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Theorem orngmul 33333
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 1201 . 2 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → 0 𝑋)
2 simp3r 1203 . 2 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → 0 𝑌)
3 simp2l 1200 . . 3 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → 𝑋𝐵)
4 simp3l 1202 . . 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 33329 . . . . 5 (𝑅 ∈ oRing ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))))
109simp3bi 1148 . . . 4 (𝑅 ∈ oRing → ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏)))
11103ad2ant1 1134 . . 3 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏)))
12 breq2 5147 . . . . . 6 (𝑎 = 𝑋 → ( 0 𝑎0 𝑋))
1312anbi1d 631 . . . . 5 (𝑎 = 𝑋 → (( 0 𝑎0 𝑏) ↔ ( 0 𝑋0 𝑏)))
14 oveq1 7438 . . . . . 6 (𝑎 = 𝑋 → (𝑎 · 𝑏) = (𝑋 · 𝑏))
1514breq2d 5155 . . . . 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 5147 . . . . . 6 (𝑏 = 𝑌 → ( 0 𝑏0 𝑌))
1817anbi2d 630 . . . . 5 (𝑏 = 𝑌 → (( 0 𝑋0 𝑏) ↔ ( 0 𝑋0 𝑌)))
19 oveq2 7439 . . . . . 6 (𝑏 = 𝑌 → (𝑋 · 𝑏) = (𝑋 · 𝑌))
2019breq2d 5155 . . . . 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 3634 . . 3 (((𝑋𝐵𝑌𝐵) ∧ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))) → (( 0 𝑋0 𝑌) → 0 (𝑋 · 𝑌)))
233, 4, 11, 22syl21anc 838 . 2 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → (( 0 𝑋0 𝑌) → 0 (𝑋 · 𝑌)))
241, 2, 23mp2and 699 1 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → 0 (𝑋 · 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  .rcmulr 17298  lecple 17304  0gc0g 17484  Ringcrg 20230  oGrpcogrp 33075  oRingcorng 33325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-orng 33327
This theorem is referenced by:  orngsqr  33334  ornglmulle  33335  orngrmulle  33336  orngmullt  33339  suborng  33345
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