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Theorem invco 16196
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 2605 . . 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 16190 . . . . . . 7 (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))
128, 11eleqtrd 2685 . . . . . 6 (𝜑𝐹 ∈ dom (𝑋𝑁𝑌))
131, 9, 4, 5, 6invfun 16189 . . . . . . 7 (𝜑 → Fun (𝑋𝑁𝑌))
14 funfvbrb 6219 . . . . . . 7 (Fun (𝑋𝑁𝑌) → (𝐹 ∈ dom (𝑋𝑁𝑌) ↔ 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)))
1513, 14syl 17 . . . . . 6 (𝜑 → (𝐹 ∈ dom (𝑋𝑁𝑌) ↔ 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)))
1612, 15mpbid 220 . . . . 5 (𝜑𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))
171, 9, 4, 5, 6, 3isinv 16185 . . . . 5 (𝜑 → (𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹) ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ∧ ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)))
1816, 17mpbid 220 . . . 4 (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ∧ ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹))
1918simpld 473 . . 3 (𝜑𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹))
20 invco.f . . . . . . 7 (𝜑𝐺 ∈ (𝑌𝐼𝑍))
211, 9, 4, 6, 7, 10isoval 16190 . . . . . . 7 (𝜑 → (𝑌𝐼𝑍) = dom (𝑌𝑁𝑍))
2220, 21eleqtrd 2685 . . . . . 6 (𝜑𝐺 ∈ dom (𝑌𝑁𝑍))
231, 9, 4, 6, 7invfun 16189 . . . . . . 7 (𝜑 → Fun (𝑌𝑁𝑍))
24 funfvbrb 6219 . . . . . . 7 (Fun (𝑌𝑁𝑍) → (𝐺 ∈ dom (𝑌𝑁𝑍) ↔ 𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺)))
2523, 24syl 17 . . . . . 6 (𝜑 → (𝐺 ∈ dom (𝑌𝑁𝑍) ↔ 𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺)))
2622, 25mpbid 220 . . . . 5 (𝜑𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺))
271, 9, 4, 6, 7, 3isinv 16185 . . . . 5 (𝜑 → (𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺) ↔ (𝐺(𝑌(Sect‘𝐶)𝑍)((𝑌𝑁𝑍)‘𝐺) ∧ ((𝑌𝑁𝑍)‘𝐺)(𝑍(Sect‘𝐶)𝑌)𝐺)))
2826, 27mpbid 220 . . . 4 (𝜑 → (𝐺(𝑌(Sect‘𝐶)𝑍)((𝑌𝑁𝑍)‘𝐺) ∧ ((𝑌𝑁𝑍)‘𝐺)(𝑍(Sect‘𝐶)𝑌)𝐺))
2928simpld 473 . . 3 (𝜑𝐺(𝑌(Sect‘𝐶)𝑍)((𝑌𝑁𝑍)‘𝐺))
301, 2, 3, 4, 5, 6, 7, 19, 29sectco 16181 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹)(𝑋(Sect‘𝐶)𝑍)(((𝑋𝑁𝑌)‘𝐹)(⟨𝑍, 𝑌· 𝑋)((𝑌𝑁𝑍)‘𝐺)))
3128simprd 477 . . 3 (𝜑 → ((𝑌𝑁𝑍)‘𝐺)(𝑍(Sect‘𝐶)𝑌)𝐺)
3218simprd 477 . . 3 (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)
331, 2, 3, 4, 7, 6, 5, 31, 32sectco 16181 . 2 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(⟨𝑍, 𝑌· 𝑋)((𝑌𝑁𝑍)‘𝐺))(𝑍(Sect‘𝐶)𝑋)(𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))
341, 9, 4, 5, 7, 3isinv 16185 . 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 958 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 194  wa 382   = wceq 1474  wcel 1975  cop 4126   class class class wbr 4573  dom cdm 5024  Fun wfun 5780  cfv 5786  (class class class)co 6523  Basecbs 15637  compcco 15722  Catccat 16090  Sectcsect 16169  Invcinv 16170  Isociso 16171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-1st 7032  df-2nd 7033  df-cat 16094  df-cid 16095  df-sect 16172  df-inv 16173  df-iso 16174
This theorem is referenced by:  isoco  16202  invisoinvl  16215
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