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Theorem grpoinvop 30054
Description: The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpasscan1.1 𝑋 = ran 𝐺
grpasscan1.2 𝑁 = (invβ€˜πΊ)
Assertion
Ref Expression
grpoinvop ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜(𝐴𝐺𝐡)) = ((π‘β€˜π΅)𝐺(π‘β€˜π΄)))
Proof of Theorem grpoinvop
StepHypRef Expression
1 simp1 1135 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ 𝐺 ∈ GrpOp)
2 simp2 1136 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ 𝐴 ∈ 𝑋)
3 simp3 1137 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ 𝐡 ∈ 𝑋)
4 grpasscan1.1 . . . . . . 7 𝑋 = ran 𝐺
5 grpasscan1.2 . . . . . . 7 𝑁 = (invβ€˜πΊ)
64, 5grpoinvcl 30045 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜π΅) ∈ 𝑋)
763adant2 1130 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜π΅) ∈ 𝑋)
84, 5grpoinvcl 30045 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) ∈ 𝑋)
983adant3 1131 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜π΄) ∈ 𝑋)
104grpocl 30021 . . . . 5 ((𝐺 ∈ GrpOp ∧ (π‘β€˜π΅) ∈ 𝑋 ∧ (π‘β€˜π΄) ∈ 𝑋) β†’ ((π‘β€˜π΅)𝐺(π‘β€˜π΄)) ∈ 𝑋)
111, 7, 9, 10syl3anc 1370 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((π‘β€˜π΅)𝐺(π‘β€˜π΄)) ∈ 𝑋)
124grpoass 30024 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ ((π‘β€˜π΅)𝐺(π‘β€˜π΄)) ∈ 𝑋)) β†’ ((𝐴𝐺𝐡)𝐺((π‘β€˜π΅)𝐺(π‘β€˜π΄))) = (𝐴𝐺(𝐡𝐺((π‘β€˜π΅)𝐺(π‘β€˜π΄)))))
131, 2, 3, 11, 12syl13anc 1371 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((𝐴𝐺𝐡)𝐺((π‘β€˜π΅)𝐺(π‘β€˜π΄))) = (𝐴𝐺(𝐡𝐺((π‘β€˜π΅)𝐺(π‘β€˜π΄)))))
14 eqid 2731 . . . . . . . 8 (GIdβ€˜πΊ) = (GIdβ€˜πΊ)
154, 14, 5grporinv 30048 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐡 ∈ 𝑋) β†’ (𝐡𝐺(π‘β€˜π΅)) = (GIdβ€˜πΊ))
16153adant2 1130 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐡𝐺(π‘β€˜π΅)) = (GIdβ€˜πΊ))
1716oveq1d 7427 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((𝐡𝐺(π‘β€˜π΅))𝐺(π‘β€˜π΄)) = ((GIdβ€˜πΊ)𝐺(π‘β€˜π΄)))
184grpoass 30024 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐡 ∈ 𝑋 ∧ (π‘β€˜π΅) ∈ 𝑋 ∧ (π‘β€˜π΄) ∈ 𝑋)) β†’ ((𝐡𝐺(π‘β€˜π΅))𝐺(π‘β€˜π΄)) = (𝐡𝐺((π‘β€˜π΅)𝐺(π‘β€˜π΄))))
191, 3, 7, 9, 18syl13anc 1371 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((𝐡𝐺(π‘β€˜π΅))𝐺(π‘β€˜π΄)) = (𝐡𝐺((π‘β€˜π΅)𝐺(π‘β€˜π΄))))
204, 14grpolid 30037 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (π‘β€˜π΄) ∈ 𝑋) β†’ ((GIdβ€˜πΊ)𝐺(π‘β€˜π΄)) = (π‘β€˜π΄))
218, 20syldan 590 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ ((GIdβ€˜πΊ)𝐺(π‘β€˜π΄)) = (π‘β€˜π΄))
22213adant3 1131 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((GIdβ€˜πΊ)𝐺(π‘β€˜π΄)) = (π‘β€˜π΄))
2317, 19, 223eqtr3d 2779 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐡𝐺((π‘β€˜π΅)𝐺(π‘β€˜π΄))) = (π‘β€˜π΄))
2423oveq2d 7428 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐺(𝐡𝐺((π‘β€˜π΅)𝐺(π‘β€˜π΄)))) = (𝐴𝐺(π‘β€˜π΄)))
254, 14, 5grporinv 30048 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (𝐴𝐺(π‘β€˜π΄)) = (GIdβ€˜πΊ))
26253adant3 1131 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐺(π‘β€˜π΄)) = (GIdβ€˜πΊ))
2713, 24, 263eqtrd 2775 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((𝐴𝐺𝐡)𝐺((π‘β€˜π΅)𝐺(π‘β€˜π΄))) = (GIdβ€˜πΊ))
284grpocl 30021 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐺𝐡) ∈ 𝑋)
294, 14, 5grpoinvid1 30049 . . 3 ((𝐺 ∈ GrpOp ∧ (𝐴𝐺𝐡) ∈ 𝑋 ∧ ((π‘β€˜π΅)𝐺(π‘β€˜π΄)) ∈ 𝑋) β†’ ((π‘β€˜(𝐴𝐺𝐡)) = ((π‘β€˜π΅)𝐺(π‘β€˜π΄)) ↔ ((𝐴𝐺𝐡)𝐺((π‘β€˜π΅)𝐺(π‘β€˜π΄))) = (GIdβ€˜πΊ)))
301, 28, 11, 29syl3anc 1370 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((π‘β€˜(𝐴𝐺𝐡)) = ((π‘β€˜π΅)𝐺(π‘β€˜π΄)) ↔ ((𝐴𝐺𝐡)𝐺((π‘β€˜π΅)𝐺(π‘β€˜π΄))) = (GIdβ€˜πΊ)))
3127, 30mpbird 257 1 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜(𝐴𝐺𝐡)) = ((π‘β€˜π΅)𝐺(π‘β€˜π΄)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ran crn 5677  β€˜cfv 6543  (class class class)co 7412  GrpOpcgr 30010  GIdcgi 30011  invcgn 30012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-grpo 30014  df-gid 30015  df-ginv 30016
This theorem is referenced by:  grpoinvdiv  30058
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