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Theorem orngmul 32679
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 𝐡 = (Baseβ€˜π‘…)
orngmul.1 ≀ = (leβ€˜π‘…)
orngmul.2 0 = (0gβ€˜π‘…)
orngmul.3 Β· = (.rβ€˜π‘…)
Assertion
Ref Expression
orngmul ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐡 ∧ 0 ≀ 𝑋) ∧ (π‘Œ ∈ 𝐡 ∧ 0 ≀ π‘Œ)) β†’ 0 ≀ (𝑋 Β· π‘Œ))
Proof of Theorem orngmul
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2r 1200 . 2 ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐡 ∧ 0 ≀ 𝑋) ∧ (π‘Œ ∈ 𝐡 ∧ 0 ≀ π‘Œ)) β†’ 0 ≀ 𝑋)
2 simp3r 1202 . 2 ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐡 ∧ 0 ≀ 𝑋) ∧ (π‘Œ ∈ 𝐡 ∧ 0 ≀ π‘Œ)) β†’ 0 ≀ π‘Œ)
3 simp2l 1199 . . 3 ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐡 ∧ 0 ≀ 𝑋) ∧ (π‘Œ ∈ 𝐡 ∧ 0 ≀ π‘Œ)) β†’ 𝑋 ∈ 𝐡)
4 simp3l 1201 . . 3 ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐡 ∧ 0 ≀ 𝑋) ∧ (π‘Œ ∈ 𝐡 ∧ 0 ≀ π‘Œ)) β†’ π‘Œ ∈ 𝐡)
5 orngmul.0 . . . . . 6 𝐡 = (Baseβ€˜π‘…)
6 orngmul.2 . . . . . 6 0 = (0gβ€˜π‘…)
7 orngmul.3 . . . . . 6 Β· = (.rβ€˜π‘…)
8 orngmul.1 . . . . . 6 ≀ = (leβ€˜π‘…)
95, 6, 7, 8isorng 32675 . . . . 5 (𝑅 ∈ oRing ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 (( 0 ≀ π‘Ž ∧ 0 ≀ 𝑏) β†’ 0 ≀ (π‘Ž Β· 𝑏))))
109simp3bi 1147 . . . 4 (𝑅 ∈ oRing β†’ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 (( 0 ≀ π‘Ž ∧ 0 ≀ 𝑏) β†’ 0 ≀ (π‘Ž Β· 𝑏)))
11103ad2ant1 1133 . . 3 ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐡 ∧ 0 ≀ 𝑋) ∧ (π‘Œ ∈ 𝐡 ∧ 0 ≀ π‘Œ)) β†’ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 (( 0 ≀ π‘Ž ∧ 0 ≀ 𝑏) β†’ 0 ≀ (π‘Ž Β· 𝑏)))
12 breq2 5152 . . . . . 6 (π‘Ž = 𝑋 β†’ ( 0 ≀ π‘Ž ↔ 0 ≀ 𝑋))
1312anbi1d 630 . . . . 5 (π‘Ž = 𝑋 β†’ (( 0 ≀ π‘Ž ∧ 0 ≀ 𝑏) ↔ ( 0 ≀ 𝑋 ∧ 0 ≀ 𝑏)))
14 oveq1 7418 . . . . . 6 (π‘Ž = 𝑋 β†’ (π‘Ž Β· 𝑏) = (𝑋 Β· 𝑏))
1514breq2d 5160 . . . . 5 (π‘Ž = 𝑋 β†’ ( 0 ≀ (π‘Ž Β· 𝑏) ↔ 0 ≀ (𝑋 Β· 𝑏)))
1613, 15imbi12d 344 . . . 4 (π‘Ž = 𝑋 β†’ ((( 0 ≀ π‘Ž ∧ 0 ≀ 𝑏) β†’ 0 ≀ (π‘Ž Β· 𝑏)) ↔ (( 0 ≀ 𝑋 ∧ 0 ≀ 𝑏) β†’ 0 ≀ (𝑋 Β· 𝑏))))
17 breq2 5152 . . . . . 6 (𝑏 = π‘Œ β†’ ( 0 ≀ 𝑏 ↔ 0 ≀ π‘Œ))
1817anbi2d 629 . . . . 5 (𝑏 = π‘Œ β†’ (( 0 ≀ 𝑋 ∧ 0 ≀ 𝑏) ↔ ( 0 ≀ 𝑋 ∧ 0 ≀ π‘Œ)))
19 oveq2 7419 . . . . . 6 (𝑏 = π‘Œ β†’ (𝑋 Β· 𝑏) = (𝑋 Β· π‘Œ))
2019breq2d 5160 . . . . 5 (𝑏 = π‘Œ β†’ ( 0 ≀ (𝑋 Β· 𝑏) ↔ 0 ≀ (𝑋 Β· π‘Œ)))
2118, 20imbi12d 344 . . . 4 (𝑏 = π‘Œ β†’ ((( 0 ≀ 𝑋 ∧ 0 ≀ 𝑏) β†’ 0 ≀ (𝑋 Β· 𝑏)) ↔ (( 0 ≀ 𝑋 ∧ 0 ≀ π‘Œ) β†’ 0 ≀ (𝑋 Β· π‘Œ))))
2216, 21rspc2va 3623 . . 3 (((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 (( 0 ≀ π‘Ž ∧ 0 ≀ 𝑏) β†’ 0 ≀ (π‘Ž Β· 𝑏))) β†’ (( 0 ≀ 𝑋 ∧ 0 ≀ π‘Œ) β†’ 0 ≀ (𝑋 Β· π‘Œ)))
233, 4, 11, 22syl21anc 836 . 2 ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐡 ∧ 0 ≀ 𝑋) ∧ (π‘Œ ∈ 𝐡 ∧ 0 ≀ π‘Œ)) β†’ (( 0 ≀ 𝑋 ∧ 0 ≀ π‘Œ) β†’ 0 ≀ (𝑋 Β· π‘Œ)))
241, 2, 23mp2and 697 1 ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐡 ∧ 0 ≀ 𝑋) ∧ (π‘Œ ∈ 𝐡 ∧ 0 ≀ π‘Œ)) β†’ 0 ≀ (𝑋 Β· π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7411  Basecbs 17148  .rcmulr 17202  lecple 17208  0gc0g 17389  Ringcrg 20127  oGrpcogrp 32474  oRingcorng 32671
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 2703  ax-nul 5306
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 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7414  df-orng 32673
This theorem is referenced by:  orngsqr  32680  ornglmulle  32681  orngrmulle  32682  orngmullt  32685  suborng  32691
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