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Theorem tsrlemax 17818
Description: Two ways of saying a number is less than or equal to the maximum of two others. (Contributed by Mario Carneiro, 9-Sep-2015.)
Hypothesis
Ref Expression
istsr.1 𝑋 = dom 𝑅
Assertion
Ref Expression
tsrlemax ((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵) ↔ (𝐴𝑅𝐵𝐴𝑅𝐶)))

Proof of Theorem tsrlemax
StepHypRef Expression
1 breq2 5061 . . 3 (𝐶 = if(𝐵𝑅𝐶, 𝐶, 𝐵) → (𝐴𝑅𝐶𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵)))
21bibi1d 345 . 2 (𝐶 = if(𝐵𝑅𝐶, 𝐶, 𝐵) → ((𝐴𝑅𝐶 ↔ (𝐴𝑅𝐵𝐴𝑅𝐶)) ↔ (𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵) ↔ (𝐴𝑅𝐵𝐴𝑅𝐶))))
3 breq2 5061 . . 3 (𝐵 = if(𝐵𝑅𝐶, 𝐶, 𝐵) → (𝐴𝑅𝐵𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵)))
43bibi1d 345 . 2 (𝐵 = if(𝐵𝑅𝐶, 𝐶, 𝐵) → ((𝐴𝑅𝐵 ↔ (𝐴𝑅𝐵𝐴𝑅𝐶)) ↔ (𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵) ↔ (𝐴𝑅𝐵𝐴𝑅𝐶))))
5 olc 862 . . 3 (𝐴𝑅𝐶 → (𝐴𝑅𝐵𝐴𝑅𝐶))
6 eqid 2818 . . . . . . . . . 10 dom 𝑅 = dom 𝑅
76istsr 17815 . . . . . . . . 9 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (dom 𝑅 × dom 𝑅) ⊆ (𝑅𝑅)))
87simplbi 498 . . . . . . . 8 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
9 pstr 17809 . . . . . . . . 9 ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)
1093expib 1114 . . . . . . . 8 (𝑅 ∈ PosetRel → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
118, 10syl 17 . . . . . . 7 (𝑅 ∈ TosetRel → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
1211adantr 481 . . . . . 6 ((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
1312expdimp 453 . . . . 5 (((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅𝐵) → (𝐵𝑅𝐶𝐴𝑅𝐶))
1413impancom 452 . . . 4 (((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐵𝑅𝐶) → (𝐴𝑅𝐵𝐴𝑅𝐶))
15 idd 24 . . . 4 (((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐵𝑅𝐶) → (𝐴𝑅𝐶𝐴𝑅𝐶))
1614, 15jaod 853 . . 3 (((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐵𝑅𝐶) → ((𝐴𝑅𝐵𝐴𝑅𝐶) → 𝐴𝑅𝐶))
175, 16impbid2 227 . 2 (((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐵𝑅𝐶) → (𝐴𝑅𝐶 ↔ (𝐴𝑅𝐵𝐴𝑅𝐶)))
18 orc 861 . . 3 (𝐴𝑅𝐵 → (𝐴𝑅𝐵𝐴𝑅𝐶))
19 idd 24 . . . 4 (((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ ¬ 𝐵𝑅𝐶) → (𝐴𝑅𝐵𝐴𝑅𝐵))
20 istsr.1 . . . . . . . 8 𝑋 = dom 𝑅
2120tsrlin 17817 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ 𝐵𝑋𝐶𝑋) → (𝐵𝑅𝐶𝐶𝑅𝐵))
22213adant3r1 1174 . . . . . 6 ((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵𝑅𝐶𝐶𝑅𝐵))
2322orcanai 996 . . . . 5 (((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ ¬ 𝐵𝑅𝐶) → 𝐶𝑅𝐵)
24 pstr 17809 . . . . . . . . . 10 ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐶𝐶𝑅𝐵) → 𝐴𝑅𝐵)
25243expib 1114 . . . . . . . . 9 (𝑅 ∈ PosetRel → ((𝐴𝑅𝐶𝐶𝑅𝐵) → 𝐴𝑅𝐵))
268, 25syl 17 . . . . . . . 8 (𝑅 ∈ TosetRel → ((𝐴𝑅𝐶𝐶𝑅𝐵) → 𝐴𝑅𝐵))
2726adantr 481 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑅𝐶𝐶𝑅𝐵) → 𝐴𝑅𝐵))
2827expdimp 453 . . . . . 6 (((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑅𝐶) → (𝐶𝑅𝐵𝐴𝑅𝐵))
2928impancom 452 . . . . 5 (((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐶𝑅𝐵) → (𝐴𝑅𝐶𝐴𝑅𝐵))
3023, 29syldan 591 . . . 4 (((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ ¬ 𝐵𝑅𝐶) → (𝐴𝑅𝐶𝐴𝑅𝐵))
3119, 30jaod 853 . . 3 (((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ ¬ 𝐵𝑅𝐶) → ((𝐴𝑅𝐵𝐴𝑅𝐶) → 𝐴𝑅𝐵))
3218, 31impbid2 227 . 2 (((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ ¬ 𝐵𝑅𝐶) → (𝐴𝑅𝐵 ↔ (𝐴𝑅𝐵𝐴𝑅𝐶)))
332, 4, 17, 32ifbothda 4500 1 ((𝑅 ∈ TosetRel ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵) ↔ (𝐴𝑅𝐵𝐴𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841  w3a 1079   = wceq 1528  wcel 2105  cun 3931  wss 3933  ifcif 4463   class class class wbr 5057   × cxp 5546  ccnv 5547  dom cdm 5548  PosetRelcps 17796   TosetRel ctsr 17797
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-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-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-res 5560  df-ps 17798  df-tsr 17799
This theorem is referenced by:  ordtbaslem  21724
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