Theorem orngmul 31175

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 = (0g‘𝑅)
orngmul.3	⊢ · = (.r‘𝑅)

Assertion

Ref	Expression
orngmul	⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → 0 ≤ (𝑋 · 𝑌))

Proof of Theorem orngmul

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2r 1202	. 2 ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → 0 ≤ 𝑋)
2		simp3r 1204	. 2 ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → 0 ≤ 𝑌)
3		simp2l 1201	. . 3 ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → 𝑋 ∈ 𝐵)
4		simp3l 1203	. . 3 ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → 𝑌 ∈ 𝐵)
5		orngmul.0	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
6		orngmul.2	. . . . . 6 ⊢ 0 = (0g‘𝑅)
7		orngmul.3	. . . . . 6 ⊢ · = (.r‘𝑅)
8		orngmul.1	. . . . . 6 ⊢ ≤ = (le‘𝑅)
9	5, 6, 7, 8	isorng 31171	. . . . 5 ⊢ (𝑅 ∈ oRing ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏))))
10	9	simp3bi 1149	. . . 4 ⊢ (𝑅 ∈ oRing → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏)))
11	10	3ad2ant1 1135	. . 3 ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏)))
12		breq2 5043	. . . . . 6 ⊢ (𝑎 = 𝑋 → ( 0 ≤ 𝑎 ↔ 0 ≤ 𝑋))
13	12	anbi1d 633	. . . . 5 ⊢ (𝑎 = 𝑋 → (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) ↔ ( 0 ≤ 𝑋 ∧ 0 ≤ 𝑏)))
14		oveq1 7198	. . . . . 6 ⊢ (𝑎 = 𝑋 → (𝑎 · 𝑏) = (𝑋 · 𝑏))
15	14	breq2d 5051	. . . . 5 ⊢ (𝑎 = 𝑋 → ( 0 ≤ (𝑎 · 𝑏) ↔ 0 ≤ (𝑋 · 𝑏)))
16	13, 15	imbi12d 348	. . . 4 ⊢ (𝑎 = 𝑋 → ((( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏)) ↔ (( 0 ≤ 𝑋 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑋 · 𝑏))))
17		breq2 5043	. . . . . 6 ⊢ (𝑏 = 𝑌 → ( 0 ≤ 𝑏 ↔ 0 ≤ 𝑌))
18	17	anbi2d 632	. . . . 5 ⊢ (𝑏 = 𝑌 → (( 0 ≤ 𝑋 ∧ 0 ≤ 𝑏) ↔ ( 0 ≤ 𝑋 ∧ 0 ≤ 𝑌)))
19		oveq2 7199	. . . . . 6 ⊢ (𝑏 = 𝑌 → (𝑋 · 𝑏) = (𝑋 · 𝑌))
20	19	breq2d 5051	. . . . 5 ⊢ (𝑏 = 𝑌 → ( 0 ≤ (𝑋 · 𝑏) ↔ 0 ≤ (𝑋 · 𝑌)))
21	18, 20	imbi12d 348	. . . 4 ⊢ (𝑏 = 𝑌 → ((( 0 ≤ 𝑋 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑋 · 𝑏)) ↔ (( 0 ≤ 𝑋 ∧ 0 ≤ 𝑌) → 0 ≤ (𝑋 · 𝑌))))
22	16, 21	rspc2va 3538	. . 3 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏))) → (( 0 ≤ 𝑋 ∧ 0 ≤ 𝑌) → 0 ≤ (𝑋 · 𝑌)))
23	3, 4, 11, 22	syl21anc 838	. 2 ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → (( 0 ≤ 𝑋 ∧ 0 ≤ 𝑌) → 0 ≤ (𝑋 · 𝑌)))
24	1, 2, 23	mp2and 699	1 ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → 0 ≤ (𝑋 · 𝑌))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112  ∀wral 3051   class class class wbr 5039  ‘cfv 6358  (class class class)co 7191  Basecbs 16666  ._rcmulr 16750  lecple 16756  0_gc0g 16898  Ringcrg 19516  oGrpcogrp 30997  oRingcorng 31167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-nul 5184

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-iota 6316  df-fv 6366  df-ov 7194  df-orng 31169

This theorem is referenced by:  orngsqr  31176  ornglmulle  31177  orngrmulle  31178  orngmullt  31181  suborng  31187