Theorem orngmul 30369

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 1214	. 2 ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → 0 ≤ 𝑋)
2		simp3r 1216	. 2 ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → 0 ≤ 𝑌)
3		simp2l 1213	. . 3 ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → 𝑋 ∈ 𝐵)
4		simp3l 1215	. . 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 30365	. . . . 5 ⊢ (𝑅 ∈ oRing ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏))))
10	9	simp3bi 1138	. . . 4 ⊢ (𝑅 ∈ oRing → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏)))
11	10	3ad2ant1 1124	. . 3 ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏)))
12		breq2 4892	. . . . . 6 ⊢ (𝑎 = 𝑋 → ( 0 ≤ 𝑎 ↔ 0 ≤ 𝑋))
13	12	anbi1d 623	. . . . 5 ⊢ (𝑎 = 𝑋 → (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) ↔ ( 0 ≤ 𝑋 ∧ 0 ≤ 𝑏)))
14		oveq1 6931	. . . . . 6 ⊢ (𝑎 = 𝑋 → (𝑎 · 𝑏) = (𝑋 · 𝑏))
15	14	breq2d 4900	. . . . 5 ⊢ (𝑎 = 𝑋 → ( 0 ≤ (𝑎 · 𝑏) ↔ 0 ≤ (𝑋 · 𝑏)))
16	13, 15	imbi12d 336	. . . 4 ⊢ (𝑎 = 𝑋 → ((( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏)) ↔ (( 0 ≤ 𝑋 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑋 · 𝑏))))
17		breq2 4892	. . . . . 6 ⊢ (𝑏 = 𝑌 → ( 0 ≤ 𝑏 ↔ 0 ≤ 𝑌))
18	17	anbi2d 622	. . . . 5 ⊢ (𝑏 = 𝑌 → (( 0 ≤ 𝑋 ∧ 0 ≤ 𝑏) ↔ ( 0 ≤ 𝑋 ∧ 0 ≤ 𝑌)))
19		oveq2 6932	. . . . . 6 ⊢ (𝑏 = 𝑌 → (𝑋 · 𝑏) = (𝑋 · 𝑌))
20	19	breq2d 4900	. . . . 5 ⊢ (𝑏 = 𝑌 → ( 0 ≤ (𝑋 · 𝑏) ↔ 0 ≤ (𝑋 · 𝑌)))
21	18, 20	imbi12d 336	. . . 4 ⊢ (𝑏 = 𝑌 → ((( 0 ≤ 𝑋 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑋 · 𝑏)) ↔ (( 0 ≤ 𝑋 ∧ 0 ≤ 𝑌) → 0 ≤ (𝑋 · 𝑌))))
22	16, 21	rspc2va 3525	. . 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 828	. 2 ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → (( 0 ≤ 𝑋 ∧ 0 ≤ 𝑌) → 0 ≤ (𝑋 · 𝑌)))
24	1, 2, 23	mp2and 689	1 ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → 0 ≤ (𝑋 · 𝑌))

Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  ∀wral 3090   class class class wbr 4888  ‘cfv 6137  (class class class)co 6924  Basecbs 16259  ._rcmulr 16343  lecple 16349  0_gc0g 16490  Ringcrg 18938  oGrpcogrp 30264  oRingcorng 30361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-nul 5027

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-iota 6101  df-fv 6145  df-ov 6927  df-orng 30363

This theorem is referenced by:  orngsqr  30370  ornglmulle  30371  orngrmulle  30372  orngmullt  30375  suborng  30381