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Theorem invco 17819
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 𝐵 = (Base‘𝐶)
invfval.n 𝑁 = (Inv‘𝐶)
invfval.c (𝜑𝐶 ∈ Cat)
invfval.x (𝜑𝑋𝐵)
invfval.y (𝜑𝑌𝐵)
isoval.n 𝐼 = (Iso‘𝐶)
invinv.f (𝜑𝐹 ∈ (𝑋𝐼𝑌))
invco.o · = (comp‘𝐶)
invco.z (𝜑𝑍𝐵)
invco.f (𝜑𝐺 ∈ (𝑌𝐼𝑍))
Assertion
Ref Expression
invco (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹)(𝑋𝑁𝑍)(((𝑋𝑁𝑌)‘𝐹)(⟨𝑍, 𝑌· 𝑋)((𝑌𝑁𝑍)‘𝐺)))

Proof of Theorem invco
StepHypRef Expression
1 invfval.b . . 3 𝐵 = (Base‘𝐶)
2 invco.o . . 3 · = (comp‘𝐶)
3 eqid 2735 . . 3 (Sect‘𝐶) = (Sect‘𝐶)
4 invfval.c . . 3 (𝜑𝐶 ∈ Cat)
5 invfval.x . . 3 (𝜑𝑋𝐵)
6 invfval.y . . 3 (𝜑𝑌𝐵)
7 invco.z . . 3 (𝜑𝑍𝐵)
8 invinv.f . . . . . . 7 (𝜑𝐹 ∈ (𝑋𝐼𝑌))
9 invfval.n . . . . . . . 8 𝑁 = (Inv‘𝐶)
10 isoval.n . . . . . . . 8 𝐼 = (Iso‘𝐶)
111, 9, 4, 5, 6, 10isoval 17813 . . . . . . 7 (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))
128, 11eleqtrd 2841 . . . . . 6 (𝜑𝐹 ∈ dom (𝑋𝑁𝑌))
131, 9, 4, 5, 6invfun 17812 . . . . . . 7 (𝜑 → Fun (𝑋𝑁𝑌))
14 funfvbrb 7071 . . . . . . 7 (Fun (𝑋𝑁𝑌) → (𝐹 ∈ dom (𝑋𝑁𝑌) ↔ 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)))
1513, 14syl 17 . . . . . 6 (𝜑 → (𝐹 ∈ dom (𝑋𝑁𝑌) ↔ 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)))
1612, 15mpbid 232 . . . . 5 (𝜑𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))
171, 9, 4, 5, 6, 3isinv 17808 . . . . 5 (𝜑 → (𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹) ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ∧ ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)))
1816, 17mpbid 232 . . . 4 (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ∧ ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹))
1918simpld 494 . . 3 (𝜑𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹))
20 invco.f . . . . . . 7 (𝜑𝐺 ∈ (𝑌𝐼𝑍))
211, 9, 4, 6, 7, 10isoval 17813 . . . . . . 7 (𝜑 → (𝑌𝐼𝑍) = dom (𝑌𝑁𝑍))
2220, 21eleqtrd 2841 . . . . . 6 (𝜑𝐺 ∈ dom (𝑌𝑁𝑍))
231, 9, 4, 6, 7invfun 17812 . . . . . . 7 (𝜑 → Fun (𝑌𝑁𝑍))
24 funfvbrb 7071 . . . . . . 7 (Fun (𝑌𝑁𝑍) → (𝐺 ∈ dom (𝑌𝑁𝑍) ↔ 𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺)))
2523, 24syl 17 . . . . . 6 (𝜑 → (𝐺 ∈ dom (𝑌𝑁𝑍) ↔ 𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺)))
2622, 25mpbid 232 . . . . 5 (𝜑𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺))
271, 9, 4, 6, 7, 3isinv 17808 . . . . 5 (𝜑 → (𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺) ↔ (𝐺(𝑌(Sect‘𝐶)𝑍)((𝑌𝑁𝑍)‘𝐺) ∧ ((𝑌𝑁𝑍)‘𝐺)(𝑍(Sect‘𝐶)𝑌)𝐺)))
2826, 27mpbid 232 . . . 4 (𝜑 → (𝐺(𝑌(Sect‘𝐶)𝑍)((𝑌𝑁𝑍)‘𝐺) ∧ ((𝑌𝑁𝑍)‘𝐺)(𝑍(Sect‘𝐶)𝑌)𝐺))
2928simpld 494 . . 3 (𝜑𝐺(𝑌(Sect‘𝐶)𝑍)((𝑌𝑁𝑍)‘𝐺))
301, 2, 3, 4, 5, 6, 7, 19, 29sectco 17804 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹)(𝑋(Sect‘𝐶)𝑍)(((𝑋𝑁𝑌)‘𝐹)(⟨𝑍, 𝑌· 𝑋)((𝑌𝑁𝑍)‘𝐺)))
3128simprd 495 . . 3 (𝜑 → ((𝑌𝑁𝑍)‘𝐺)(𝑍(Sect‘𝐶)𝑌)𝐺)
3218simprd 495 . . 3 (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)
331, 2, 3, 4, 7, 6, 5, 31, 32sectco 17804 . 2 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(⟨𝑍, 𝑌· 𝑋)((𝑌𝑁𝑍)‘𝐺))(𝑍(Sect‘𝐶)𝑋)(𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))
341, 9, 4, 5, 7, 3isinv 17808 . 2 (𝜑 → ((𝐺(⟨𝑋, 𝑌· 𝑍)𝐹)(𝑋𝑁𝑍)(((𝑋𝑁𝑌)‘𝐹)(⟨𝑍, 𝑌· 𝑋)((𝑌𝑁𝑍)‘𝐺)) ↔ ((𝐺(⟨𝑋, 𝑌· 𝑍)𝐹)(𝑋(Sect‘𝐶)𝑍)(((𝑋𝑁𝑌)‘𝐹)(⟨𝑍, 𝑌· 𝑋)((𝑌𝑁𝑍)‘𝐺)) ∧ (((𝑋𝑁𝑌)‘𝐹)(⟨𝑍, 𝑌· 𝑋)((𝑌𝑁𝑍)‘𝐺))(𝑍(Sect‘𝐶)𝑋)(𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))))
3530, 33, 34mpbir2and 713 1 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹)(𝑋𝑁𝑍)(((𝑋𝑁𝑌)‘𝐹)(⟨𝑍, 𝑌· 𝑋)((𝑌𝑁𝑍)‘𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  cop 4637   class class class wbr 5148  dom cdm 5689  Fun wfun 6557  cfv 6563  (class class class)co 7431  Basecbs 17245  compcco 17310  Catccat 17709  Sectcsect 17792  Invcinv 17793  Isociso 17794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-cat 17713  df-cid 17714  df-sect 17795  df-inv 17796  df-iso 17797
This theorem is referenced by:  isoco  17825  invisoinvl  17838
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