MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  comfffval2 Structured version   Visualization version   GIF version

Theorem comfffval2 17581
Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
comfffval2.o 𝑂 = (compf𝐶)
comfffval2.b 𝐵 = (Base‘𝐶)
comfffval2.h 𝐻 = (Homf𝐶)
comfffval2.x · = (comp‘𝐶)
Assertion
Ref Expression
comfffval2 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
Distinct variable groups:   𝑓,𝑔,𝑥,𝑦,𝐵   𝐶,𝑓,𝑔,𝑥,𝑦   · ,𝑓,𝑔,𝑥
Allowed substitution hints:   · (𝑦)   𝐻(𝑥,𝑦,𝑓,𝑔)   𝑂(𝑥,𝑦,𝑓,𝑔)

Proof of Theorem comfffval2
StepHypRef Expression
1 comfffval2.o . . 3 𝑂 = (compf𝐶)
2 comfffval2.b . . 3 𝐵 = (Base‘𝐶)
3 eqid 2736 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
4 comfffval2.x . . 3 · = (comp‘𝐶)
51, 2, 3, 4comfffval 17578 . 2 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)𝑦), 𝑓 ∈ ((Hom ‘𝐶)‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
6 comfffval2.h . . . . 5 𝐻 = (Homf𝐶)
7 xp2nd 7954 . . . . . 6 (𝑥 ∈ (𝐵 × 𝐵) → (2nd𝑥) ∈ 𝐵)
87adantr 481 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (2nd𝑥) ∈ 𝐵)
9 simpr 485 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → 𝑦𝐵)
106, 2, 3, 8, 9homfval 17572 . . . 4 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → ((2nd𝑥)𝐻𝑦) = ((2nd𝑥)(Hom ‘𝐶)𝑦))
11 xp1st 7953 . . . . . . . 8 (𝑥 ∈ (𝐵 × 𝐵) → (1st𝑥) ∈ 𝐵)
1211adantr 481 . . . . . . 7 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (1st𝑥) ∈ 𝐵)
136, 2, 3, 12, 8homfval 17572 . . . . . 6 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → ((1st𝑥)𝐻(2nd𝑥)) = ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)))
14 df-ov 7360 . . . . . 6 ((1st𝑥)𝐻(2nd𝑥)) = (𝐻‘⟨(1st𝑥), (2nd𝑥)⟩)
15 df-ov 7360 . . . . . 6 ((1st𝑥)(Hom ‘𝐶)(2nd𝑥)) = ((Hom ‘𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩)
1613, 14, 153eqtr3g 2799 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (𝐻‘⟨(1st𝑥), (2nd𝑥)⟩) = ((Hom ‘𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩))
17 1st2nd2 7960 . . . . . . 7 (𝑥 ∈ (𝐵 × 𝐵) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
1817adantr 481 . . . . . 6 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
1918fveq2d 6846 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (𝐻𝑥) = (𝐻‘⟨(1st𝑥), (2nd𝑥)⟩))
2018fveq2d 6846 . . . . 5 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → ((Hom ‘𝐶)‘𝑥) = ((Hom ‘𝐶)‘⟨(1st𝑥), (2nd𝑥)⟩))
2116, 19, 203eqtr4d 2786 . . . 4 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (𝐻𝑥) = ((Hom ‘𝐶)‘𝑥))
22 eqidd 2737 . . . 4 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (𝑔(𝑥 · 𝑦)𝑓) = (𝑔(𝑥 · 𝑦)𝑓))
2310, 21, 22mpoeq123dv 7432 . . 3 ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦𝐵) → (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)) = (𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)𝑦), 𝑓 ∈ ((Hom ‘𝐶)‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
2423mpoeq3ia 7435 . 2 (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)𝑦), 𝑓 ∈ ((Hom ‘𝐶)‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
255, 24eqtr4i 2767 1 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wcel 2106  cop 4592   × cxp 5631  cfv 6496  (class class class)co 7357  cmpo 7359  1st c1st 7919  2nd c2nd 7920  Basecbs 17083  Hom chom 17144  compcco 17145  Homf chomf 17546  compfccomf 17547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-homf 17550  df-comf 17551
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator