MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isocoinvid Structured version   Visualization version   GIF version

Theorem isocoinvid 17745
Description: The inverse of an isomorphism composed with the isomorphism is the identity. (Contributed by AV, 10-Apr-2020.)
Hypotheses
Ref Expression
invisoinv.b 𝐡 = (Baseβ€˜πΆ)
invisoinv.i 𝐼 = (Isoβ€˜πΆ)
invisoinv.n 𝑁 = (Invβ€˜πΆ)
invisoinv.c (πœ‘ β†’ 𝐢 ∈ Cat)
invisoinv.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
invisoinv.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
invisoinv.f (πœ‘ β†’ 𝐹 ∈ (π‘‹πΌπ‘Œ))
invcoisoid.1 1 = (Idβ€˜πΆ)
isocoinvid.o ⚬ = (βŸ¨π‘Œ, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)
Assertion
Ref Expression
isocoinvid (πœ‘ β†’ (𝐹 ⚬ ((π‘‹π‘π‘Œ)β€˜πΉ)) = ( 1 β€˜π‘Œ))

Proof of Theorem isocoinvid
StepHypRef Expression
1 invisoinv.b . . . 4 𝐡 = (Baseβ€˜πΆ)
2 invisoinv.i . . . 4 𝐼 = (Isoβ€˜πΆ)
3 invisoinv.n . . . 4 𝑁 = (Invβ€˜πΆ)
4 invisoinv.c . . . 4 (πœ‘ β†’ 𝐢 ∈ Cat)
5 invisoinv.x . . . 4 (πœ‘ β†’ 𝑋 ∈ 𝐡)
6 invisoinv.y . . . 4 (πœ‘ β†’ π‘Œ ∈ 𝐡)
7 invisoinv.f . . . 4 (πœ‘ β†’ 𝐹 ∈ (π‘‹πΌπ‘Œ))
81, 2, 3, 4, 5, 6, 7invisoinvl 17742 . . 3 (πœ‘ β†’ ((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œπ‘π‘‹)𝐹)
9 eqid 2731 . . . . 5 (Sectβ€˜πΆ) = (Sectβ€˜πΆ)
101, 3, 4, 6, 5, 9isinv 17712 . . . 4 (πœ‘ β†’ (((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œπ‘π‘‹)𝐹 ↔ (((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹 ∧ 𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ))))
11 simpl 482 . . . 4 ((((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹 ∧ 𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ)) β†’ ((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹)
1210, 11syl6bi 252 . . 3 (πœ‘ β†’ (((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œπ‘π‘‹)𝐹 β†’ ((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹))
138, 12mpd 15 . 2 (πœ‘ β†’ ((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹)
14 eqid 2731 . . . 4 (Hom β€˜πΆ) = (Hom β€˜πΆ)
15 eqid 2731 . . . 4 (compβ€˜πΆ) = (compβ€˜πΆ)
16 invcoisoid.1 . . . 4 1 = (Idβ€˜πΆ)
171, 14, 2, 4, 6, 5isohom 17728 . . . . 5 (πœ‘ β†’ (π‘ŒπΌπ‘‹) βŠ† (π‘Œ(Hom β€˜πΆ)𝑋))
181, 3, 4, 5, 6, 2invf 17720 . . . . . 6 (πœ‘ β†’ (π‘‹π‘π‘Œ):(π‘‹πΌπ‘Œ)⟢(π‘ŒπΌπ‘‹))
1918, 7ffvelcdmd 7088 . . . . 5 (πœ‘ β†’ ((π‘‹π‘π‘Œ)β€˜πΉ) ∈ (π‘ŒπΌπ‘‹))
2017, 19sseldd 3984 . . . 4 (πœ‘ β†’ ((π‘‹π‘π‘Œ)β€˜πΉ) ∈ (π‘Œ(Hom β€˜πΆ)𝑋))
211, 14, 2, 4, 5, 6isohom 17728 . . . . 5 (πœ‘ β†’ (π‘‹πΌπ‘Œ) βŠ† (𝑋(Hom β€˜πΆ)π‘Œ))
2221, 7sseldd 3984 . . . 4 (πœ‘ β†’ 𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ))
231, 14, 15, 16, 9, 4, 6, 5, 20, 22issect2 17706 . . 3 (πœ‘ β†’ (((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹 ↔ (𝐹(βŸ¨π‘Œ, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ)) = ( 1 β€˜π‘Œ)))
24 isocoinvid.o . . . . . . 7 ⚬ = (βŸ¨π‘Œ, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)
2524a1i 11 . . . . . 6 (πœ‘ β†’ ⚬ = (βŸ¨π‘Œ, π‘‹βŸ©(compβ€˜πΆ)π‘Œ))
2625eqcomd 2737 . . . . 5 (πœ‘ β†’ (βŸ¨π‘Œ, π‘‹βŸ©(compβ€˜πΆ)π‘Œ) = ⚬ )
2726oveqd 7429 . . . 4 (πœ‘ β†’ (𝐹(βŸ¨π‘Œ, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ)) = (𝐹 ⚬ ((π‘‹π‘π‘Œ)β€˜πΉ)))
2827eqeq1d 2733 . . 3 (πœ‘ β†’ ((𝐹(βŸ¨π‘Œ, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ)) = ( 1 β€˜π‘Œ) ↔ (𝐹 ⚬ ((π‘‹π‘π‘Œ)β€˜πΉ)) = ( 1 β€˜π‘Œ)))
2923, 28bitrd 278 . 2 (πœ‘ β†’ (((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹 ↔ (𝐹 ⚬ ((π‘‹π‘π‘Œ)β€˜πΉ)) = ( 1 β€˜π‘Œ)))
3013, 29mpbid 231 1 (πœ‘ β†’ (𝐹 ⚬ ((π‘‹π‘π‘Œ)β€˜πΉ)) = ( 1 β€˜π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  βŸ¨cop 4635   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7412  Basecbs 17149  Hom chom 17213  compcco 17214  Catccat 17613  Idccid 17614  Sectcsect 17696  Invcinv 17697  Isociso 17698
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
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-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7978  df-2nd 7979  df-cat 17617  df-cid 17618  df-sect 17699  df-inv 17700  df-iso 17701
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator