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Theorem omndadd 29515
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 29510 . . . 4 (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐))))
54simp3bi 1076 . . 3 (𝑀 ∈ oMnd → ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐)))
6 breq1 4621 . . . . 5 (𝑎 = 𝑋 → (𝑎 𝑏𝑋 𝑏))
7 oveq1 6617 . . . . . 6 (𝑎 = 𝑋 → (𝑎 + 𝑐) = (𝑋 + 𝑐))
87breq1d 4628 . . . . 5 (𝑎 = 𝑋 → ((𝑎 + 𝑐) (𝑏 + 𝑐) ↔ (𝑋 + 𝑐) (𝑏 + 𝑐)))
96, 8imbi12d 334 . . . 4 (𝑎 = 𝑋 → ((𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐)) ↔ (𝑋 𝑏 → (𝑋 + 𝑐) (𝑏 + 𝑐))))
10 breq2 4622 . . . . 5 (𝑏 = 𝑌 → (𝑋 𝑏𝑋 𝑌))
11 oveq1 6617 . . . . . 6 (𝑏 = 𝑌 → (𝑏 + 𝑐) = (𝑌 + 𝑐))
1211breq2d 4630 . . . . 5 (𝑏 = 𝑌 → ((𝑋 + 𝑐) (𝑏 + 𝑐) ↔ (𝑋 + 𝑐) (𝑌 + 𝑐)))
1310, 12imbi12d 334 . . . 4 (𝑏 = 𝑌 → ((𝑋 𝑏 → (𝑋 + 𝑐) (𝑏 + 𝑐)) ↔ (𝑋 𝑌 → (𝑋 + 𝑐) (𝑌 + 𝑐))))
14 oveq2 6618 . . . . . 6 (𝑐 = 𝑍 → (𝑋 + 𝑐) = (𝑋 + 𝑍))
15 oveq2 6618 . . . . . 6 (𝑐 = 𝑍 → (𝑌 + 𝑐) = (𝑌 + 𝑍))
1614, 15breq12d 4631 . . . . 5 (𝑐 = 𝑍 → ((𝑋 + 𝑐) (𝑌 + 𝑐) ↔ (𝑋 + 𝑍) (𝑌 + 𝑍)))
1716imbi2d 330 . . . 4 (𝑐 = 𝑍 → ((𝑋 𝑌 → (𝑋 + 𝑐) (𝑌 + 𝑐)) ↔ (𝑋 𝑌 → (𝑋 + 𝑍) (𝑌 + 𝑍))))
189, 13, 17rspc3v 3313 . . 3 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐)) → (𝑋 𝑌 → (𝑋 + 𝑍) (𝑌 + 𝑍))))
195, 18mpan9 486 . 2 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 → (𝑋 + 𝑍) (𝑌 + 𝑍)))
20193impia 1258 1 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑋 + 𝑍) (𝑌 + 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1987  wral 2907   class class class wbr 4618  cfv 5852  (class class class)co 6610  Basecbs 15792  +gcplusg 15873  lecple 15880  Tosetctos 16965  Mndcmnd 17226  oMndcomnd 29506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4754
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5815  df-fv 5860  df-ov 6613  df-omnd 29508
This theorem is referenced by:  omndaddr  29516  omndadd2d  29517  omndadd2rd  29518  submomnd  29519  omndmul2  29521  omndmul3  29522  ogrpinvOLD  29524  ogrpinv0le  29525  ogrpsub  29526  ogrpaddlt  29527  orngsqr  29613  ornglmulle  29614  orngrmulle  29615
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