Theorem omndadd 20042

Description: In an ordered monoid, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 30-Jan-2018.)

Hypotheses

Ref	Expression
omndadd.0	⊢ 𝐵 = (Base‘𝑀)
omndadd.1	⊢ ≤ = (le‘𝑀)
omndadd.2	⊢ + = (+g‘𝑀)

Assertion

Ref	Expression
omndadd	⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 + 𝑍) ≤ (𝑌 + 𝑍))

Proof of Theorem omndadd

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

Step	Hyp	Ref	Expression
1		omndadd.0	. . . . 5 ⊢ 𝐵 = (Base‘𝑀)
2		omndadd.2	. . . . 5 ⊢ + = (+g‘𝑀)
3		omndadd.1	. . . . 5 ⊢ ≤ = (le‘𝑀)
4	1, 2, 3	isomnd 20037	. . . 4 ⊢ (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))))
5	4	simp3bi 1147	. . 3 ⊢ (𝑀 ∈ oMnd → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)))
6		breq1 5096	. . . . 5 ⊢ (𝑎 = 𝑋 → (𝑎 ≤ 𝑏 ↔ 𝑋 ≤ 𝑏))
7		oveq1 7359	. . . . . 6 ⊢ (𝑎 = 𝑋 → (𝑎 + 𝑐) = (𝑋 + 𝑐))
8	7	breq1d 5103	. . . . 5 ⊢ (𝑎 = 𝑋 → ((𝑎 + 𝑐) ≤ (𝑏 + 𝑐) ↔ (𝑋 + 𝑐) ≤ (𝑏 + 𝑐)))
9	6, 8	imbi12d 344	. . . 4 ⊢ (𝑎 = 𝑋 → ((𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)) ↔ (𝑋 ≤ 𝑏 → (𝑋 + 𝑐) ≤ (𝑏 + 𝑐))))
10		breq2 5097	. . . . 5 ⊢ (𝑏 = 𝑌 → (𝑋 ≤ 𝑏 ↔ 𝑋 ≤ 𝑌))
11		oveq1 7359	. . . . . 6 ⊢ (𝑏 = 𝑌 → (𝑏 + 𝑐) = (𝑌 + 𝑐))
12	11	breq2d 5105	. . . . 5 ⊢ (𝑏 = 𝑌 → ((𝑋 + 𝑐) ≤ (𝑏 + 𝑐) ↔ (𝑋 + 𝑐) ≤ (𝑌 + 𝑐)))
13	10, 12	imbi12d 344	. . . 4 ⊢ (𝑏 = 𝑌 → ((𝑋 ≤ 𝑏 → (𝑋 + 𝑐) ≤ (𝑏 + 𝑐)) ↔ (𝑋 ≤ 𝑌 → (𝑋 + 𝑐) ≤ (𝑌 + 𝑐))))
14		oveq2 7360	. . . . . 6 ⊢ (𝑐 = 𝑍 → (𝑋 + 𝑐) = (𝑋 + 𝑍))
15		oveq2 7360	. . . . . 6 ⊢ (𝑐 = 𝑍 → (𝑌 + 𝑐) = (𝑌 + 𝑍))
16	14, 15	breq12d 5106	. . . . 5 ⊢ (𝑐 = 𝑍 → ((𝑋 + 𝑐) ≤ (𝑌 + 𝑐) ↔ (𝑋 + 𝑍) ≤ (𝑌 + 𝑍)))
17	16	imbi2d 340	. . . 4 ⊢ (𝑐 = 𝑍 → ((𝑋 ≤ 𝑌 → (𝑋 + 𝑐) ≤ (𝑌 + 𝑐)) ↔ (𝑋 ≤ 𝑌 → (𝑋 + 𝑍) ≤ (𝑌 + 𝑍))))
18	9, 13, 17	rspc3v 3589	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)) → (𝑋 ≤ 𝑌 → (𝑋 + 𝑍) ≤ (𝑌 + 𝑍))))
19	5, 18	mpan9 506	. 2 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑌 → (𝑋 + 𝑍) ≤ (𝑌 + 𝑍)))
20	19	3impia 1117	1 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 + 𝑍) ≤ (𝑌 + 𝑍))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3048   class class class wbr 5093  ‘cfv 6486  (class class class)co 7352  Basecbs 17122  +_gcplusg 17163  lecple 17170  Tosetctos 18322  Mndcmnd 18644  oMndcomnd 20033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-nul 5246

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-iota 6442  df-fv 6494  df-ov 7355  df-omnd 20035

This theorem is referenced by:  omndaddr  20043  omndadd2d  20044  omndadd2rd  20045  submomnd  20046  omndmul2  20047  omndmul3  20048  ogrpinv0le  20050  ogrpsub  20051  ogrpaddlt  20052  orngsqr  20783  ornglmulle  20784  orngrmulle  20785