Theorem invco 17139

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)
invfval.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
invfval.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 2738	. . 3 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
4		invfval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
5		invfval.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		invfval.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 17133	. . . . . . 7 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))
12	8, 11	eleqtrd 2835	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ dom (𝑋𝑁𝑌))
13	1, 9, 4, 5, 6	invfun 17132	. . . . . . 7 ⊢ (𝜑 → Fun (𝑋𝑁𝑌))
14		funfvbrb 6822	. . . . . . 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 235	. . . . 5 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))
17	1, 9, 4, 5, 6, 3	isinv 17128	. . . . 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 498	. . 3 ⊢ (𝜑 → 𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹))
20		invco.f	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐼𝑍))
21	1, 9, 4, 6, 7, 10	isoval 17133	. . . . . . 7 ⊢ (𝜑 → (𝑌𝐼𝑍) = dom (𝑌𝑁𝑍))
22	20, 21	eleqtrd 2835	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ dom (𝑌𝑁𝑍))
23	1, 9, 4, 6, 7	invfun 17132	. . . . . . 7 ⊢ (𝜑 → Fun (𝑌𝑁𝑍))
24		funfvbrb 6822	. . . . . . 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 235	. . . . 5 ⊢ (𝜑 → 𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺))
27	1, 9, 4, 6, 7, 3	isinv 17128	. . . . 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 498	. . 3 ⊢ (𝜑 → 𝐺(𝑌(Sect‘𝐶)𝑍)((𝑌𝑁𝑍)‘𝐺))
30	1, 2, 3, 4, 5, 6, 7, 19, 29	sectco 17124	. 2 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)(𝑋(Sect‘𝐶)𝑍)(((𝑋𝑁𝑌)‘𝐹)(⟨𝑍, 𝑌⟩ · 𝑋)((𝑌𝑁𝑍)‘𝐺)))
31	28	simprd 499	. . 3 ⊢ (𝜑 → ((𝑌𝑁𝑍)‘𝐺)(𝑍(Sect‘𝐶)𝑌)𝐺)
32	18	simprd 499	. . 3 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)
33	1, 2, 3, 4, 7, 6, 5, 31, 32	sectco 17124	. 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 17128	. 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 209   ∧ wa 399   = wceq 1542   ∈ wcel 2113  ⟨cop 4519   class class class wbr 5027  dom cdm 5519  Fun wfun 6327  ‘cfv 6333  (class class class)co 7164  Basecbs 16579  compcco 16673  Catccat 17031  Sectcsect 17112  Invcinv 17113  Isociso 17114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-oprab 7168  df-mpo 7169  df-1st 7707  df-2nd 7708  df-cat 17035  df-cid 17036  df-sect 17115  df-inv 17116  df-iso 17117

This theorem is referenced by:  isoco  17145  invisoinvl  17158