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Theorem catcocl 16731
 Description: Closure of a composition arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
catcocl.b 𝐵 = (Base‘𝐶)
catcocl.h 𝐻 = (Hom ‘𝐶)
catcocl.o · = (comp‘𝐶)
catcocl.c (𝜑𝐶 ∈ Cat)
catcocl.x (𝜑𝑋𝐵)
catcocl.y (𝜑𝑌𝐵)
catcocl.z (𝜑𝑍𝐵)
catcocl.f (𝜑𝐹 ∈ (𝑋𝐻𝑌))
catcocl.g (𝜑𝐺 ∈ (𝑌𝐻𝑍))
Assertion
Ref Expression
catcocl (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐻𝑍))

Proof of Theorem catcocl
Dummy variables 𝑓 𝑔 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catcocl.c . . 3 (𝜑𝐶 ∈ Cat)
2 catcocl.b . . . . 5 𝐵 = (Base‘𝐶)
3 catcocl.h . . . . 5 𝐻 = (Hom ‘𝐶)
4 catcocl.o . . . . 5 · = (comp‘𝐶)
52, 3, 4iscat 16718 . . . 4 (𝐶 ∈ Cat → (𝐶 ∈ Cat ↔ ∀𝑥𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓) ∧ ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑣 ∈ (𝑧𝐻𝑤)((𝑣(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑣(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))))))
65ibi 259 . . 3 (𝐶 ∈ Cat → ∀𝑥𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓) ∧ ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑣 ∈ (𝑧𝐻𝑤)((𝑣(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑣(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))))
7 simpl 476 . . . . . . 7 (((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑣 ∈ (𝑧𝐻𝑤)((𝑣(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑣(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
872ralimi 3134 . . . . . 6 (∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑣 ∈ (𝑧𝐻𝑤)((𝑣(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑣(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))) → ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
982ralimi 3134 . . . . 5 (∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑣 ∈ (𝑧𝐻𝑤)((𝑣(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑣(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓))) → ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
109adantl 475 . . . 4 ((∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓) ∧ ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑣 ∈ (𝑧𝐻𝑤)((𝑣(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑣(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))) → ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
1110ralimi 3133 . . 3 (∀𝑥𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥· 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥· 𝑦)𝑔) = 𝑓) ∧ ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤𝐵𝑣 ∈ (𝑧𝐻𝑤)((𝑣(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑣(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))) → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
121, 6, 113syl 18 . 2 (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
13 catcocl.x . . 3 (𝜑𝑋𝐵)
14 catcocl.y . . . . 5 (𝜑𝑌𝐵)
1514adantr 474 . . . 4 ((𝜑𝑥 = 𝑋) → 𝑌𝐵)
16 catcocl.z . . . . . 6 (𝜑𝑍𝐵)
1716ad2antrr 716 . . . . 5 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑍𝐵)
18 catcocl.f . . . . . . . 8 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
1918ad3antrrr 720 . . . . . . 7 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝐹 ∈ (𝑋𝐻𝑌))
20 simpllr 766 . . . . . . . 8 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑥 = 𝑋)
21 simplr 759 . . . . . . . 8 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑦 = 𝑌)
2220, 21oveq12d 6940 . . . . . . 7 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
2319, 22eleqtrrd 2861 . . . . . 6 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝐹 ∈ (𝑥𝐻𝑦))
24 catcocl.g . . . . . . . . . 10 (𝜑𝐺 ∈ (𝑌𝐻𝑍))
2524ad3antrrr 720 . . . . . . . . 9 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝐺 ∈ (𝑌𝐻𝑍))
26 simpr 479 . . . . . . . . . 10 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑧 = 𝑍)
2721, 26oveq12d 6940 . . . . . . . . 9 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (𝑦𝐻𝑧) = (𝑌𝐻𝑍))
2825, 27eleqtrrd 2861 . . . . . . . 8 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝐺 ∈ (𝑦𝐻𝑧))
2928adantr 474 . . . . . . 7 (((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → 𝐺 ∈ (𝑦𝐻𝑧))
30 simp-5r 776 . . . . . . . . . . 11 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑥 = 𝑋)
31 simp-4r 774 . . . . . . . . . . 11 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑦 = 𝑌)
3230, 31opeq12d 4644 . . . . . . . . . 10 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩)
33 simpllr 766 . . . . . . . . . 10 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑧 = 𝑍)
3432, 33oveq12d 6940 . . . . . . . . 9 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (⟨𝑥, 𝑦· 𝑧) = (⟨𝑋, 𝑌· 𝑍))
35 simpr 479 . . . . . . . . 9 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
36 simplr 759 . . . . . . . . 9 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
3734, 35, 36oveq123d 6943 . . . . . . . 8 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) = (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))
3830, 33oveq12d 6940 . . . . . . . 8 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑥𝐻𝑧) = (𝑋𝐻𝑍))
3937, 38eleq12d 2852 . . . . . . 7 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ↔ (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐻𝑍)))
4029, 39rspcdv 3513 . . . . . 6 (((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → (∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐻𝑍)))
4123, 40rspcimdv 3511 . . . . 5 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐻𝑍)))
4217, 41rspcimdv 3511 . . . 4 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (∀𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐻𝑍)))
4315, 42rspcimdv 3511 . . 3 ((𝜑𝑥 = 𝑋) → (∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐻𝑍)))
4413, 43rspcimdv 3511 . 2 (𝜑 → (∀𝑥𝐵𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧) → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐻𝑍)))
4512, 44mpd 15 1 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐻𝑍))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2106  ∀wral 3089  ∃wrex 3090  ⟨cop 4403  ‘cfv 6135  (class class class)co 6922  Basecbs 16255  Hom chom 16349  compcco 16350  Catccat 16710 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-nul 5025 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-iota 6099  df-fv 6143  df-ov 6925  df-cat 16714 This theorem is referenced by:  oppccatid  16764  ismon2  16779  isepi2  16786  sectco  16801  monsect  16828  catsubcat  16884  issubc3  16894  fullsubc  16895  idfucl  16926  cofucl  16933  fthsect  16970  fthmon  16972  fuccocl  17009  invfuc  17019  2initoinv  17045  initoeu2lem0  17048  initoeu2lem1  17049  initoeu2  17051  2termoinv  17052  coahom  17105  catcisolem  17141  xpccatid  17214  1stfcl  17223  2ndfcl  17224  prfcl  17229  evlfcllem  17247  evlfcl  17248  curf1cl  17254  curfcl  17258  hofcllem  17284  hofcl  17285  yon12  17291  hofpropd  17293  yonedalem4c  17303  srhmsubc  43073  srhmsubcALTV  43091
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