Theorem invco 17400

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 17394	. . . . . . 7 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))
12	8, 11	eleqtrd 2841	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ dom (𝑋𝑁𝑌))
13	1, 9, 4, 5, 6	invfun 17393	. . . . . . 7 ⊢ (𝜑 → Fun (𝑋𝑁𝑌))
14		funfvbrb 6910	. . . . . . 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 231	. . . . 5 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))
17	1, 9, 4, 5, 6, 3	isinv 17389	. . . . 5 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹) ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ∧ ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)))
18	16, 17	mpbid 231	. . . 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 17394	. . . . . . 7 ⊢ (𝜑 → (𝑌𝐼𝑍) = dom (𝑌𝑁𝑍))
22	20, 21	eleqtrd 2841	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ dom (𝑌𝑁𝑍))
23	1, 9, 4, 6, 7	invfun 17393	. . . . . . 7 ⊢ (𝜑 → Fun (𝑌𝑁𝑍))
24		funfvbrb 6910	. . . . . . 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 231	. . . . 5 ⊢ (𝜑 → 𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺))
27	1, 9, 4, 6, 7, 3	isinv 17389	. . . . 5 ⊢ (𝜑 → (𝐺(𝑌𝑁𝑍)((𝑌𝑁𝑍)‘𝐺) ↔ (𝐺(𝑌(Sect‘𝐶)𝑍)((𝑌𝑁𝑍)‘𝐺) ∧ ((𝑌𝑁𝑍)‘𝐺)(𝑍(Sect‘𝐶)𝑌)𝐺)))
28	26, 27	mpbid 231	. . . 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 17385	. 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 17385	. 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 17389	. 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 709	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)(𝑋𝑁𝑍)(((𝑋𝑁𝑌)‘𝐹)(⟨𝑍, 𝑌⟩ · 𝑋)((𝑌𝑁𝑍)‘𝐺)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ⟨cop 4564   class class class wbr 5070  dom cdm 5580  Fun wfun 6412  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  compcco 16900  Catccat 17290  Sectcsect 17373  Invcinv 17374  Isociso 17375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-cat 17294  df-cid 17295  df-sect 17376  df-inv 17377  df-iso 17378

This theorem is referenced by:  isoco  17406  invisoinvl  17419