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Theorem omndadd 32662
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 + = (+g𝑀)
Assertion
Ref Expression
omndadd ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑋 + 𝑍) (𝑌 + 𝑍))

Proof of Theorem omndadd
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omndadd.0 . . . . 5 𝐵 = (Base‘𝑀)
2 omndadd.2 . . . . 5 + = (+g𝑀)
3 omndadd.1 . . . . 5 = (le‘𝑀)
41, 2, 3isomnd 32657 . . . 4 (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐))))
54simp3bi 1146 . . 3 (𝑀 ∈ oMnd → ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐)))
6 breq1 5151 . . . . 5 (𝑎 = 𝑋 → (𝑎 𝑏𝑋 𝑏))
7 oveq1 7419 . . . . . 6 (𝑎 = 𝑋 → (𝑎 + 𝑐) = (𝑋 + 𝑐))
87breq1d 5158 . . . . 5 (𝑎 = 𝑋 → ((𝑎 + 𝑐) (𝑏 + 𝑐) ↔ (𝑋 + 𝑐) (𝑏 + 𝑐)))
96, 8imbi12d 344 . . . 4 (𝑎 = 𝑋 → ((𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐)) ↔ (𝑋 𝑏 → (𝑋 + 𝑐) (𝑏 + 𝑐))))
10 breq2 5152 . . . . 5 (𝑏 = 𝑌 → (𝑋 𝑏𝑋 𝑌))
11 oveq1 7419 . . . . . 6 (𝑏 = 𝑌 → (𝑏 + 𝑐) = (𝑌 + 𝑐))
1211breq2d 5160 . . . . 5 (𝑏 = 𝑌 → ((𝑋 + 𝑐) (𝑏 + 𝑐) ↔ (𝑋 + 𝑐) (𝑌 + 𝑐)))
1310, 12imbi12d 344 . . . 4 (𝑏 = 𝑌 → ((𝑋 𝑏 → (𝑋 + 𝑐) (𝑏 + 𝑐)) ↔ (𝑋 𝑌 → (𝑋 + 𝑐) (𝑌 + 𝑐))))
14 oveq2 7420 . . . . . 6 (𝑐 = 𝑍 → (𝑋 + 𝑐) = (𝑋 + 𝑍))
15 oveq2 7420 . . . . . 6 (𝑐 = 𝑍 → (𝑌 + 𝑐) = (𝑌 + 𝑍))
1614, 15breq12d 5161 . . . . 5 (𝑐 = 𝑍 → ((𝑋 + 𝑐) (𝑌 + 𝑐) ↔ (𝑋 + 𝑍) (𝑌 + 𝑍)))
1716imbi2d 340 . . . 4 (𝑐 = 𝑍 → ((𝑋 𝑌 → (𝑋 + 𝑐) (𝑌 + 𝑐)) ↔ (𝑋 𝑌 → (𝑋 + 𝑍) (𝑌 + 𝑍))))
189, 13, 17rspc3v 3627 . . 3 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐)) → (𝑋 𝑌 → (𝑋 + 𝑍) (𝑌 + 𝑍))))
195, 18mpan9 506 . 2 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 → (𝑋 + 𝑍) (𝑌 + 𝑍)))
20193impia 1116 1 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑋 + 𝑍) (𝑌 + 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1540  wcel 2105  wral 3060   class class class wbr 5148  cfv 6543  (class class class)co 7412  Basecbs 17151  +gcplusg 17204  lecple 17211  Tosetctos 18379  Mndcmnd 18665  oMndcomnd 32653
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7415  df-omnd 32655
This theorem is referenced by:  omndaddr  32663  omndadd2d  32664  omndadd2rd  32665  submomnd  32666  omndmul2  32668  omndmul3  32669  ogrpinv0le  32671  ogrpsub  32672  ogrpaddlt  32673  orngsqr  32860  ornglmulle  32861  orngrmulle  32862
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