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Theorem ogrpinvOLD 29689
Description: In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 30-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
ogrpsub.0 𝐵 = (Base‘𝐺)
ogrpsub.1 = (le‘𝐺)
ogrpinv.2 𝐼 = (invg𝐺)
ogrpinv.3 0 = (0g𝐺)
Assertion
Ref Expression
ogrpinvOLD ((𝐺 ∈ oGrp ∧ 𝑋𝐵0 𝑋) → (𝐼𝑋) 0 )

Proof of Theorem ogrpinvOLD
StepHypRef Expression
1 isogrp 29676 . . . . 5 (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
21simprbi 480 . . . 4 (𝐺 ∈ oGrp → 𝐺 ∈ oMnd)
323ad2ant1 1080 . . 3 ((𝐺 ∈ oGrp ∧ 𝑋𝐵0 𝑋) → 𝐺 ∈ oMnd)
41simplbi 476 . . . . 5 (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
543ad2ant1 1080 . . . 4 ((𝐺 ∈ oGrp ∧ 𝑋𝐵0 𝑋) → 𝐺 ∈ Grp)
6 ogrpsub.0 . . . . 5 𝐵 = (Base‘𝐺)
7 ogrpinv.3 . . . . 5 0 = (0g𝐺)
86, 7grpidcl 17431 . . . 4 (𝐺 ∈ Grp → 0𝐵)
95, 8syl 17 . . 3 ((𝐺 ∈ oGrp ∧ 𝑋𝐵0 𝑋) → 0𝐵)
10 simp2 1060 . . 3 ((𝐺 ∈ oGrp ∧ 𝑋𝐵0 𝑋) → 𝑋𝐵)
11 ogrpinv.2 . . . . 5 𝐼 = (invg𝐺)
126, 11grpinvcl 17448 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝐼𝑋) ∈ 𝐵)
135, 10, 12syl2anc 692 . . 3 ((𝐺 ∈ oGrp ∧ 𝑋𝐵0 𝑋) → (𝐼𝑋) ∈ 𝐵)
14 simp3 1061 . . 3 ((𝐺 ∈ oGrp ∧ 𝑋𝐵0 𝑋) → 0 𝑋)
15 ogrpsub.1 . . . 4 = (le‘𝐺)
16 eqid 2620 . . . 4 (+g𝐺) = (+g𝐺)
176, 15, 16omndadd 29680 . . 3 ((𝐺 ∈ oMnd ∧ ( 0𝐵𝑋𝐵 ∧ (𝐼𝑋) ∈ 𝐵) ∧ 0 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) (𝑋(+g𝐺)(𝐼𝑋)))
183, 9, 10, 13, 14, 17syl131anc 1337 . 2 ((𝐺 ∈ oGrp ∧ 𝑋𝐵0 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) (𝑋(+g𝐺)(𝐼𝑋)))
196, 16, 7grplid 17433 . . 3 ((𝐺 ∈ Grp ∧ (𝐼𝑋) ∈ 𝐵) → ( 0 (+g𝐺)(𝐼𝑋)) = (𝐼𝑋))
205, 13, 19syl2anc 692 . 2 ((𝐺 ∈ oGrp ∧ 𝑋𝐵0 𝑋) → ( 0 (+g𝐺)(𝐼𝑋)) = (𝐼𝑋))
216, 16, 7, 11grprinv 17450 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋(+g𝐺)(𝐼𝑋)) = 0 )
225, 10, 21syl2anc 692 . 2 ((𝐺 ∈ oGrp ∧ 𝑋𝐵0 𝑋) → (𝑋(+g𝐺)(𝐼𝑋)) = 0 )
2318, 20, 223brtr3d 4675 1 ((𝐺 ∈ oGrp ∧ 𝑋𝐵0 𝑋) → (𝐼𝑋) 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1481  wcel 1988   class class class wbr 4644  cfv 5876  (class class class)co 6635  Basecbs 15838  +gcplusg 15922  lecple 15929  0gc0g 16081  Grpcgrp 17403  invgcminusg 17404  oMndcomnd 29671  oGrpcogrp 29672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-0g 16083  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-grp 17406  df-minusg 17407  df-omnd 29673  df-ogrp 29674
This theorem is referenced by: (None)
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