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Theorem soltmin 5690
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 5204 . . . . . 6 (𝑅 Or 𝑋𝑅 Po 𝑋)
21ad2antrr 764 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝑅 Po 𝑋)
3 simplr1 1261 . . . . . 6 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴𝑋)
4 simplr2 1263 . . . . . . 7 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐵𝑋)
5 simplr3 1265 . . . . . . 7 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐶𝑋)
64, 5ifcld 4275 . . . . . 6 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋)
73, 6, 43jca 1123 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → (𝐴𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋𝐵𝑋))
8 simpr 479 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
9 simpll 807 . . . . . 6 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝑅 Or 𝑋)
10 somin1 5687 . . . . . 6 ((𝑅 Or 𝑋 ∧ (𝐵𝑋𝐶𝑋)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐵)
119, 4, 5, 10syl12anc 1475 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐵)
12 poltletr 5686 . . . . . 6 ((𝑅 Po 𝑋 ∧ (𝐴𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋𝐵𝑋)) → ((𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐵) → 𝐴𝑅𝐵))
1312imp 444 . . . . 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 1478 . . . 4 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴𝑅𝐵)
153, 6, 53jca 1123 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → (𝐴𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋𝐶𝑋))
16 somin2 5689 . . . . . 6 ((𝑅 Or 𝑋 ∧ (𝐵𝑋𝐶𝑋)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐶)
179, 4, 5, 16syl12anc 1475 . . . . 5 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐶)
18 poltletr 5686 . . . . . 6 ((𝑅 Po 𝑋 ∧ (𝐴𝑋 ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶) ∈ 𝑋𝐶𝑋)) → ((𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ∧ if(𝐵𝑅𝐶, 𝐵, 𝐶)(𝑅 ∪ I )𝐶) → 𝐴𝑅𝐶))
1918imp 444 . . . . 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 1478 . . . 4 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → 𝐴𝑅𝐶)
2114, 20jca 555 . . 3 (((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)) → (𝐴𝑅𝐵𝐴𝑅𝐶))
2221ex 449 . 2 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐴𝑅𝐵𝐴𝑅𝐶)))
23 breq2 4808 . . 3 (𝐵 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐴𝑅𝐵𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
24 breq2 4808 . . 3 (𝐶 = if(𝐵𝑅𝐶, 𝐵, 𝐶) → (𝐴𝑅𝐶𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶)))
2523, 24ifboth 4268 . 2 ((𝐴𝑅𝐵𝐴𝑅𝐶) → 𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶))
2622, 25impbid1 215 1 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ (𝐴𝑅𝐵𝐴𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072  wcel 2139  cun 3713  ifcif 4230   class class class wbr 4804   I cid 5173   Po wpo 5185   Or wor 5186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-id 5174  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273
This theorem is referenced by:  wemaplem2  8619
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