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Theorem lefld 17434
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 10308 . . 3 Rel ≤
2 relfld 5804 . . 3 (Rel ≤ → ≤ = (dom ≤ ∪ ran ≤ ))
31, 2ax-mp 5 . 2 ≤ = (dom ≤ ∪ ran ≤ )
4 ledm 17432 . . 3 * = dom ≤
5 lern 17433 . . 3 * = ran ≤
64, 5uneq12i 3916 . 2 (ℝ* ∪ ℝ*) = (dom ≤ ∪ ran ≤ )
7 unidm 3907 . 2 (ℝ* ∪ ℝ*) = ℝ*
83, 6, 73eqtr2ri 2800 1 * =
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  cun 3721   cuni 4575  dom cdm 5250  ran crn 5251  Rel wrel 5255  *cxr 10279  cle 10281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-pre-lttri 10216  ax-pre-lttrn 10217
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-po 5171  df-so 5172  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-er 7900  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286
This theorem is referenced by:  letsr  17435
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