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Theorem ogrpaddltbi 30239
Description: In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.)
Hypotheses
Ref Expression
ogrpaddlt.0 𝐵 = (Base‘𝐺)
ogrpaddlt.1 < = (lt‘𝐺)
ogrpaddlt.2 + = (+g𝐺)
Assertion
Ref Expression
ogrpaddltbi ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 < 𝑌 ↔ (𝑋 + 𝑍) < (𝑌 + 𝑍)))

Proof of Theorem ogrpaddltbi
StepHypRef Expression
1 ogrpaddlt.0 . . . 4 𝐵 = (Base‘𝐺)
2 ogrpaddlt.1 . . . 4 < = (lt‘𝐺)
3 ogrpaddlt.2 . . . 4 + = (+g𝐺)
41, 2, 3ogrpaddlt 30238 . . 3 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
543expa 1148 . 2 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
6 simpll 784 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝐺 ∈ oGrp)
7 ogrpgrp 30223 . . . . . 6 (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
86, 7syl 17 . . . . 5 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝐺 ∈ Grp)
9 simplr1 1276 . . . . 5 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑋𝐵)
10 simplr3 1280 . . . . 5 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑍𝐵)
111, 3grpcl 17750 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑍𝐵) → (𝑋 + 𝑍) ∈ 𝐵)
128, 9, 10, 11syl3anc 1491 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + 𝑍) ∈ 𝐵)
13 simplr2 1278 . . . . 5 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑌𝐵)
141, 3grpcl 17750 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝐵𝑍𝐵) → (𝑌 + 𝑍) ∈ 𝐵)
158, 13, 10, 14syl3anc 1491 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑌 + 𝑍) ∈ 𝐵)
16 eqid 2803 . . . . . 6 (invg𝐺) = (invg𝐺)
171, 16grpinvcl 17787 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ((invg𝐺)‘𝑍) ∈ 𝐵)
188, 10, 17syl2anc 580 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((invg𝐺)‘𝑍) ∈ 𝐵)
19 simpr 478 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
201, 2, 3ogrpaddlt 30238 . . . 4 ((𝐺 ∈ oGrp ∧ ((𝑋 + 𝑍) ∈ 𝐵 ∧ (𝑌 + 𝑍) ∈ 𝐵 ∧ ((invg𝐺)‘𝑍) ∈ 𝐵) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑋 + 𝑍) + ((invg𝐺)‘𝑍)) < ((𝑌 + 𝑍) + ((invg𝐺)‘𝑍)))
216, 12, 15, 18, 19, 20syl131anc 1503 . . 3 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑋 + 𝑍) + ((invg𝐺)‘𝑍)) < ((𝑌 + 𝑍) + ((invg𝐺)‘𝑍)))
221, 3grpass 17751 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑍𝐵 ∧ ((invg𝐺)‘𝑍) ∈ 𝐵)) → ((𝑋 + 𝑍) + ((invg𝐺)‘𝑍)) = (𝑋 + (𝑍 + ((invg𝐺)‘𝑍))))
238, 9, 10, 18, 22syl13anc 1492 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑋 + 𝑍) + ((invg𝐺)‘𝑍)) = (𝑋 + (𝑍 + ((invg𝐺)‘𝑍))))
24 eqid 2803 . . . . . . 7 (0g𝐺) = (0g𝐺)
251, 3, 24, 16grprinv 17789 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → (𝑍 + ((invg𝐺)‘𝑍)) = (0g𝐺))
268, 10, 25syl2anc 580 . . . . 5 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑍 + ((invg𝐺)‘𝑍)) = (0g𝐺))
2726oveq2d 6898 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + (𝑍 + ((invg𝐺)‘𝑍))) = (𝑋 + (0g𝐺)))
281, 3, 24grprid 17773 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (0g𝐺)) = 𝑋)
298, 9, 28syl2anc 580 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑋 + (0g𝐺)) = 𝑋)
3023, 27, 293eqtrd 2841 . . 3 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑋 + 𝑍) + ((invg𝐺)‘𝑍)) = 𝑋)
311, 3grpass 17751 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑌𝐵𝑍𝐵 ∧ ((invg𝐺)‘𝑍) ∈ 𝐵)) → ((𝑌 + 𝑍) + ((invg𝐺)‘𝑍)) = (𝑌 + (𝑍 + ((invg𝐺)‘𝑍))))
328, 13, 10, 18, 31syl13anc 1492 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑌 + 𝑍) + ((invg𝐺)‘𝑍)) = (𝑌 + (𝑍 + ((invg𝐺)‘𝑍))))
3326oveq2d 6898 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑌 + (𝑍 + ((invg𝐺)‘𝑍))) = (𝑌 + (0g𝐺)))
341, 3, 24grprid 17773 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑌 + (0g𝐺)) = 𝑌)
358, 13, 34syl2anc 580 . . . 4 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → (𝑌 + (0g𝐺)) = 𝑌)
3632, 33, 353eqtrd 2841 . . 3 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → ((𝑌 + 𝑍) + ((invg𝐺)‘𝑍)) = 𝑌)
3721, 30, 363brtr3d 4878 . 2 (((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 + 𝑍) < (𝑌 + 𝑍)) → 𝑋 < 𝑌)
385, 37impbida 836 1 ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 < 𝑌 ↔ (𝑋 + 𝑍) < (𝑌 + 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157   class class class wbr 4847  cfv 6105  (class class class)co 6882  Basecbs 16188  +gcplusg 16271  0gc0g 16419  ltcplt 17260  Grpcgrp 17742  invgcminusg 17743  oGrpcogrp 30218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2379  ax-ext 2781  ax-rep 4968  ax-sep 4979  ax-nul 4987  ax-pow 5039  ax-pr 5101
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2593  df-eu 2611  df-clab 2790  df-cleq 2796  df-clel 2799  df-nfc 2934  df-ne 2976  df-ral 3098  df-rex 3099  df-reu 3100  df-rmo 3101  df-rab 3102  df-v 3391  df-sbc 3638  df-csb 3733  df-dif 3776  df-un 3778  df-in 3780  df-ss 3787  df-nul 4120  df-if 4282  df-sn 4373  df-pr 4375  df-op 4379  df-uni 4633  df-iun 4716  df-br 4848  df-opab 4910  df-mpt 4927  df-id 5224  df-xp 5322  df-rel 5323  df-cnv 5324  df-co 5325  df-dm 5326  df-rn 5327  df-res 5328  df-ima 5329  df-iota 6068  df-fun 6107  df-fn 6108  df-f 6109  df-f1 6110  df-fo 6111  df-f1o 6112  df-fv 6113  df-riota 6843  df-ov 6885  df-0g 16421  df-plt 17277  df-mgm 17561  df-sgrp 17603  df-mnd 17614  df-grp 17745  df-minusg 17746  df-omnd 30219  df-ogrp 30220
This theorem is referenced by:  ogrpinvlt  30244
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