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Theorem omndadd 33066
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 33061 . . . 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 7438 . . . . . 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 7438 . . . . . 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 7439 . . . . . 6 (𝑐 = 𝑍 → (𝑋 + 𝑐) = (𝑋 + 𝑍))
15 oveq2 7439 . . . . . 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 3638 . . 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 1537  wcel 2106  wral 3059   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  lecple 17305  Tosetctos 18474  Mndcmnd 18760  oMndcomnd 33057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-omnd 33059
This theorem is referenced by:  omndaddr  33067  omndadd2d  33068  omndadd2rd  33069  submomnd  33070  omndmul2  33072  omndmul3  33073  ogrpinv0le  33075  ogrpsub  33076  ogrpaddlt  33077  orngsqr  33314  ornglmulle  33315  orngrmulle  33316
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