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Theorem omndadd 32807
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 32802 . . . 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 1144 . . 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 5155 . . . . 5 (π‘Ž = 𝑋 β†’ (π‘Ž ≀ 𝑏 ↔ 𝑋 ≀ 𝑏))
7 oveq1 7433 . . . . . 6 (π‘Ž = 𝑋 β†’ (π‘Ž + 𝑐) = (𝑋 + 𝑐))
87breq1d 5162 . . . . 5 (π‘Ž = 𝑋 β†’ ((π‘Ž + 𝑐) ≀ (𝑏 + 𝑐) ↔ (𝑋 + 𝑐) ≀ (𝑏 + 𝑐)))
96, 8imbi12d 343 . . . 4 (π‘Ž = 𝑋 β†’ ((π‘Ž ≀ 𝑏 β†’ (π‘Ž + 𝑐) ≀ (𝑏 + 𝑐)) ↔ (𝑋 ≀ 𝑏 β†’ (𝑋 + 𝑐) ≀ (𝑏 + 𝑐))))
10 breq2 5156 . . . . 5 (𝑏 = π‘Œ β†’ (𝑋 ≀ 𝑏 ↔ 𝑋 ≀ π‘Œ))
11 oveq1 7433 . . . . . 6 (𝑏 = π‘Œ β†’ (𝑏 + 𝑐) = (π‘Œ + 𝑐))
1211breq2d 5164 . . . . 5 (𝑏 = π‘Œ β†’ ((𝑋 + 𝑐) ≀ (𝑏 + 𝑐) ↔ (𝑋 + 𝑐) ≀ (π‘Œ + 𝑐)))
1310, 12imbi12d 343 . . . 4 (𝑏 = π‘Œ β†’ ((𝑋 ≀ 𝑏 β†’ (𝑋 + 𝑐) ≀ (𝑏 + 𝑐)) ↔ (𝑋 ≀ π‘Œ β†’ (𝑋 + 𝑐) ≀ (π‘Œ + 𝑐))))
14 oveq2 7434 . . . . . 6 (𝑐 = 𝑍 β†’ (𝑋 + 𝑐) = (𝑋 + 𝑍))
15 oveq2 7434 . . . . . 6 (𝑐 = 𝑍 β†’ (π‘Œ + 𝑐) = (π‘Œ + 𝑍))
1614, 15breq12d 5165 . . . . 5 (𝑐 = 𝑍 β†’ ((𝑋 + 𝑐) ≀ (π‘Œ + 𝑐) ↔ (𝑋 + 𝑍) ≀ (π‘Œ + 𝑍)))
1716imbi2d 339 . . . 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 3627 . . 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 505 . 2 ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝑋 ≀ π‘Œ β†’ (𝑋 + 𝑍) ≀ (π‘Œ + 𝑍)))
20193impia 1114 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 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058   class class class wbr 5152  β€˜cfv 6553  (class class class)co 7426  Basecbs 17187  +gcplusg 17240  lecple 17247  Tosetctos 18415  Mndcmnd 18701  oMndcomnd 32798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-nul 5310
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7429  df-omnd 32800
This theorem is referenced by:  omndaddr  32808  omndadd2d  32809  omndadd2rd  32810  submomnd  32811  omndmul2  32813  omndmul3  32814  ogrpinv0le  32816  ogrpsub  32817  ogrpaddlt  32818  orngsqr  33043  ornglmulle  33044  orngrmulle  33045
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