Theorem invco 17733

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	⊢ (𝜑 → 𝐶 ∈ Cat)
invss.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
invss.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
isoval.n	⊢ 𝐼 = (Iso‘𝐶)
invinv.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))
invco.o	⊢ · = (comp‘𝐶)
invco.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
invco.f	⊢ (𝜑 → 𝐺 ∈ (𝑌𝐼𝑍))

Assertion

Ref	Expression
invco	⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)(𝑋𝑁𝑍)(((𝑋𝑁𝑌)‘𝐹)(⟨𝑍, 𝑌⟩ · 𝑋)((𝑌𝑁𝑍)‘𝐺)))

Proof of Theorem invco

Step	Hyp	Ref	Expression
1		invfval.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		invco.o	. . 3 ⊢ · = (comp‘𝐶)
3		eqid 2729	. . 3 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
4		invfval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
5		invss.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		invss.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
7		invco.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
8		invinv.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))
9		invfval.n	. . . . . . . 8 ⊢ 𝑁 = (Inv‘𝐶)
10		isoval.n	. . . . . . . 8 ⊢ 𝐼 = (Iso‘𝐶)
11	1, 9, 4, 5, 6, 10	isoval 17727	. . . . . . 7 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))
12	8, 11	eleqtrd 2830	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ dom (𝑋𝑁𝑌))
13	1, 9, 4, 5, 6	invfun 17726	. . . . . . 7 ⊢ (𝜑 → Fun (𝑋𝑁𝑌))
14		funfvbrb 7023	. . . . . . 7 ⊢ (Fun (𝑋𝑁𝑌) → (𝐹 ∈ dom (𝑋𝑁𝑌) ↔ 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)))
15	13, 14	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋𝑁𝑌) ↔ 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)))
16	12, 15	mpbid 232	. . . . 5 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))
17	1, 9, 4, 5, 6, 3	isinv 17722	. . . . 5 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹) ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ∧ ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)))
18	16, 17	mpbid 232	. . . 4 ⊢ (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ∧ ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹))
19	18	simpld 494	. . 3 ⊢ (𝜑 → 𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹))
20		invco.f	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐼𝑍))
21	1, 9, 4, 6, 7, 10	isoval 17727	. . . . . . 7 ⊢ (𝜑 → (𝑌𝐼𝑍) = dom (𝑌𝑁𝑍))
22	20, 21	eleqtrd 2830	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ dom (𝑌𝑁𝑍))
23	1, 9, 4, 6, 7	invfun 17726	. . . . . . 7 ⊢ (𝜑 → Fun (𝑌𝑁𝑍))
24		funfvbrb 7023	. . . . . . 7 ⊢ (Fun (𝑌𝑁𝑍) → (𝐺 ∈ dom (𝑌𝑁𝑍) ↔ 𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺)))
25	23, 24	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐺 ∈ dom (𝑌𝑁𝑍) ↔ 𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺)))
26	22, 25	mpbid 232	. . . . 5 ⊢ (𝜑 → 𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺))
27	1, 9, 4, 6, 7, 3	isinv 17722	. . . . 5 ⊢ (𝜑 → (𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺) ↔ (𝐺(𝑌(Sect‘𝐶)𝑍)((𝑌𝑁𝑍)‘𝐺) ∧ ((𝑌𝑁𝑍)‘𝐺)(𝑍(Sect‘𝐶)𝑌)𝐺)))
28	26, 27	mpbid 232	. . . 4 ⊢ (𝜑 → (𝐺(𝑌(Sect‘𝐶)𝑍)((𝑌𝑁𝑍)‘𝐺) ∧ ((𝑌𝑁𝑍)‘𝐺)(𝑍(Sect‘𝐶)𝑌)𝐺))
29	28	simpld 494	. . 3 ⊢ (𝜑 → 𝐺(𝑌(Sect‘𝐶)𝑍)((𝑌𝑁𝑍)‘𝐺))
30	1, 2, 3, 4, 5, 6, 7, 19, 29	sectco 17718	. 2 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)(𝑋(Sect‘𝐶)𝑍)(((𝑋𝑁𝑌)‘𝐹)(⟨𝑍, 𝑌⟩ · 𝑋)((𝑌𝑁𝑍)‘𝐺)))
31	28	simprd 495	. . 3 ⊢ (𝜑 → ((𝑌𝑁𝑍)‘𝐺)(𝑍(Sect‘𝐶)𝑌)𝐺)
32	18	simprd 495	. . 3 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)
33	1, 2, 3, 4, 7, 6, 5, 31, 32	sectco 17718	. 2 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(⟨𝑍, 𝑌⟩ · 𝑋)((𝑌𝑁𝑍)‘𝐺))(𝑍(Sect‘𝐶)𝑋)(𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))
34	1, 9, 4, 5, 7, 3	isinv 17722	. 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))))
35	30, 33, 34	mpbir2and 713	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)(𝑋𝑁𝑍)(((𝑋𝑁𝑌)‘𝐹)(⟨𝑍, 𝑌⟩ · 𝑋)((𝑌𝑁𝑍)‘𝐺)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ⟨cop 4595   class class class wbr 5107  dom cdm 5638  Fun wfun 6505  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  compcco 17232  Catccat 17625  Sectcsect 17706  Invcinv 17707  Isociso 17708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-cat 17629  df-cid 17630  df-sect 17709  df-inv 17710  df-iso 17711

This theorem is referenced by:  isoco  17739  invisoinvl  17752