MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fuclid Structured version   Visualization version   GIF version

Theorem fuclid 17815
Description: Left identity of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
fuclid.q 𝑄 = (𝐶 FuncCat 𝐷)
fuclid.n 𝑁 = (𝐶 Nat 𝐷)
fuclid.x = (comp‘𝑄)
fuclid.1 1 = (Id‘𝐷)
fuclid.r (𝜑𝑅 ∈ (𝐹𝑁𝐺))
Assertion
Ref Expression
fuclid (𝜑 → (( 1 ∘ (1st𝐺))(⟨𝐹, 𝐺 𝐺)𝑅) = 𝑅)

Proof of Theorem fuclid
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2738 . . . . . . 7 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2738 . . . . . . 7 (Base‘𝐷) = (Base‘𝐷)
3 relfunc 17708 . . . . . . . 8 Rel (𝐶 Func 𝐷)
4 fuclid.r . . . . . . . . . 10 (𝜑𝑅 ∈ (𝐹𝑁𝐺))
5 fuclid.n . . . . . . . . . . 11 𝑁 = (𝐶 Nat 𝐷)
65natrcl 17797 . . . . . . . . . 10 (𝑅 ∈ (𝐹𝑁𝐺) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
74, 6syl 17 . . . . . . . . 9 (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
87simprd 497 . . . . . . . 8 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
9 1st2ndbr 7967 . . . . . . . 8 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
103, 8, 9sylancr 588 . . . . . . 7 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
111, 2, 10funcf1 17712 . . . . . 6 (𝜑 → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐷))
12 fvco3 6938 . . . . . 6 (((1st𝐺):(Base‘𝐶)⟶(Base‘𝐷) ∧ 𝑥 ∈ (Base‘𝐶)) → (( 1 ∘ (1st𝐺))‘𝑥) = ( 1 ‘((1st𝐺)‘𝑥)))
1311, 12sylan 581 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → (( 1 ∘ (1st𝐺))‘𝑥) = ( 1 ‘((1st𝐺)‘𝑥)))
1413oveq1d 7367 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((( 1 ∘ (1st𝐺))‘𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐺)‘𝑥))(𝑅𝑥)) = (( 1 ‘((1st𝐺)‘𝑥))(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐺)‘𝑥))(𝑅𝑥)))
15 eqid 2738 . . . . 5 (Hom ‘𝐷) = (Hom ‘𝐷)
16 fuclid.1 . . . . 5 1 = (Id‘𝐷)
177simpld 496 . . . . . . . 8 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
18 funcrcl 17709 . . . . . . . 8 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1917, 18syl 17 . . . . . . 7 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
2019simprd 497 . . . . . 6 (𝜑𝐷 ∈ Cat)
2120adantr 482 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
22 1st2ndbr 7967 . . . . . . . 8 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
233, 17, 22sylancr 588 . . . . . . 7 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
241, 2, 23funcf1 17712 . . . . . 6 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
2524ffvelcdmda 7032 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
26 eqid 2738 . . . . 5 (comp‘𝐷) = (comp‘𝐷)
2711ffvelcdmda 7032 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
285, 4nat1st2nd 17798 . . . . . . 7 (𝜑𝑅 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
2928adantr 482 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
30 simpr 486 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
315, 29, 1, 15, 30natcl 17800 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑅𝑥) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)))
322, 15, 16, 21, 25, 26, 27, 31catlid 17523 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → (( 1 ‘((1st𝐺)‘𝑥))(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐺)‘𝑥))(𝑅𝑥)) = (𝑅𝑥))
3314, 32eqtrd 2778 . . 3 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((( 1 ∘ (1st𝐺))‘𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐺)‘𝑥))(𝑅𝑥)) = (𝑅𝑥))
3433mpteq2dva 5204 . 2 (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ((( 1 ∘ (1st𝐺))‘𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐺)‘𝑥))(𝑅𝑥))) = (𝑥 ∈ (Base‘𝐶) ↦ (𝑅𝑥)))
35 fuclid.q . . 3 𝑄 = (𝐶 FuncCat 𝐷)
36 fuclid.x . . 3 = (comp‘𝑄)
3735, 5, 16, 8fucidcl 17814 . . 3 (𝜑 → ( 1 ∘ (1st𝐺)) ∈ (𝐺𝑁𝐺))
3835, 5, 1, 26, 36, 4, 37fucco 17811 . 2 (𝜑 → (( 1 ∘ (1st𝐺))(⟨𝐹, 𝐺 𝐺)𝑅) = (𝑥 ∈ (Base‘𝐶) ↦ ((( 1 ∘ (1st𝐺))‘𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐺)‘𝑥))(𝑅𝑥))))
395, 28, 1natfn 17801 . . 3 (𝜑𝑅 Fn (Base‘𝐶))
40 dffn5 6899 . . 3 (𝑅 Fn (Base‘𝐶) ↔ 𝑅 = (𝑥 ∈ (Base‘𝐶) ↦ (𝑅𝑥)))
4139, 40sylib 217 . 2 (𝜑𝑅 = (𝑥 ∈ (Base‘𝐶) ↦ (𝑅𝑥)))
4234, 38, 413eqtr4d 2788 1 (𝜑 → (( 1 ∘ (1st𝐺))(⟨𝐹, 𝐺 𝐺)𝑅) = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  cop 4591   class class class wbr 5104  cmpt 5187  ccom 5636  Rel wrel 5637   Fn wfn 6489  wf 6490  cfv 6494  (class class class)co 7352  1st c1st 7912  2nd c2nd 7913  Basecbs 17043  Hom chom 17104  compcco 17105  Catccat 17504  Idccid 17505   Func cfunc 17700   Nat cnat 17788   FuncCat cfuc 17789
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7665  ax-cnex 11066  ax-resscn 11067  ax-1cn 11068  ax-icn 11069  ax-addcl 11070  ax-addrcl 11071  ax-mulcl 11072  ax-mulrcl 11073  ax-mulcom 11074  ax-addass 11075  ax-mulass 11076  ax-distr 11077  ax-i2m1 11078  ax-1ne0 11079  ax-1rid 11080  ax-rnegex 11081  ax-rrecex 11082  ax-cnre 11083  ax-pre-lttri 11084  ax-pre-lttrn 11085  ax-pre-ltadd 11086  ax-pre-mulgt0 11087
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-tr 5222  df-id 5530  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5587  df-we 5589  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6252  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7308  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7796  df-1st 7914  df-2nd 7915  df-frecs 8205  df-wrecs 8236  df-recs 8310  df-rdg 8349  df-1o 8405  df-er 8607  df-map 8726  df-ixp 8795  df-en 8843  df-dom 8844  df-sdom 8845  df-fin 8846  df-pnf 11150  df-mnf 11151  df-xr 11152  df-ltxr 11153  df-le 11154  df-sub 11346  df-neg 11347  df-nn 12113  df-2 12175  df-3 12176  df-4 12177  df-5 12178  df-6 12179  df-7 12180  df-8 12181  df-9 12182  df-n0 12373  df-z 12459  df-dec 12578  df-uz 12723  df-fz 13380  df-struct 16979  df-slot 17014  df-ndx 17026  df-base 17044  df-hom 17117  df-cco 17118  df-cat 17508  df-cid 17509  df-func 17704  df-nat 17790  df-fuc 17791
This theorem is referenced by:  fuccatid  17818
  Copyright terms: Public domain W3C validator