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Theorem orngmul 29134
 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 𝐵 = (Base‘𝑅)
orngmul.1 = (le‘𝑅)
orngmul.2 0 = (0g𝑅)
orngmul.3 · = (.r𝑅)
Assertion
Ref Expression
orngmul ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → 0 (𝑋 · 𝑌))

Proof of Theorem orngmul
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2r 1081 . 2 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → 0 𝑋)
2 simp3r 1083 . 2 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → 0 𝑌)
3 simp2l 1080 . . 3 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → 𝑋𝐵)
4 simp3l 1082 . . 3 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → 𝑌𝐵)
5 orngmul.0 . . . . . 6 𝐵 = (Base‘𝑅)
6 orngmul.2 . . . . . 6 0 = (0g𝑅)
7 orngmul.3 . . . . . 6 · = (.r𝑅)
8 orngmul.1 . . . . . 6 = (le‘𝑅)
95, 6, 7, 8isorng 29130 . . . . 5 (𝑅 ∈ oRing ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))))
109simp3bi 1071 . . . 4 (𝑅 ∈ oRing → ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏)))
11103ad2ant1 1075 . . 3 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏)))
12 breq2 4587 . . . . . 6 (𝑎 = 𝑋 → ( 0 𝑎0 𝑋))
1312anbi1d 737 . . . . 5 (𝑎 = 𝑋 → (( 0 𝑎0 𝑏) ↔ ( 0 𝑋0 𝑏)))
14 oveq1 6556 . . . . . 6 (𝑎 = 𝑋 → (𝑎 · 𝑏) = (𝑋 · 𝑏))
1514breq2d 4595 . . . . 5 (𝑎 = 𝑋 → ( 0 (𝑎 · 𝑏) ↔ 0 (𝑋 · 𝑏)))
1613, 15imbi12d 333 . . . 4 (𝑎 = 𝑋 → ((( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏)) ↔ (( 0 𝑋0 𝑏) → 0 (𝑋 · 𝑏))))
17 breq2 4587 . . . . . 6 (𝑏 = 𝑌 → ( 0 𝑏0 𝑌))
1817anbi2d 736 . . . . 5 (𝑏 = 𝑌 → (( 0 𝑋0 𝑏) ↔ ( 0 𝑋0 𝑌)))
19 oveq2 6557 . . . . . 6 (𝑏 = 𝑌 → (𝑋 · 𝑏) = (𝑋 · 𝑌))
2019breq2d 4595 . . . . 5 (𝑏 = 𝑌 → ( 0 (𝑋 · 𝑏) ↔ 0 (𝑋 · 𝑌)))
2118, 20imbi12d 333 . . . 4 (𝑏 = 𝑌 → ((( 0 𝑋0 𝑏) → 0 (𝑋 · 𝑏)) ↔ (( 0 𝑋0 𝑌) → 0 (𝑋 · 𝑌))))
2216, 21rspc2va 3294 . . 3 (((𝑋𝐵𝑌𝐵) ∧ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))) → (( 0 𝑋0 𝑌) → 0 (𝑋 · 𝑌)))
233, 4, 11, 22syl21anc 1317 . 2 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → (( 0 𝑋0 𝑌) → 0 (𝑋 · 𝑌)))
241, 2, 23mp2and 711 1 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → 0 (𝑋 · 𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  .rcmulr 15769  lecple 15775  0gc0g 15923  Ringcrg 18370  oGrpcogrp 29029  oRingcorng 29126 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-orng 29128 This theorem is referenced by:  orngsqr  29135  ornglmulle  29136  orngrmulle  29137  orngmullt  29140  suborng  29146
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