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Theorem orngmul 30369
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 1214 . 2 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → 0 𝑋)
2 simp3r 1216 . 2 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → 0 𝑌)
3 simp2l 1213 . . 3 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → 𝑋𝐵)
4 simp3l 1215 . . 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 30365 . . . . 5 (𝑅 ∈ oRing ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))))
109simp3bi 1138 . . . 4 (𝑅 ∈ oRing → ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏)))
11103ad2ant1 1124 . . 3 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏)))
12 breq2 4892 . . . . . 6 (𝑎 = 𝑋 → ( 0 𝑎0 𝑋))
1312anbi1d 623 . . . . 5 (𝑎 = 𝑋 → (( 0 𝑎0 𝑏) ↔ ( 0 𝑋0 𝑏)))
14 oveq1 6931 . . . . . 6 (𝑎 = 𝑋 → (𝑎 · 𝑏) = (𝑋 · 𝑏))
1514breq2d 4900 . . . . 5 (𝑎 = 𝑋 → ( 0 (𝑎 · 𝑏) ↔ 0 (𝑋 · 𝑏)))
1613, 15imbi12d 336 . . . 4 (𝑎 = 𝑋 → ((( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏)) ↔ (( 0 𝑋0 𝑏) → 0 (𝑋 · 𝑏))))
17 breq2 4892 . . . . . 6 (𝑏 = 𝑌 → ( 0 𝑏0 𝑌))
1817anbi2d 622 . . . . 5 (𝑏 = 𝑌 → (( 0 𝑋0 𝑏) ↔ ( 0 𝑋0 𝑌)))
19 oveq2 6932 . . . . . 6 (𝑏 = 𝑌 → (𝑋 · 𝑏) = (𝑋 · 𝑌))
2019breq2d 4900 . . . . 5 (𝑏 = 𝑌 → ( 0 (𝑋 · 𝑏) ↔ 0 (𝑋 · 𝑌)))
2118, 20imbi12d 336 . . . 4 (𝑏 = 𝑌 → ((( 0 𝑋0 𝑏) → 0 (𝑋 · 𝑏)) ↔ (( 0 𝑋0 𝑌) → 0 (𝑋 · 𝑌))))
2216, 21rspc2va 3525 . . 3 (((𝑋𝐵𝑌𝐵) ∧ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))) → (( 0 𝑋0 𝑌) → 0 (𝑋 · 𝑌)))
233, 4, 11, 22syl21anc 828 . 2 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → (( 0 𝑋0 𝑌) → 0 (𝑋 · 𝑌)))
241, 2, 23mp2and 689 1 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → 0 (𝑋 · 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2107  wral 3090   class class class wbr 4888  cfv 6137  (class class class)co 6924  Basecbs 16259  .rcmulr 16343  lecple 16349  0gc0g 16490  Ringcrg 18938  oGrpcogrp 30264  oRingcorng 30361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-nul 5027
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-iota 6101  df-fv 6145  df-ov 6927  df-orng 30363
This theorem is referenced by:  orngsqr  30370  ornglmulle  30371  orngrmulle  30372  orngmullt  30375  suborng  30381
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