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Theorem oppgid 17702
Description: Zero in a monoid is a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.)
Hypotheses
Ref Expression
oppgbas.1 𝑂 = (oppg𝑅)
oppgid.2 0 = (0g𝑅)
Assertion
Ref Expression
oppgid 0 = (0g𝑂)

Proof of Theorem oppgid
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ancom 466 . . . . . 6 (((𝑥(+g𝑅)𝑦) = 𝑦 ∧ (𝑦(+g𝑅)𝑥) = 𝑦) ↔ ((𝑦(+g𝑅)𝑥) = 𝑦 ∧ (𝑥(+g𝑅)𝑦) = 𝑦))
2 eqid 2626 . . . . . . . . 9 (+g𝑅) = (+g𝑅)
3 oppgbas.1 . . . . . . . . 9 𝑂 = (oppg𝑅)
4 eqid 2626 . . . . . . . . 9 (+g𝑂) = (+g𝑂)
52, 3, 4oppgplus 17695 . . . . . . . 8 (𝑥(+g𝑂)𝑦) = (𝑦(+g𝑅)𝑥)
65eqeq1i 2631 . . . . . . 7 ((𝑥(+g𝑂)𝑦) = 𝑦 ↔ (𝑦(+g𝑅)𝑥) = 𝑦)
72, 3, 4oppgplus 17695 . . . . . . . 8 (𝑦(+g𝑂)𝑥) = (𝑥(+g𝑅)𝑦)
87eqeq1i 2631 . . . . . . 7 ((𝑦(+g𝑂)𝑥) = 𝑦 ↔ (𝑥(+g𝑅)𝑦) = 𝑦)
96, 8anbi12i 732 . . . . . 6 (((𝑥(+g𝑂)𝑦) = 𝑦 ∧ (𝑦(+g𝑂)𝑥) = 𝑦) ↔ ((𝑦(+g𝑅)𝑥) = 𝑦 ∧ (𝑥(+g𝑅)𝑦) = 𝑦))
101, 9bitr4i 267 . . . . 5 (((𝑥(+g𝑅)𝑦) = 𝑦 ∧ (𝑦(+g𝑅)𝑥) = 𝑦) ↔ ((𝑥(+g𝑂)𝑦) = 𝑦 ∧ (𝑦(+g𝑂)𝑥) = 𝑦))
1110ralbii 2979 . . . 4 (∀𝑦 ∈ (Base‘𝑅)((𝑥(+g𝑅)𝑦) = 𝑦 ∧ (𝑦(+g𝑅)𝑥) = 𝑦) ↔ ∀𝑦 ∈ (Base‘𝑅)((𝑥(+g𝑂)𝑦) = 𝑦 ∧ (𝑦(+g𝑂)𝑥) = 𝑦))
1211anbi2i 729 . . 3 ((𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(+g𝑅)𝑦) = 𝑦 ∧ (𝑦(+g𝑅)𝑥) = 𝑦)) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(+g𝑂)𝑦) = 𝑦 ∧ (𝑦(+g𝑂)𝑥) = 𝑦)))
1312iotabii 5835 . 2 (℩𝑥(𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(+g𝑅)𝑦) = 𝑦 ∧ (𝑦(+g𝑅)𝑥) = 𝑦))) = (℩𝑥(𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(+g𝑂)𝑦) = 𝑦 ∧ (𝑦(+g𝑂)𝑥) = 𝑦)))
14 eqid 2626 . . 3 (Base‘𝑅) = (Base‘𝑅)
15 oppgid.2 . . 3 0 = (0g𝑅)
1614, 2, 15grpidval 17176 . 2 0 = (℩𝑥(𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(+g𝑅)𝑦) = 𝑦 ∧ (𝑦(+g𝑅)𝑥) = 𝑦)))
173, 14oppgbas 17697 . . 3 (Base‘𝑅) = (Base‘𝑂)
18 eqid 2626 . . 3 (0g𝑂) = (0g𝑂)
1917, 4, 18grpidval 17176 . 2 (0g𝑂) = (℩𝑥(𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(+g𝑂)𝑦) = 𝑦 ∧ (𝑦(+g𝑂)𝑥) = 𝑦)))
2013, 16, 193eqtr4i 2658 1 0 = (0g𝑂)
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wcel 1992  wral 2912  cio 5811  cfv 5850  (class class class)co 6605  Basecbs 15776  +gcplusg 15857  0gc0g 16016  oppgcoppg 17691
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-tpos 7298  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-plusg 15870  df-0g 16018  df-oppg 17692
This theorem is referenced by:  oppggrp  17703  oppginv  17705  oppgsubm  17708  gsumwrev  17712  lsmdisj2r  18014  gsumzoppg  18260  tgpconncomp  21821
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