Theorem orngmul 20883

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 1210	. 2 ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → 0 ≤ 𝑋)
2		simp3r 1212	. 2 ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → 0 ≤ 𝑌)
3		simp2l 1209	. . 3 ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → 𝑋 ∈ 𝐵)
4		simp3l 1211	. . 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 20879	. . . . 5 ⊢ (𝑅 ∈ oRing ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏))))
10	9	simp3bi 1156	. . . 4 ⊢ (𝑅 ∈ oRing → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏)))
11	10	3ad2ant1 1142	. . 3 ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏)))
12		breq2 5094	. . . . . 6 ⊢ (𝑎 = 𝑋 → ( 0 ≤ 𝑎 ↔ 0 ≤ 𝑋))
13	12	anbi1d 639	. . . . 5 ⊢ (𝑎 = 𝑋 → (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) ↔ ( 0 ≤ 𝑋 ∧ 0 ≤ 𝑏)))
14		oveq1 7388	. . . . . 6 ⊢ (𝑎 = 𝑋 → (𝑎 · 𝑏) = (𝑋 · 𝑏))
15	14	breq2d 5102	. . . . 5 ⊢ (𝑎 = 𝑋 → ( 0 ≤ (𝑎 · 𝑏) ↔ 0 ≤ (𝑋 · 𝑏)))
16	13, 15	imbi12d 346	. . . 4 ⊢ (𝑎 = 𝑋 → ((( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏)) ↔ (( 0 ≤ 𝑋 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑋 · 𝑏))))
17		breq2 5094	. . . . . 6 ⊢ (𝑏 = 𝑌 → ( 0 ≤ 𝑏 ↔ 0 ≤ 𝑌))
18	17	anbi2d 638	. . . . 5 ⊢ (𝑏 = 𝑌 → (( 0 ≤ 𝑋 ∧ 0 ≤ 𝑏) ↔ ( 0 ≤ 𝑋 ∧ 0 ≤ 𝑌)))
19		oveq2 7389	. . . . . 6 ⊢ (𝑏 = 𝑌 → (𝑋 · 𝑏) = (𝑋 · 𝑌))
20	19	breq2d 5102	. . . . 5 ⊢ (𝑏 = 𝑌 → ( 0 ≤ (𝑋 · 𝑏) ↔ 0 ≤ (𝑋 · 𝑌)))
21	18, 20	imbi12d 346	. . . 4 ⊢ (𝑏 = 𝑌 → ((( 0 ≤ 𝑋 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑋 · 𝑏)) ↔ (( 0 ≤ 𝑋 ∧ 0 ≤ 𝑌) → 0 ≤ (𝑋 · 𝑌))))
22	16, 21	rspc2va 3584	. . 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 846	. 2 ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → (( 0 ≤ 𝑋 ∧ 0 ≤ 𝑌) → 0 ≤ (𝑋 · 𝑌)))
24	1, 2, 23	mp2and 707	1 ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → 0 ≤ (𝑋 · 𝑌))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1095   = wceq 1550   ∈ wcel 2132  ∀wral 3066   class class class wbr 5090  ‘cfv 6506  (class class class)co 7381  Basecbs 17217  ._rcmulr 17259  lecple 17265  0_gc0g 17440  oGrpcogrp 20132  Ringcrg 20251  oRingcorng 20875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-ext 2724  ax-nul 5246

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-ne 2948  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-sbc 3736  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-br 5091  df-iota 6462  df-fv 6514  df-ov 7384  df-orng 20877

This theorem is referenced by:  orngsqr  20884  ornglmulle  20885  orngrmulle  20886  orngmullt  20889  suborng  20894