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Theorem ogrpaddlt 30711
Description: In an ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 20-Jan-2018.)
Hypotheses
Ref Expression
ogrpaddlt.0 𝐵 = (Base‘𝐺)
ogrpaddlt.1 < = (lt‘𝐺)
ogrpaddlt.2 + = (+g𝐺)
Assertion
Ref Expression
ogrpaddlt ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))

Proof of Theorem ogrpaddlt
StepHypRef Expression
1 isogrp 30696 . . . . 5 (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
21simprbi 499 . . . 4 (𝐺 ∈ oGrp → 𝐺 ∈ oMnd)
323ad2ant1 1127 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝐺 ∈ oMnd)
4 simp2 1131 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋𝐵𝑌𝐵𝑍𝐵))
5 simp1 1130 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝐺 ∈ oGrp)
6 simp21 1200 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑋𝐵)
7 simp22 1201 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑌𝐵)
8 simp3 1132 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑋 < 𝑌)
9 eqid 2819 . . . . . 6 (le‘𝐺) = (le‘𝐺)
10 ogrpaddlt.1 . . . . . 6 < = (lt‘𝐺)
119, 10pltle 17563 . . . . 5 ((𝐺 ∈ oGrp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌𝑋(le‘𝐺)𝑌))
1211imp 409 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝑋(le‘𝐺)𝑌)
135, 6, 7, 8, 12syl31anc 1367 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑋(le‘𝐺)𝑌)
14 ogrpaddlt.0 . . . 4 𝐵 = (Base‘𝐺)
15 ogrpaddlt.2 . . . 4 + = (+g𝐺)
1614, 9, 15omndadd 30700 . . 3 ((𝐺 ∈ oMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋(le‘𝐺)𝑌) → (𝑋 + 𝑍)(le‘𝐺)(𝑌 + 𝑍))
173, 4, 13, 16syl3anc 1365 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍)(le‘𝐺)(𝑌 + 𝑍))
1810pltne 17564 . . . . 5 ((𝐺 ∈ oGrp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌𝑋𝑌))
1918imp 409 . . . 4 (((𝐺 ∈ oGrp ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → 𝑋𝑌)
205, 6, 7, 8, 19syl31anc 1367 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → 𝑋𝑌)
21 ogrpgrp 30697 . . . . . 6 (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
2214, 15grprcan 18129 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) ↔ 𝑋 = 𝑌))
2322biimpd 231 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) → 𝑋 = 𝑌))
2421, 23sylan 582 . . . . 5 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) → 𝑋 = 𝑌))
2524necon3d 3035 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝑌 → (𝑋 + 𝑍) ≠ (𝑌 + 𝑍)))
26253impia 1111 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝑌) → (𝑋 + 𝑍) ≠ (𝑌 + 𝑍))
275, 4, 20, 26syl3anc 1365 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) ≠ (𝑌 + 𝑍))
28 ovex 7181 . . . 4 (𝑋 + 𝑍) ∈ V
29 ovex 7181 . . . 4 (𝑌 + 𝑍) ∈ V
309, 10pltval 17562 . . . 4 ((𝐺 ∈ oGrp ∧ (𝑋 + 𝑍) ∈ V ∧ (𝑌 + 𝑍) ∈ V) → ((𝑋 + 𝑍) < (𝑌 + 𝑍) ↔ ((𝑋 + 𝑍)(le‘𝐺)(𝑌 + 𝑍) ∧ (𝑋 + 𝑍) ≠ (𝑌 + 𝑍))))
3128, 29, 30mp3an23 1446 . . 3 (𝐺 ∈ oGrp → ((𝑋 + 𝑍) < (𝑌 + 𝑍) ↔ ((𝑋 + 𝑍)(le‘𝐺)(𝑌 + 𝑍) ∧ (𝑋 + 𝑍) ≠ (𝑌 + 𝑍))))
32313ad2ant1 1127 . 2 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → ((𝑋 + 𝑍) < (𝑌 + 𝑍) ↔ ((𝑋 + 𝑍)(le‘𝐺)(𝑌 + 𝑍) ∧ (𝑋 + 𝑍) ≠ (𝑌 + 𝑍))))
3317, 27, 32mpbir2and 711 1 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1081   = wceq 1530  wcel 2107  wne 3014  Vcvv 3493   class class class wbr 5057  cfv 6348  (class class class)co 7148  Basecbs 16475  +gcplusg 16557  lecple 16564  ltcplt 17543  Grpcgrp 18095  oMndcomnd 30691  oGrpcogrp 30692
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-riota 7106  df-ov 7151  df-0g 16707  df-plt 17560  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-omnd 30693  df-ogrp 30694
This theorem is referenced by:  ogrpaddltbi  30712  ogrpaddltrd  30713  ogrpinv0lt  30716  isarchi3  30809  archirngz  30811  archiabllem1b  30814  archiabllem2c  30817  ofldchr  30880
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