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Theorem omndadd 31963
Description: In an ordered monoid, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 30-Jan-2018.)
Hypotheses
Ref Expression
omndadd.0 𝐡 = (Baseβ€˜π‘€)
omndadd.1 ≀ = (leβ€˜π‘€)
omndadd.2 + = (+gβ€˜π‘€)
Assertion
Ref Expression
omndadd ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) ∧ 𝑋 ≀ π‘Œ) β†’ (𝑋 + 𝑍) ≀ (π‘Œ + 𝑍))

Proof of Theorem omndadd
Dummy variables π‘Ž 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omndadd.0 . . . . 5 𝐡 = (Baseβ€˜π‘€)
2 omndadd.2 . . . . 5 + = (+gβ€˜π‘€)
3 omndadd.1 . . . . 5 ≀ = (leβ€˜π‘€)
41, 2, 3isomnd 31958 . . . 4 (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 βˆ€π‘ ∈ 𝐡 (π‘Ž ≀ 𝑏 β†’ (π‘Ž + 𝑐) ≀ (𝑏 + 𝑐))))
54simp3bi 1148 . . 3 (𝑀 ∈ oMnd β†’ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 βˆ€π‘ ∈ 𝐡 (π‘Ž ≀ 𝑏 β†’ (π‘Ž + 𝑐) ≀ (𝑏 + 𝑐)))
6 breq1 5109 . . . . 5 (π‘Ž = 𝑋 β†’ (π‘Ž ≀ 𝑏 ↔ 𝑋 ≀ 𝑏))
7 oveq1 7365 . . . . . 6 (π‘Ž = 𝑋 β†’ (π‘Ž + 𝑐) = (𝑋 + 𝑐))
87breq1d 5116 . . . . 5 (π‘Ž = 𝑋 β†’ ((π‘Ž + 𝑐) ≀ (𝑏 + 𝑐) ↔ (𝑋 + 𝑐) ≀ (𝑏 + 𝑐)))
96, 8imbi12d 345 . . . 4 (π‘Ž = 𝑋 β†’ ((π‘Ž ≀ 𝑏 β†’ (π‘Ž + 𝑐) ≀ (𝑏 + 𝑐)) ↔ (𝑋 ≀ 𝑏 β†’ (𝑋 + 𝑐) ≀ (𝑏 + 𝑐))))
10 breq2 5110 . . . . 5 (𝑏 = π‘Œ β†’ (𝑋 ≀ 𝑏 ↔ 𝑋 ≀ π‘Œ))
11 oveq1 7365 . . . . . 6 (𝑏 = π‘Œ β†’ (𝑏 + 𝑐) = (π‘Œ + 𝑐))
1211breq2d 5118 . . . . 5 (𝑏 = π‘Œ β†’ ((𝑋 + 𝑐) ≀ (𝑏 + 𝑐) ↔ (𝑋 + 𝑐) ≀ (π‘Œ + 𝑐)))
1310, 12imbi12d 345 . . . 4 (𝑏 = π‘Œ β†’ ((𝑋 ≀ 𝑏 β†’ (𝑋 + 𝑐) ≀ (𝑏 + 𝑐)) ↔ (𝑋 ≀ π‘Œ β†’ (𝑋 + 𝑐) ≀ (π‘Œ + 𝑐))))
14 oveq2 7366 . . . . . 6 (𝑐 = 𝑍 β†’ (𝑋 + 𝑐) = (𝑋 + 𝑍))
15 oveq2 7366 . . . . . 6 (𝑐 = 𝑍 β†’ (π‘Œ + 𝑐) = (π‘Œ + 𝑍))
1614, 15breq12d 5119 . . . . 5 (𝑐 = 𝑍 β†’ ((𝑋 + 𝑐) ≀ (π‘Œ + 𝑐) ↔ (𝑋 + 𝑍) ≀ (π‘Œ + 𝑍)))
1716imbi2d 341 . . . 4 (𝑐 = 𝑍 β†’ ((𝑋 ≀ π‘Œ β†’ (𝑋 + 𝑐) ≀ (π‘Œ + 𝑐)) ↔ (𝑋 ≀ π‘Œ β†’ (𝑋 + 𝑍) ≀ (π‘Œ + 𝑍))))
189, 13, 17rspc3v 3592 . . 3 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) β†’ (βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 βˆ€π‘ ∈ 𝐡 (π‘Ž ≀ 𝑏 β†’ (π‘Ž + 𝑐) ≀ (𝑏 + 𝑐)) β†’ (𝑋 ≀ π‘Œ β†’ (𝑋 + 𝑍) ≀ (π‘Œ + 𝑍))))
195, 18mpan9 508 . 2 ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝑋 ≀ π‘Œ β†’ (𝑋 + 𝑍) ≀ (π‘Œ + 𝑍)))
20193impia 1118 1 ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) ∧ 𝑋 ≀ π‘Œ) β†’ (𝑋 + 𝑍) ≀ (π‘Œ + 𝑍))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  lecple 17145  Tosetctos 18310  Mndcmnd 18561  oMndcomnd 31954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5264
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361  df-omnd 31956
This theorem is referenced by:  omndaddr  31964  omndadd2d  31965  omndadd2rd  31966  submomnd  31967  omndmul2  31969  omndmul3  31970  ogrpinv0le  31972  ogrpsub  31973  ogrpaddlt  31974  orngsqr  32146  ornglmulle  32147  orngrmulle  32148
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