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Theorem invco 16371
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 2621 . . 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 16365 . . . . . . 7 (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))
128, 11eleqtrd 2700 . . . . . 6 (𝜑𝐹 ∈ dom (𝑋𝑁𝑌))
131, 9, 4, 5, 6invfun 16364 . . . . . . 7 (𝜑 → Fun (𝑋𝑁𝑌))
14 funfvbrb 6296 . . . . . . 7 (Fun (𝑋𝑁𝑌) → (𝐹 ∈ dom (𝑋𝑁𝑌) ↔ 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)))
1513, 14syl 17 . . . . . 6 (𝜑 → (𝐹 ∈ dom (𝑋𝑁𝑌) ↔ 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)))
1612, 15mpbid 222 . . . . 5 (𝜑𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))
171, 9, 4, 5, 6, 3isinv 16360 . . . . 5 (𝜑 → (𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹) ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ∧ ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)))
1816, 17mpbid 222 . . . 4 (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ∧ ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹))
1918simpld 475 . . 3 (𝜑𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹))
20 invco.f . . . . . . 7 (𝜑𝐺 ∈ (𝑌𝐼𝑍))
211, 9, 4, 6, 7, 10isoval 16365 . . . . . . 7 (𝜑 → (𝑌𝐼𝑍) = dom (𝑌𝑁𝑍))
2220, 21eleqtrd 2700 . . . . . 6 (𝜑𝐺 ∈ dom (𝑌𝑁𝑍))
231, 9, 4, 6, 7invfun 16364 . . . . . . 7 (𝜑 → Fun (𝑌𝑁𝑍))
24 funfvbrb 6296 . . . . . . 7 (Fun (𝑌𝑁𝑍) → (𝐺 ∈ dom (𝑌𝑁𝑍) ↔ 𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺)))
2523, 24syl 17 . . . . . 6 (𝜑 → (𝐺 ∈ dom (𝑌𝑁𝑍) ↔ 𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺)))
2622, 25mpbid 222 . . . . 5 (𝜑𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺))
271, 9, 4, 6, 7, 3isinv 16360 . . . . 5 (𝜑 → (𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺) ↔ (𝐺(𝑌(Sect‘𝐶)𝑍)((𝑌𝑁𝑍)‘𝐺) ∧ ((𝑌𝑁𝑍)‘𝐺)(𝑍(Sect‘𝐶)𝑌)𝐺)))
2826, 27mpbid 222 . . . 4 (𝜑 → (𝐺(𝑌(Sect‘𝐶)𝑍)((𝑌𝑁𝑍)‘𝐺) ∧ ((𝑌𝑁𝑍)‘𝐺)(𝑍(Sect‘𝐶)𝑌)𝐺))
2928simpld 475 . . 3 (𝜑𝐺(𝑌(Sect‘𝐶)𝑍)((𝑌𝑁𝑍)‘𝐺))
301, 2, 3, 4, 5, 6, 7, 19, 29sectco 16356 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹)(𝑋(Sect‘𝐶)𝑍)(((𝑋𝑁𝑌)‘𝐹)(⟨𝑍, 𝑌· 𝑋)((𝑌𝑁𝑍)‘𝐺)))
3128simprd 479 . . 3 (𝜑 → ((𝑌𝑁𝑍)‘𝐺)(𝑍(Sect‘𝐶)𝑌)𝐺)
3218simprd 479 . . 3 (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)
331, 2, 3, 4, 7, 6, 5, 31, 32sectco 16356 . 2 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(⟨𝑍, 𝑌· 𝑋)((𝑌𝑁𝑍)‘𝐺))(𝑍(Sect‘𝐶)𝑋)(𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))
341, 9, 4, 5, 7, 3isinv 16360 . 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 956 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 196  wa 384   = wceq 1480  wcel 1987  cop 4161   class class class wbr 4623  dom cdm 5084  Fun wfun 5851  cfv 5857  (class class class)co 6615  Basecbs 15800  compcco 15893  Catccat 16265  Sectcsect 16344  Invcinv 16345  Isociso 16346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-cat 16269  df-cid 16270  df-sect 16347  df-inv 16348  df-iso 16349
This theorem is referenced by:  isoco  16377  invisoinvl  16390
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