MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  invco Structured version   Visualization version   GIF version
Theorem invco 17723
Description: The composition of two isomorphisms is an isomorphism, and the inverse is the composition of the individual inverses. Proposition 3.14(2) of [Adamek] p. 29. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
invfval.b 𝐡 = (Baseβ€˜πΆ)
invfval.n 𝑁 = (Invβ€˜πΆ)
invfval.c (πœ‘ β†’ 𝐢 ∈ Cat)
invfval.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
invfval.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
isoval.n 𝐼 = (Isoβ€˜πΆ)
invinv.f (πœ‘ β†’ 𝐹 ∈ (π‘‹πΌπ‘Œ))
invco.o Β· = (compβ€˜πΆ)
invco.z (πœ‘ β†’ 𝑍 ∈ 𝐡)
invco.f (πœ‘ β†’ 𝐺 ∈ (π‘ŒπΌπ‘))
Assertion
Ref Expression
invco (πœ‘ β†’ (𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍)𝐹)(𝑋𝑁𝑍)(((π‘‹π‘π‘Œ)β€˜πΉ)(βŸ¨π‘, π‘ŒβŸ© Β· 𝑋)((π‘Œπ‘π‘)β€˜πΊ)))
Proof of Theorem invco
StepHypRef Expression
1 invfval.b . . 3 𝐡 = (Baseβ€˜πΆ)
2 invco.o . . 3 Β· = (compβ€˜πΆ)
3 eqid 2731 . . 3 (Sectβ€˜πΆ) = (Sectβ€˜πΆ)
4 invfval.c . . 3 (πœ‘ β†’ 𝐢 ∈ Cat)
5 invfval.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝐡)
6 invfval.y . . 3 (πœ‘ β†’ π‘Œ ∈ 𝐡)
7 invco.z . . 3 (πœ‘ β†’ 𝑍 ∈ 𝐡)
8 invinv.f . . . . . . 7 (πœ‘ β†’ 𝐹 ∈ (π‘‹πΌπ‘Œ))
9 invfval.n . . . . . . . 8 𝑁 = (Invβ€˜πΆ)
10 isoval.n . . . . . . . 8 𝐼 = (Isoβ€˜πΆ)
111, 9, 4, 5, 6, 10isoval 17717 . . . . . . 7 (πœ‘ β†’ (π‘‹πΌπ‘Œ) = dom (π‘‹π‘π‘Œ))
128, 11eleqtrd 2834 . . . . . 6 (πœ‘ β†’ 𝐹 ∈ dom (π‘‹π‘π‘Œ))
131, 9, 4, 5, 6invfun 17716 . . . . . . 7 (πœ‘ β†’ Fun (π‘‹π‘π‘Œ))
14 funfvbrb 7052 . . . . . . 7 (Fun (π‘‹π‘π‘Œ) β†’ (𝐹 ∈ dom (π‘‹π‘π‘Œ) ↔ 𝐹(π‘‹π‘π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ)))
1513, 14syl 17 . . . . . 6 (πœ‘ β†’ (𝐹 ∈ dom (π‘‹π‘π‘Œ) ↔ 𝐹(π‘‹π‘π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ)))
1612, 15mpbid 231 . . . . 5 (πœ‘ β†’ 𝐹(π‘‹π‘π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ))
171, 9, 4, 5, 6, 3isinv 17712 . . . . 5 (πœ‘ β†’ (𝐹(π‘‹π‘π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ) ↔ (𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ) ∧ ((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹)))
1816, 17mpbid 231 . . . 4 (πœ‘ β†’ (𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ) ∧ ((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹))
1918simpld 494 . . 3 (πœ‘ β†’ 𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ))
20 invco.f . . . . . . 7 (πœ‘ β†’ 𝐺 ∈ (π‘ŒπΌπ‘))
