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Theorem lefld 17815
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 10683 . . 3 Rel ≤
2 relfld 6102 . . 3 (Rel ≤ → ≤ = (dom ≤ ∪ ran ≤ ))
31, 2ax-mp 5 . 2 ≤ = (dom ≤ ∪ ran ≤ )
4 ledm 17813 . . 3 * = dom ≤
5 lern 17814 . . 3 * = ran ≤
64, 5uneq12i 4116 . 2 (ℝ* ∪ ℝ*) = (dom ≤ ∪ ran ≤ )
7 unidm 4107 . 2 (ℝ* ∪ ℝ*) = ℝ*
83, 6, 73eqtr2ri 2850 1 * =
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  cun 3911   cuni 4814  dom cdm 5531  ran crn 5532  Rel wrel 5536  *cxr 10652  cle 10654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-cnex 10571  ax-resscn 10572  ax-pre-lttri 10589  ax-pre-lttrn 10590
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-po 5450  df-so 5451  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-er 8267  df-en 8488  df-dom 8489  df-sdom 8490  df-pnf 10655  df-mnf 10656  df-xr 10657  df-ltxr 10658  df-le 10659
This theorem is referenced by:  letsr  17816
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