Theorem invco 17483

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 17477	. . . . . . 7 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))
12	8, 11	eleqtrd 2841	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ dom (𝑋𝑁𝑌))
13	1, 9, 4, 5, 6	invfun 17476	. . . . . . 7 ⊢ (𝜑 → Fun (𝑋𝑁𝑌))
14		funfvbrb 6928	. . . . . . 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 17472	. . . . 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 495	. . 3 ⊢ (𝜑 → 𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹))
20		invco.f	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐼𝑍))
21	1, 9, 4, 6, 7, 10	isoval 17477	. . . . . . 7 ⊢ (𝜑 → (𝑌𝐼𝑍) = dom (𝑌𝑁𝑍))
22	20, 21	eleqtrd 2841	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ dom (𝑌𝑁𝑍))
23	1, 9, 4, 6, 7	invfun 17476	. . . . . . 7 ⊢ (𝜑 → Fun (𝑌𝑁𝑍))
24		funfvbrb 6928	. . . . . . 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 17472	. . . . 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 495	. . 3 ⊢ (𝜑 → 𝐺(𝑌(Sect‘𝐶)𝑍)((𝑌𝑁𝑍)‘𝐺))
30	1, 2, 3, 4, 5, 6, 7, 19, 29	sectco 17468	. 2 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)(𝑋(Sect‘𝐶)𝑍)(((𝑋𝑁𝑌)‘𝐹)(⟨𝑍, 𝑌⟩ · 𝑋)((𝑌𝑁𝑍)‘𝐺)))
31	28	simprd 496	. . 3 ⊢ (𝜑 → ((𝑌𝑁𝑍)‘𝐺)(𝑍(Sect‘𝐶)𝑌)𝐺)
32	18	simprd 496	. . 3 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)
33	1, 2, 3, 4, 7, 6, 5, 31, 32	sectco 17468	. 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 17472	. 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 710	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)(𝑋𝑁𝑍)(((𝑋𝑁𝑌)‘𝐹)(⟨𝑍, 𝑌⟩ · 𝑋)((𝑌𝑁𝑍)‘𝐺)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ⟨cop 4567   class class class wbr 5074  dom cdm 5589  Fun wfun 6427  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  compcco 16974  Catccat 17373  Sectcsect 17456  Invcinv 17457  Isociso 17458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-cat 17377  df-cid 17378  df-sect 17459  df-inv 17460  df-iso 17461

This theorem is referenced by:  isoco  17489  invisoinvl  17502