211, 9, 4, 6, 7, 10isoval 17717 . . . . . . 7 (πœ‘ β†’ (π‘ŒπΌπ‘) = dom (π‘Œπ‘π‘))
2220, 21eleqtrd 2834 . . . . . 6 (πœ‘ β†’ 𝐺 ∈ dom (π‘Œπ‘π‘))
231, 9, 4, 6, 7invfun 17716 . . . . . . 7 (πœ‘ β†’ Fun (π‘Œπ‘π‘))
24 funfvbrb 7052 . . . . . . 7 (Fun (π‘Œπ‘π‘) β†’ (𝐺 ∈ dom (π‘Œπ‘π‘) ↔ 𝐺(π‘Œπ‘π‘)((π‘Œπ‘π‘)β€˜πΊ)))
2523, 24syl 17 . . . . . 6 (πœ‘ β†’ (𝐺 ∈ dom (π‘Œπ‘π‘) ↔ 𝐺(π‘Œπ‘π‘)((π‘Œπ‘π‘)β€˜πΊ)))
2622, 25mpbid 231 . . . . 5 (πœ‘ β†’ 𝐺(π‘Œπ‘π‘)((π‘Œπ‘π‘)β€˜πΊ))
271, 9, 4, 6, 7, 3isinv 17712 . . . . 5 (πœ‘ β†’ (𝐺(π‘Œπ‘π‘)((π‘Œπ‘π‘)β€˜πΊ) ↔ (𝐺(π‘Œ(Sectβ€˜πΆ)𝑍)((π‘Œπ‘π‘)β€˜πΊ) ∧ ((π‘Œπ‘π‘)β€˜πΊ)(𝑍(Sectβ€˜πΆ)π‘Œ)𝐺)))
2826, 27mpbid 231 . . . 4 (πœ‘ β†’ (𝐺(π‘Œ(Sectβ€˜πΆ)𝑍)((π‘Œπ‘π‘)β€˜πΊ) ∧ ((π‘Œπ‘π‘)β€˜πΊ)(𝑍(Sectβ€˜πΆ)π‘Œ)𝐺))
2928simpld 494 . . 3 (πœ‘ β†’ 𝐺(π‘Œ(Sectβ€˜πΆ)𝑍)((π‘Œπ‘π‘)β€˜πΊ))
301, 2, 3, 4, 5, 6, 7, 19, 29sectco 17708 . 2 (πœ‘ β†’ (𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍)𝐹)(𝑋(Sectβ€˜πΆ)𝑍)(((π‘‹π‘π‘Œ)β€˜πΉ)(βŸ¨π‘, π‘ŒβŸ© Β· 𝑋)((π‘Œπ‘π‘)β€˜πΊ)))
3128simprd 495 . . 3 (πœ‘ β†’ ((π‘Œπ‘π‘)β€˜πΊ)(𝑍(Sectβ€˜πΆ)π‘Œ)𝐺)
3218simprd 495 . . 3 (πœ‘ β†’ ((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹)
331, 2, 3, 4, 7, 6, 5, 31, 32sectco 17708 . 2 (πœ‘ β†’ (((π‘‹π‘π‘Œ)β€˜πΉ)(βŸ¨π‘, π‘ŒβŸ© Β· 𝑋)((π‘Œπ‘π‘)β€˜πΊ))(𝑍(Sectβ€˜πΆ)𝑋)(𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍)𝐹))
341, 9, 4, 5, 7, 3isinv 17712 . 2 (πœ‘ β†’ ((𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍)𝐹)(𝑋𝑁𝑍)(((π‘‹π‘π‘Œ)β€˜πΉ)(βŸ¨π‘, π‘ŒβŸ© Β· 𝑋)((π‘Œπ‘π‘)β€˜πΊ)) ↔ ((𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍)𝐹)(𝑋(Sectβ€˜πΆ)𝑍)(((π‘‹π‘π‘Œ)β€˜πΉ)(βŸ¨π‘, π‘ŒβŸ© Β· 𝑋)((π‘Œπ‘π‘)β€˜πΊ)) ∧ (((π‘‹π‘π‘Œ)β€˜πΉ)(βŸ¨π‘, π‘ŒβŸ© Β· 𝑋)((π‘Œπ‘π‘)β€˜πΊ))(𝑍(Sectβ€˜πΆ)𝑋)(𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍)𝐹))))
3530, 33, 34mpbir2and 710 1 (πœ‘ β†’ (𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍)𝐹)(𝑋𝑁𝑍)(((π‘‹π‘π‘Œ)β€˜πΉ)(βŸ¨π‘, π‘ŒβŸ© Β· 𝑋)((π‘Œπ‘π‘)β€˜πΊ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  βŸ¨cop 4634   class class class wbr 5148  dom cdm 5676  Fun wfun 6537  β€˜cfv 6543  (class class class)co 7412  Basecbs 17149  compcco 17214  Catccat 17613  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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  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-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:  isoco  17729  invisoinvl  17742
  Copyright terms: Public domain W3C validator