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Theorem invco 16783
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 2825 . . 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 16777 . . . . . . 7 (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))
128, 11eleqtrd 2908 . . . . . 6 (𝜑𝐹 ∈ dom (𝑋𝑁𝑌))
131, 9, 4, 5, 6invfun 16776 . . . . . . 7 (𝜑 → Fun (𝑋𝑁𝑌))
14 funfvbrb 6579 . . . . . . 7 (Fun (𝑋𝑁𝑌) → (𝐹 ∈ dom (𝑋𝑁𝑌) ↔ 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)))
1513, 14syl 17 . . . . . 6 (𝜑 → (𝐹 ∈ dom (𝑋𝑁𝑌) ↔ 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)))
1612, 15mpbid 224 . . . . 5 (𝜑𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))
171, 9, 4, 5, 6, 3isinv 16772 . . . . 5 (𝜑 → (𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹) ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ∧ ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)))
1816, 17mpbid 224 . . . 4 (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ∧ ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹))
1918simpld 490 . . 3 (𝜑𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹))
20 invco.f . . . . . . 7 (𝜑𝐺 ∈ (𝑌𝐼𝑍))
211, 9, 4, 6, 7, 10isoval 16777 . . . . . . 7 (𝜑 → (𝑌𝐼𝑍) = dom (𝑌𝑁𝑍))
2220, 21eleqtrd 2908 . . . . . 6 (𝜑𝐺 ∈ dom (𝑌𝑁𝑍))
231, 9, 4, 6, 7invfun 16776 . . . . . . 7 (𝜑 → Fun (𝑌𝑁𝑍))
24 funfvbrb 6579 . . . . . . 7 (Fun (𝑌𝑁𝑍) → (𝐺 ∈ dom (𝑌𝑁𝑍) ↔ 𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺)))
2523, 24syl 17 . . . . . 6 (𝜑 → (𝐺 ∈ dom (𝑌𝑁𝑍) ↔ 𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺)))
2622, 25mpbid 224 . . . . 5 (𝜑𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺))
271, 9, 4, 6, 7, 3isinv 16772 . . . . 5 (𝜑 → (𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺) ↔ (𝐺(𝑌(Sect‘𝐶)𝑍)((𝑌𝑁𝑍)‘𝐺) ∧ ((𝑌𝑁𝑍)‘𝐺)(𝑍(Sect‘𝐶)𝑌)𝐺)))
2826, 27mpbid 224 . . . 4 (𝜑 → (𝐺(𝑌(Sect‘𝐶)𝑍)((𝑌𝑁𝑍)‘𝐺) ∧ ((𝑌𝑁𝑍)‘𝐺)(𝑍(Sect‘𝐶)𝑌)𝐺))
2928simpld 490 . . 3 (𝜑𝐺(𝑌(Sect‘𝐶)𝑍)((𝑌𝑁𝑍)‘𝐺))
301, 2, 3, 4, 5, 6, 7, 19, 29sectco 16768 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹)(𝑋(Sect‘𝐶)𝑍)(((𝑋𝑁𝑌)‘𝐹)(⟨𝑍, 𝑌· 𝑋)((𝑌𝑁𝑍)‘𝐺)))
3128simprd 491 . . 3 (𝜑 → ((𝑌𝑁𝑍)‘𝐺)(𝑍(Sect‘𝐶)𝑌)𝐺)
3218simprd 491 . . 3 (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)
331, 2, 3, 4, 7, 6, 5, 31, 32sectco 16768 . 2 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(⟨𝑍, 𝑌· 𝑋)((𝑌𝑁𝑍)‘𝐺))(𝑍(Sect‘𝐶)𝑋)(𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))
341, 9, 4, 5, 7, 3isinv 16772 . 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 706 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 198  wa 386   = wceq 1658  wcel 2166  cop 4403   class class class wbr 4873  dom cdm 5342  Fun wfun 6117  cfv 6123  (class class class)co 6905  Basecbs 16222  compcco 16317  Catccat 16677  Sectcsect 16756  Invcinv 16757  Isociso 16758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-cat 16681  df-cid 16682  df-sect 16759  df-inv 16760  df-iso 16761
This theorem is referenced by:  isoco  16789  invisoinvl  16802
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