Theorem invco 17823

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 2763	. . 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 17817	. . . . . . 7 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))
12	8, 11	eleqtrd 2865	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ dom (𝑋𝑁𝑌))
13	1, 9, 4, 5, 6	invfun 17816	. . . . . . 7 ⊢ (𝜑 → Fun (𝑋𝑁𝑌))
14		funfvbrb 7046	. . . . . . 7 ⊢ (Fun (𝑋𝑁𝑌) → (𝐹 ∈ dom (𝑋𝑁𝑌) ↔ 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)))
15	13, 14	syl 18	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋𝑁𝑌) ↔ 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)))
16	12, 15	mpbid 235	. . . . 5 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))
17	1, 9, 4, 5, 6, 3	isinv 17812	. . . . 5 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹) ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ∧ ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)))
18	16, 17	mpbid 235	. . . 4 ⊢ (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ∧ ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹))
19	18	simpld 499	. . 3 ⊢ (𝜑 → 𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹))
20		invco.f	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐼𝑍))
21	1, 9, 4, 6, 7, 10	isoval 17817	. . . . . . 7 ⊢ (𝜑 → (𝑌𝐼𝑍) = dom (𝑌𝑁𝑍))
22	20, 21	eleqtrd 2865	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ dom (𝑌𝑁𝑍))
23	1, 9, 4, 6, 7	invfun 17816	. . . . . . 7 ⊢ (𝜑 → Fun (𝑌𝑁𝑍))
24		funfvbrb 7046	. . . . . . 7 ⊢ (Fun (𝑌𝑁𝑍) → (𝐺 ∈ dom (𝑌𝑁𝑍) ↔ 𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺)))
25	23, 24	syl 18	. . . . . 6 ⊢ (𝜑 → (𝐺 ∈ dom (𝑌𝑁𝑍) ↔ 𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺)))
26	22, 25	mpbid 235	. . . . 5 ⊢ (𝜑 → 𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺))
27	1, 9, 4, 6, 7, 3	isinv 17812	. . . . 5 ⊢ (𝜑 → (𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺) ↔ (𝐺(𝑌(Sect‘𝐶)𝑍)((𝑌𝑁𝑍)‘𝐺) ∧ ((𝑌𝑁𝑍)‘𝐺)(𝑍(Sect‘𝐶)𝑌)𝐺)))
28	26, 27	mpbid 235	. . . 4 ⊢ (𝜑 → (𝐺(𝑌(Sect‘𝐶)𝑍)((𝑌𝑁𝑍)‘𝐺) ∧ ((𝑌𝑁𝑍)‘𝐺)(𝑍(Sect‘𝐶)𝑌)𝐺))
29	28	simpld 499	. . 3 ⊢ (𝜑 → 𝐺(𝑌(Sect‘𝐶)𝑍)((𝑌𝑁𝑍)‘𝐺))
30	1, 2, 3, 4, 5, 6, 7, 19, 29	sectco 17808	. 2 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)(𝑋(Sect‘𝐶)𝑍)(((𝑋𝑁𝑌)‘𝐹)(⟨𝑍, 𝑌⟩ · 𝑋)((𝑌𝑁𝑍)‘𝐺)))
31	28	simprd 500	. . 3 ⊢ (𝜑 → ((𝑌𝑁𝑍)‘𝐺)(𝑍(Sect‘𝐶)𝑌)𝐺)
32	18	simprd 500	. . 3 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)
33	1, 2, 3, 4, 7, 6, 5, 31, 32	sectco 17808	. 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 17812	. 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 725	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)(𝑋𝑁𝑍)(((𝑋𝑁𝑌)‘𝐹)(⟨𝑍, 𝑌⟩ · 𝑋)((𝑌𝑁𝑍)‘𝐺)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ⟨cop 4595   class class class wbr 5109  dom cdm 5661  Fun wfun 6530  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  compcco 17317  Catccat 17715  Sectcsect 17796  Invcinv 17797  Isociso 17798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-cat 17719  df-cid 17720  df-sect 17799  df-inv 17800  df-iso 17801

This theorem is referenced by:  isoco  17829  invisoinvl  17842