Theorem orngmul 20979

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 1218	. 2 ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → 0 ≤ 𝑋)
2		simp3r 1220	. 2 ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → 0 ≤ 𝑌)
3		simp2l 1217	. . 3 ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → 𝑋 ∈ 𝐵)
4		simp3l 1219	. . 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 20975	. . . . 5 ⊢ (𝑅 ∈ oRing ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏))))
10	9	simp3bi 1164	. . . 4 ⊢ (𝑅 ∈ oRing → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏)))
11	10	3ad2ant1 1150	. . 3 ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏)))
12		breq2 5112	. . . . . 6 ⊢ (𝑎 = 𝑋 → ( 0 ≤ 𝑎 ↔ 0 ≤ 𝑋))
13	12	anbi1d 642	. . . . 5 ⊢ (𝑎 = 𝑋 → (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) ↔ ( 0 ≤ 𝑋 ∧ 0 ≤ 𝑏)))
14		oveq1 7419	. . . . . 6 ⊢ (𝑎 = 𝑋 → (𝑎 · 𝑏) = (𝑋 · 𝑏))
15	14	breq2d 5120	. . . . 5 ⊢ (𝑎 = 𝑋 → ( 0 ≤ (𝑎 · 𝑏) ↔ 0 ≤ (𝑋 · 𝑏)))
16	13, 15	imbi12d 347	. . . 4 ⊢ (𝑎 = 𝑋 → ((( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏)) ↔ (( 0 ≤ 𝑋 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑋 · 𝑏))))
17		breq2 5112	. . . . . 6 ⊢ (𝑏 = 𝑌 → ( 0 ≤ 𝑏 ↔ 0 ≤ 𝑌))
18	17	anbi2d 641	. . . . 5 ⊢ (𝑏 = 𝑌 → (( 0 ≤ 𝑋 ∧ 0 ≤ 𝑏) ↔ ( 0 ≤ 𝑋 ∧ 0 ≤ 𝑌)))
19		oveq2 7420	. . . . . 6 ⊢ (𝑏 = 𝑌 → (𝑋 · 𝑏) = (𝑋 · 𝑌))
20	19	breq2d 5120	. . . . 5 ⊢ (𝑏 = 𝑌 → ( 0 ≤ (𝑋 · 𝑏) ↔ 0 ≤ (𝑋 · 𝑌)))
21	18, 20	imbi12d 347	. . . 4 ⊢ (𝑏 = 𝑌 → ((( 0 ≤ 𝑋 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑋 · 𝑏)) ↔ (( 0 ≤ 𝑋 ∧ 0 ≤ 𝑌) → 0 ≤ (𝑋 · 𝑌))))
22	16, 21	rspc2va 3592	. . 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 850	. 2 ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → (( 0 ≤ 𝑋 ∧ 0 ≤ 𝑌) → 0 ≤ (𝑋 · 𝑌)))
24	1, 2, 23	mp2and 711	1 ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → 0 ≤ (𝑋 · 𝑌))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  ∀wral 3078   class class class wbr 5108  ‘cfv 6536  (class class class)co 7412  Basecbs 17275  ._rcmulr 17317  lecple 17323  0_gc0g 17498  oGrpcogrp 20196  Ringcrg 20321  oRingcorng 20971

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-nul 5268

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-iota 6492  df-fv 6544  df-ov 7415  df-orng 20973

This theorem is used by:  orngsqr  20980  ornglmulle  20981  orngrmulle  20982  orngmullt  20985  suborng  20990