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Theorem orngmul 29630
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 1086 . 2 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → 0 𝑋)
2 simp3r 1088 . 2 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → 0 𝑌)
3 simp2l 1085 . . 3 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → 𝑋𝐵)
4 simp3l 1087 . . 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 29626 . . . . 5 (𝑅 ∈ oRing ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))))
109simp3bi 1076 . . . 4 (𝑅 ∈ oRing → ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏)))
11103ad2ant1 1080 . . 3 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏)))
12 breq2 4627 . . . . . 6 (𝑎 = 𝑋 → ( 0 𝑎0 𝑋))
1312anbi1d 740 . . . . 5 (𝑎 = 𝑋 → (( 0 𝑎0 𝑏) ↔ ( 0 𝑋0 𝑏)))
14 oveq1 6622 . . . . . 6 (𝑎 = 𝑋 → (𝑎 · 𝑏) = (𝑋 · 𝑏))
1514breq2d 4635 . . . . 5 (𝑎 = 𝑋 → ( 0 (𝑎 · 𝑏) ↔ 0 (𝑋 · 𝑏)))
1613, 15imbi12d 334 . . . 4 (𝑎 = 𝑋 → ((( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏)) ↔ (( 0 𝑋0 𝑏) → 0 (𝑋 · 𝑏))))
17 breq2 4627 . . . . . 6 (𝑏 = 𝑌 → ( 0 𝑏0 𝑌))
1817anbi2d 739 . . . . 5 (𝑏 = 𝑌 → (( 0 𝑋0 𝑏) ↔ ( 0 𝑋0 𝑌)))
19 oveq2 6623 . . . . . 6 (𝑏 = 𝑌 → (𝑋 · 𝑏) = (𝑋 · 𝑌))
2019breq2d 4635 . . . . 5 (𝑏 = 𝑌 → ( 0 (𝑋 · 𝑏) ↔ 0 (𝑋 · 𝑌)))
2118, 20imbi12d 334 . . . 4 (𝑏 = 𝑌 → ((( 0 𝑋0 𝑏) → 0 (𝑋 · 𝑏)) ↔ (( 0 𝑋0 𝑌) → 0 (𝑋 · 𝑌))))
2216, 21rspc2va 3312 . . 3 (((𝑋𝐵𝑌𝐵) ∧ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))) → (( 0 𝑋0 𝑌) → 0 (𝑋 · 𝑌)))
233, 4, 11, 22syl21anc 1322 . 2 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → (( 0 𝑋0 𝑌) → 0 (𝑋 · 𝑌)))
241, 2, 23mp2and 714 1 ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → 0 (𝑋 · 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908   class class class wbr 4623  cfv 5857  (class class class)co 6615  Basecbs 15800  .rcmulr 15882  lecple 15888  0gc0g 16040  Ringcrg 18487  oGrpcogrp 29525  oRingcorng 29622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4759
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-iota 5820  df-fv 5865  df-ov 6618  df-orng 29624
This theorem is referenced by:  orngsqr  29631  ornglmulle  29632  orngrmulle  29633  orngmullt  29636  suborng  29642
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