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

Theorem ofco 7706
Description: The composition of a function operation with another function. (Contributed by Mario Carneiro, 19-Dec-2014.)
Hypotheses
Ref Expression
ofco.1 (𝜑𝐹 Fn 𝐴)
ofco.2 (𝜑𝐺 Fn 𝐵)
ofco.3 (𝜑𝐻:𝐷𝐶)
ofco.4 (𝜑𝐴𝑉)
ofco.5 (𝜑𝐵𝑊)
ofco.6 (𝜑𝐷𝑋)
ofco.7 (𝐴𝐵) = 𝐶
Assertion
Ref Expression
ofco (𝜑 → ((𝐹f 𝑅𝐺) ∘ 𝐻) = ((𝐹𝐻) ∘f 𝑅(𝐺𝐻)))

Proof of Theorem ofco
Dummy variables 𝑦 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ofco.3 . . . 4 (𝜑𝐻:𝐷𝐶)
21ffvelcdmda 7090 . . 3 ((𝜑𝑥𝐷) → (𝐻𝑥) ∈ 𝐶)
31feqmptd 6963 . . 3 (𝜑𝐻 = (𝑥𝐷 ↦ (𝐻𝑥)))
4 ofco.1 . . . 4 (𝜑𝐹 Fn 𝐴)
5 ofco.2 . . . 4 (𝜑𝐺 Fn 𝐵)
6 ofco.4 . . . 4 (𝜑𝐴𝑉)
7 ofco.5 . . . 4 (𝜑𝐵𝑊)
8 ofco.7 . . . 4 (𝐴𝐵) = 𝐶
9 eqidd 2727 . . . 4 ((𝜑𝑦𝐴) → (𝐹𝑦) = (𝐹𝑦))
10 eqidd 2727 . . . 4 ((𝜑𝑦𝐵) → (𝐺𝑦) = (𝐺𝑦))
114, 5, 6, 7, 8, 9, 10offval 7691 . . 3 (𝜑 → (𝐹f 𝑅𝐺) = (𝑦𝐶 ↦ ((𝐹𝑦)𝑅(𝐺𝑦))))
12 fveq2 6893 . . . 4 (𝑦 = (𝐻𝑥) → (𝐹𝑦) = (𝐹‘(𝐻𝑥)))
13 fveq2 6893 . . . 4 (𝑦 = (𝐻𝑥) → (𝐺𝑦) = (𝐺‘(𝐻𝑥)))
1412, 13oveq12d 7434 . . 3 (𝑦 = (𝐻𝑥) → ((𝐹𝑦)𝑅(𝐺𝑦)) = ((𝐹‘(𝐻𝑥))𝑅(𝐺‘(𝐻𝑥))))
152, 3, 11, 14fmptco 7135 . 2 (𝜑 → ((𝐹f 𝑅𝐺) ∘ 𝐻) = (𝑥𝐷 ↦ ((𝐹‘(𝐻𝑥))𝑅(𝐺‘(𝐻𝑥)))))
16 inss1 4227 . . . . . 6 (𝐴𝐵) ⊆ 𝐴
178, 16eqsstrri 4014 . . . . 5 𝐶𝐴
18 fss 6736 . . . . 5 ((𝐻:𝐷𝐶𝐶𝐴) → 𝐻:𝐷𝐴)
191, 17, 18sylancl 584 . . . 4 (𝜑𝐻:𝐷𝐴)
20 fnfco 6759 . . . 4 ((𝐹 Fn 𝐴𝐻:𝐷𝐴) → (𝐹𝐻) Fn 𝐷)
214, 19, 20syl2anc 582 . . 3 (𝜑 → (𝐹𝐻) Fn 𝐷)
22 inss2 4228 . . . . . 6 (𝐴𝐵) ⊆ 𝐵
238, 22eqsstrri 4014 . . . . 5 𝐶𝐵
24 fss 6736 . . . . 5 ((𝐻:𝐷𝐶𝐶𝐵) → 𝐻:𝐷𝐵)
251, 23, 24sylancl 584 . . . 4 (𝜑𝐻:𝐷𝐵)
26 fnfco 6759 . . . 4 ((𝐺 Fn 𝐵𝐻:𝐷𝐵) → (𝐺𝐻) Fn 𝐷)
275, 25, 26syl2anc 582 . . 3 (𝜑 → (𝐺𝐻) Fn 𝐷)
28 ofco.6 . . 3 (𝜑𝐷𝑋)
29 inidm 4217 . . 3 (𝐷𝐷) = 𝐷
301ffnd 6721 . . . 4 (𝜑𝐻 Fn 𝐷)
31 fvco2 6991 . . . 4 ((𝐻 Fn 𝐷𝑥𝐷) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
3230, 31sylan 578 . . 3 ((𝜑𝑥𝐷) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
33 fvco2 6991 . . . 4 ((𝐻 Fn 𝐷𝑥𝐷) → ((𝐺𝐻)‘𝑥) = (𝐺‘(𝐻𝑥)))
3430, 33sylan 578 . . 3 ((𝜑𝑥𝐷) → ((𝐺𝐻)‘𝑥) = (𝐺‘(𝐻𝑥)))
3521, 27, 28, 28, 29, 32, 34offval 7691 . 2 (𝜑 → ((𝐹𝐻) ∘f 𝑅(𝐺𝐻)) = (𝑥𝐷 ↦ ((𝐹‘(𝐻𝑥))𝑅(𝐺‘(𝐻𝑥)))))
3615, 35eqtr4d 2769 1 (𝜑 → ((𝐹f 𝑅𝐺) ∘ 𝐻) = ((𝐹𝐻) ∘f 𝑅(𝐺𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  cin 3945  wss 3946  cmpt 5228  ccom 5678   Fn wfn 6541  wf 6542  cfv 6546  (class class class)co 7416  f cof 7680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7682
This theorem is referenced by:  gsumzaddlem  19915  coe1add  22251  pf1ind  22343  1arithidomlem2  33417  mendring  42890
  Copyright terms: Public domain W3C validator