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Theorem lefld 18308
Description: The field of the 'less or equal to' relationship on the extended real. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 4-May-2015.)
Assertion
Ref Expression
lefld * =

Proof of Theorem lefld
StepHypRef Expression
1 lerel 11040 . . 3 Rel ≤
2 relfld 6177 . . 3 (Rel ≤ → ≤ = (dom ≤ ∪ ran ≤ ))
31, 2ax-mp 5 . 2 ≤ = (dom ≤ ∪ ran ≤ )
4 ledm 18306 . . 3 * = dom ≤
5 lern 18307 . . 3 * = ran ≤
64, 5uneq12i 4100 . 2 (ℝ* ∪ ℝ*) = (dom ≤ ∪ ran ≤ )
7 unidm 4091 . 2 (ℝ* ∪ ℝ*) = ℝ*
83, 6, 73eqtr2ri 2775 1 * =
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  cun 3890   cuni 4845  dom cdm 5590  ran crn 5591  Rel wrel 5595  *cxr 11009  cle 11011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-cnex 10928  ax-resscn 10929  ax-pre-lttri 10946  ax-pre-lttrn 10947
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-po 5504  df-so 5505  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-er 8481  df-en 8717  df-dom 8718  df-sdom 8719  df-pnf 11012  df-mnf 11013  df-xr 11014  df-ltxr 11015  df-le 11016
This theorem is referenced by:  letsr  18309
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