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Theorem orngmul 32042
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 1200 . 2 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → 0 𝑋)
2 simp3r 1202 . 2 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → 0 𝑌)
3 simp2l 1199 . . 3 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → 𝑋𝐵)
4 simp3l 1201 . . 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 32038 . . . . 5 (𝑅 ∈ oRing ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))))
109simp3bi 1147 . . . 4 (𝑅 ∈ oRing → ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏)))
11103ad2ant1 1133 . . 3 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏)))
12 breq2 5109 . . . . . 6 (𝑎 = 𝑋 → ( 0 𝑎0 𝑋))
1312anbi1d 630 . . . . 5 (𝑎 = 𝑋 → (( 0 𝑎0 𝑏) ↔ ( 0 𝑋0 𝑏)))
14 oveq1 7363 . . . . . 6 (𝑎 = 𝑋 → (𝑎 · 𝑏) = (𝑋 · 𝑏))
1514breq2d 5117 . . . . 5 (𝑎 = 𝑋 → ( 0 (𝑎 · 𝑏) ↔ 0 (𝑋 · 𝑏)))
1613, 15imbi12d 344 . . . 4 (𝑎 = 𝑋 → ((( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏)) ↔ (( 0 𝑋0 𝑏) → 0 (𝑋 · 𝑏))))
17 breq2 5109 . . . . . 6 (𝑏 = 𝑌 → ( 0 𝑏0 𝑌))
1817anbi2d 629 . . . . 5 (𝑏 = 𝑌 → (( 0 𝑋0 𝑏) ↔ ( 0 𝑋0 𝑌)))
19 oveq2 7364 . . . . . 6 (𝑏 = 𝑌 → (𝑋 · 𝑏) = (𝑋 · 𝑌))
2019breq2d 5117 . . . . 5 (𝑏 = 𝑌 → ( 0 (𝑋 · 𝑏) ↔ 0 (𝑋 · 𝑌)))
2118, 20imbi12d 344 . . . 4 (𝑏 = 𝑌 → ((( 0 𝑋0 𝑏) → 0 (𝑋 · 𝑏)) ↔ (( 0 𝑋0 𝑌) → 0 (𝑋 · 𝑌))))
2216, 21rspc2va 3591 . . 3 (((𝑋𝐵𝑌𝐵) ∧ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))) → (( 0 𝑋0 𝑌) → 0 (𝑋 · 𝑌)))
233, 4, 11, 22syl21anc 836 . 2 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → (( 0 𝑋0 𝑌) → 0 (𝑋 · 𝑌)))
241, 2, 23mp2and 697 1 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → 0 (𝑋 · 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064   class class class wbr 5105  cfv 6496  (class class class)co 7356  Basecbs 17082  .rcmulr 17133  lecple 17139  0gc0g 17320  Ringcrg 19962  oGrpcogrp 31850  oRingcorng 32034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-nul 5263
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-iota 6448  df-fv 6504  df-ov 7359  df-orng 32036
This theorem is referenced by:  orngsqr  32043  ornglmulle  32044  orngrmulle  32045  orngmullt  32048  suborng  32054
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