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Theorem soltmin 5437
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 4965 . . . . . 6 (𝑅 Or 𝑋𝑅 Po 𝑋)
21ad2antrr 757 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝑅 Po 𝑋)
3 simplr1 1095 . . . . . 6 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴𝑋)
4 simplr2 1096 . . . . . . 7 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐵𝑋)
5 simplr3 1097 . . . . . . 7 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐶𝑋)
64, 5ifcld 4080 . . . . . 6 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋)
73, 6, 43jca 1234 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → (𝐴𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋𝐵𝑋))
8 simpr 475 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
9 simpll 785 . . . . . 6 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝑅 Or 𝑋)
10 somin1 5434 . . . . . 6 ((𝑅 Or 𝑋 ∧ (𝐵𝑋𝐶𝑋)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐵)
119, 4, 5, 10syl12anc 1315 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐵)
12 poltletr 5433 . . . . . 6 ((𝑅 Po 𝑋 ∧ (𝐴𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋𝐵𝑋)) → ((𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐵) → 𝐴𝑅𝐵))
1312imp 443 . . . . 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 1318 . . . 4 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴𝑅𝐵)
153, 6, 53jca 1234 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → (𝐴𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋𝐶𝑋))
16 somin2 5436 . . . . . 6 ((𝑅 Or 𝑋 ∧ (𝐵𝑋𝐶𝑋)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐶)
179, 4, 5, 16syl12anc 1315 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐶)
18 poltletr 5433 . . . . . 6 ((𝑅 Po 𝑋 ∧ (𝐴𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋𝐶𝑋)) → ((𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐶) → 𝐴𝑅𝐶))
1918imp 443 . . . . 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 1318 . . . 4 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴𝑅𝐶)
2114, 20jca 552 . . 3 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → (𝐴𝑅𝐵𝐴𝑅𝐶))
2221ex 448 . 2 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐴𝑅𝐵𝐴𝑅𝐶)))
23 breq2 4581 . . 3 (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐴𝑅𝐵𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
24 breq2 4581 . . 3 (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐴𝑅𝐶𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
2523, 24ifboth 4073 . 2 ((𝐴𝑅𝐵𝐴𝑅𝐶) → 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
2622, 25impbid1 213 1 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ (𝐴𝑅𝐵𝐴𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030  wcel 1976  cun 3537  ifcif 4035   class class class wbr 4577   I cid 4937   Po wpo 4946   Or wor 4947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-id 4942  df-po 4948  df-so 4949  df-xp 5033  df-rel 5034
This theorem is referenced by:  wemaplem2  8312
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