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Theorem invco 17589
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 2738 . . 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 17583 . . . . . . 7 (πœ‘ β†’ (π‘‹πΌπ‘Œ) = dom (π‘‹π‘π‘Œ))
128, 11eleqtrd 2841 . . . . . 6 (πœ‘ β†’ 𝐹 ∈ dom (π‘‹π‘π‘Œ))
131, 9, 4, 5, 6invfun 17582 . . . . . . 7 (πœ‘ β†’ Fun (π‘‹π‘π‘Œ))
14 funfvbrb 6997 . . . . . . 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 17578 . . . . 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 496 . . 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 17583 . . . . . . 7 (πœ‘ β†’ (π‘ŒπΌπ‘) = dom (π‘Œπ‘π‘))
2220, 21eleqtrd 2841 . . . . . 6 (πœ‘ β†’ 𝐺 ∈ dom (π‘Œπ‘π‘))
231, 9, 4, 6, 7invfun 17582 . . . . . . 7 (πœ‘ β†’ Fun (π‘Œπ‘π‘))
24 funfvbrb 6997 . . . . . . 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 17578 . . . . 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 496 . . 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 17574 . 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 497 . . 3 (πœ‘ β†’ ((π‘Œπ‘π‘)β€˜πΊ)(𝑍(Sectβ€˜πΆ)π‘Œ)𝐺)
3218simprd 497 . . 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 17574 . 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 17578 . 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 712 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 397   = wceq 1542   ∈ wcel 2107  βŸ¨cop 4591   class class class wbr 5104  dom cdm 5631  Fun wfun 6486  β€˜cfv 6492  (class class class)co 7350  Basecbs 17018  compcco 17080  Catccat 17479  Sectcsect 17562  Invcinv 17563  Isociso 17564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7663
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7306  df-ov 7353  df-oprab 7354  df-mpo 7355  df-1st 7912  df-2nd 7913  df-cat 17483  df-cid 17484  df-sect 17565  df-inv 17566  df-iso 17567
This theorem is referenced by:  isoco  17595  invisoinvl  17608
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