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Theorem soltmin 5983
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 5479 . . . . . 6 (𝑅 Or 𝑋𝑅 Po 𝑋)
21ad2antrr 725 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝑅 Po 𝑋)
3 simplr1 1212 . . . . . 6 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴𝑋)
4 simplr2 1213 . . . . . . 7 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐵𝑋)
5 simplr3 1214 . . . . . . 7 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐶𝑋)
64, 5ifcld 4494 . . . . . 6 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋)
73, 6, 43jca 1125 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → (𝐴𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋𝐵𝑋))
8 simpr 488 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
9 simpll 766 . . . . . 6 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝑅 Or 𝑋)
10 somin1 5980 . . . . . 6 ((𝑅 Or 𝑋 ∧ (𝐵𝑋𝐶𝑋)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐵)
119, 4, 5, 10syl12anc 835 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐵)
12 poltletr 5979 . . . . . 6 ((𝑅 Po 𝑋 ∧ (𝐴𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋𝐵𝑋)) → ((𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐵) → 𝐴𝑅𝐵))
1312imp 410 . . . . 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 837 . . . 4 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴𝑅𝐵)
153, 6, 53jca 1125 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → (𝐴𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋𝐶𝑋))
16 somin2 5982 . . . . . 6 ((𝑅 Or 𝑋 ∧ (𝐵𝑋𝐶𝑋)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐶)
179, 4, 5, 16syl12anc 835 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐶)
18 poltletr 5979 . . . . . 6 ((𝑅 Po 𝑋 ∧ (𝐴𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋𝐶𝑋)) → ((𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐶) → 𝐴𝑅𝐶))
1918imp 410 . . . . 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 837 . . . 4 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴𝑅𝐶)
2114, 20jca 515 . . 3 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → (𝐴𝑅𝐵𝐴𝑅𝐶))
2221ex 416 . 2 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐴𝑅𝐵𝐴𝑅𝐶)))
23 breq2 5056 . . 3 (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐴𝑅𝐵𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
24 breq2 5056 . . 3 (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐴𝑅𝐶𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
2523, 24ifboth 4487 . 2 ((𝐴𝑅𝐵𝐴𝑅𝐶) → 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
2622, 25impbid1 228 1 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ (𝐴𝑅𝐵𝐴𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084  wcel 2115  cun 3917  ifcif 4449   class class class wbr 5052   I cid 5446   Po wpo 5459   Or wor 5460
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-sn 4550  df-pr 4552  df-op 4556  df-br 5053  df-opab 5115  df-id 5447  df-po 5461  df-so 5462  df-xp 5548  df-rel 5549
This theorem is referenced by:  wemaplem2  9002
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