𝐶𝑋)) → ((𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐶) → 𝐴𝑅𝐶))
1918imp 405 . . . . 5 (((𝑅 Po 𝑋 ∧ (𝐴𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋𝐶𝑋)) ∧ (𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐶)) → 𝐴𝑅𝐶)
202, 15, 8, 17, 19syl22anc 835 . . . 4 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴𝑅𝐶)
2114, 20jca 510 . . 3 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → (𝐴𝑅𝐵𝐴𝑅𝐶))
2221ex 411 . 2 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐴𝑅𝐵𝐴𝑅𝐶)))
23 breq2 5151 . . 3 (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐴𝑅𝐵𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
24 breq2 5151 . . 3 (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐴𝑅𝐶𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
2523, 24ifboth 4566 . 2 ((𝐴𝑅𝐵𝐴𝑅𝐶) → 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
2622, 25impbid1 224 1 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ (𝐴𝑅𝐵𝐴𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1085  wcel 2104  cun 3945  ifcif 4527   class class class wbr 5147   I cid 5572   Po wpo 5585   Or wor 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-po 5587  df-so 5588  df-xp 5681  df-rel 5682
This theorem is referenced by:  wemaplem2  9544
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