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Theorem orngmul 28232
 Description: In an ordered ring, the ordering is compatible with the ring multiplication operation. (Contributed by Thierry Arnoux, 20-Jan-2018.)
Hypotheses
Ref Expression
orngmul.0
orngmul.1
orngmul.2
orngmul.3
Assertion
Ref Expression
orngmul oRing

Proof of Theorem orngmul
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2r 1024 . 2 oRing
2 simp3r 1026 . 2 oRing
3 simp2l 1023 . . 3 oRing
4 simp3l 1025 . . 3 oRing
5 orngmul.0 . . . . . 6
6 orngmul.2 . . . . . 6
7 orngmul.3 . . . . . 6
8 orngmul.1 . . . . . 6
95, 6, 7, 8isorng 28228 . . . . 5 oRing oGrp
109simp3bi 1014 . . . 4 oRing
11103ad2ant1 1018 . . 3 oRing
12 breq2 4398 . . . . . 6
1312anbi1d 703 . . . . 5
14 oveq1 6284 . . . . . 6
1514breq2d 4406 . . . . 5
1613, 15imbi12d 318 . . . 4
17 breq2 4398 . . . . . 6
1817anbi2d 702 . . . . 5
19 oveq2 6285 . . . . . 6
2019breq2d 4406 . . . . 5
2118, 20imbi12d 318 . . . 4
2216, 21rspc2va 3169 . . 3
233, 4, 11, 22syl21anc 1229 . 2 oRing
241, 2, 23mp2and 677 1 oRing
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 367   w3a 974   wceq 1405   wcel 1842  wral 2753   class class class wbr 4394  cfv 5568  (class class class)co 6277  cbs 14839  cmulr 14908  cple 14914  c0g 15052  crg 17516  oGrpcogrp 28126  oRingcorng 28224 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-nul 4524 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-iota 5532  df-fv 5576  df-ov 6280  df-orng 28226 This theorem is referenced by:  orngsqr  28233  ornglmulle  28234  orngrmulle  28235  orngmullt  28238  suborng  28244
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