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Theorem soltmin 5989
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 5485 . . . . . 6 (𝑅 Or 𝑋𝑅 Po 𝑋)
21ad2antrr 722 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝑅 Po 𝑋)
3 simplr1 1207 . . . . . 6 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴𝑋)
4 simplr2 1208 . . . . . . 7 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐵𝑋)
5 simplr3 1209 . . . . . . 7 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐶𝑋)
64, 5ifcld 4508 . . . . . 6 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋)
73, 6, 43jca 1120 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → (𝐴𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋𝐵𝑋))
8 simpr 485 . . . . 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 5986 . . . . . 6 ((𝑅 Or 𝑋 ∧ (𝐵𝑋𝐶𝑋)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐵)
119, 4, 5, 10syl12anc 832 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐵)
12 poltletr 5985 . . . . . 6 ((𝑅 Po 𝑋 ∧ (𝐴𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋𝐵𝑋)) → ((𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐵) → 𝐴𝑅𝐵))
1312imp 407 . . . . 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 834 . . . 4 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴𝑅𝐵)
153, 6, 53jca 1120 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → (𝐴𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋𝐶𝑋))
16 somin2 5988 . . . . . 6 ((𝑅 Or 𝑋 ∧ (𝐵𝑋𝐶𝑋)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐶)
179, 4, 5, 16syl12anc 832 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐶)
18 poltletr 5985 . . . . . 6 ((𝑅 Po 𝑋 ∧ (𝐴𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋𝐶𝑋)) → ((𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐶) → 𝐴𝑅𝐶))
1918imp 407 . . . . 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 834 . . . 4 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴𝑅𝐶)
2114, 20jca 512 . . 3 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → (𝐴𝑅𝐵𝐴𝑅𝐶))
2221ex 413 . 2 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐴𝑅𝐵𝐴𝑅𝐶)))
23 breq2 5061 . . 3 (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐴𝑅𝐵𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
24 breq2 5061 . . 3 (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐴𝑅𝐶𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
2523, 24ifboth 4501 . 2 ((𝐴𝑅𝐵𝐴𝑅𝐶) → 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
2622, 25impbid1 226 1 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ (𝐴𝑅𝐵𝐴𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079  wcel 2105  cun 3931  ifcif 4463   class class class wbr 5057   I cid 5452   Po wpo 5465   Or wor 5466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555
This theorem is referenced by:  wemaplem2  8999
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