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Theorem omndadd 30046
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 30041 . . . 4 (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐))))
54simp3bi 1141 . . 3 (𝑀 ∈ oMnd → ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐)))
6 breq1 4789 . . . . 5 (𝑎 = 𝑋 → (𝑎 𝑏𝑋 𝑏))
7 oveq1 6800 . . . . . 6 (𝑎 = 𝑋 → (𝑎 + 𝑐) = (𝑋 + 𝑐))
87breq1d 4796 . . . . 5 (𝑎 = 𝑋 → ((𝑎 + 𝑐) (𝑏 + 𝑐) ↔ (𝑋 + 𝑐) (𝑏 + 𝑐)))
96, 8imbi12d 333 . . . 4 (𝑎 = 𝑋 → ((𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐)) ↔ (𝑋 𝑏 → (𝑋 + 𝑐) (𝑏 + 𝑐))))
10 breq2 4790 . . . . 5 (𝑏 = 𝑌 → (𝑋 𝑏𝑋 𝑌))
11 oveq1 6800 . . . . . 6 (𝑏 = 𝑌 → (𝑏 + 𝑐) = (𝑌 + 𝑐))
1211breq2d 4798 . . . . 5 (𝑏 = 𝑌 → ((𝑋 + 𝑐) (𝑏 + 𝑐) ↔ (𝑋 + 𝑐) (𝑌 + 𝑐)))
1310, 12imbi12d 333 . . . 4 (𝑏 = 𝑌 → ((𝑋 𝑏 → (𝑋 + 𝑐) (𝑏 + 𝑐)) ↔ (𝑋 𝑌 → (𝑋 + 𝑐) (𝑌 + 𝑐))))
14 oveq2 6801 . . . . . 6 (𝑐 = 𝑍 → (𝑋 + 𝑐) = (𝑋 + 𝑍))
15 oveq2 6801 . . . . . 6 (𝑐 = 𝑍 → (𝑌 + 𝑐) = (𝑌 + 𝑍))
1614, 15breq12d 4799 . . . . 5 (𝑐 = 𝑍 → ((𝑋 + 𝑐) (𝑌 + 𝑐) ↔ (𝑋 + 𝑍) (𝑌 + 𝑍)))
1716imbi2d 329 . . . 4 (𝑐 = 𝑍 → ((𝑋 𝑌 → (𝑋 + 𝑐) (𝑌 + 𝑐)) ↔ (𝑋 𝑌 → (𝑋 + 𝑍) (𝑌 + 𝑍))))
189, 13, 17rspc3v 3475 . . 3 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐)) → (𝑋 𝑌 → (𝑋 + 𝑍) (𝑌 + 𝑍))))
195, 18mpan9 496 . 2 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 → (𝑋 + 𝑍) (𝑌 + 𝑍)))
20193impia 1109 1 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑋 + 𝑍) (𝑌 + 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071   = wceq 1631  wcel 2145  wral 3061   class class class wbr 4786  cfv 6031  (class class class)co 6793  Basecbs 16064  +gcplusg 16149  lecple 16156  Tosetctos 17241  Mndcmnd 17502  oMndcomnd 30037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4923
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6796  df-omnd 30039
This theorem is referenced by:  omndaddr  30047  omndadd2d  30048  omndadd2rd  30049  submomnd  30050  omndmul2  30052  omndmul3  30053  ogrpinvOLD  30055  ogrpinv0le  30056  ogrpsub  30057  ogrpaddlt  30058  orngsqr  30144  ornglmulle  30145  orngrmulle  30146
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