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Theorem pcoval 22714
Description: The concatenation of two paths. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypotheses
Ref Expression
pcoval.2 (𝜑𝐹 ∈ (II Cn 𝐽))
pcoval.3 (𝜑𝐺 ∈ (II Cn 𝐽))
Assertion
Ref Expression
pcoval (𝜑 → (𝐹(*𝑝𝐽)𝐺) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1)))))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐺   𝜑,𝑥   𝑥,𝐽

Proof of Theorem pcoval
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pcoval.2 . 2 (𝜑𝐹 ∈ (II Cn 𝐽))
2 pcoval.3 . 2 (𝜑𝐺 ∈ (II Cn 𝐽))
3 fveq1 6149 . . . . . 6 (𝑓 = 𝐹 → (𝑓‘(2 · 𝑥)) = (𝐹‘(2 · 𝑥)))
43adantr 481 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓‘(2 · 𝑥)) = (𝐹‘(2 · 𝑥)))
5 fveq1 6149 . . . . . 6 (𝑔 = 𝐺 → (𝑔‘((2 · 𝑥) − 1)) = (𝐺‘((2 · 𝑥) − 1)))
65adantl 482 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑔‘((2 · 𝑥) − 1)) = (𝐺‘((2 · 𝑥) − 1)))
74, 6ifeq12d 4083 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → if(𝑥 ≤ (1 / 2), (𝑓‘(2 · 𝑥)), (𝑔‘((2 · 𝑥) − 1))) = if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))))
87mpteq2dv 4710 . . 3 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝑓‘(2 · 𝑥)), (𝑔‘((2 · 𝑥) − 1)))) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1)))))
9 pcofval 22713 . . 3 (*𝑝𝐽) = (𝑓 ∈ (II Cn 𝐽), 𝑔 ∈ (II Cn 𝐽) ↦ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝑓‘(2 · 𝑥)), (𝑔‘((2 · 𝑥) − 1)))))
10 ovex 6633 . . . 4 (0[,]1) ∈ V
1110mptex 6441 . . 3 (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1)))) ∈ V
128, 9, 11ovmpt2a 6745 . 2 ((𝐹 ∈ (II Cn 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽)) → (𝐹(*𝑝𝐽)𝐺) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1)))))
131, 2, 12syl2anc 692 1 (𝜑 → (𝐹(*𝑝𝐽)𝐺) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  ifcif 4063   class class class wbr 4618  cmpt 4678  cfv 5850  (class class class)co 6605  0cc0 9881  1c1 9882   · cmul 9886  cle 10020  cmin 10211   / cdiv 10629  2c2 11015  [,]cicc 12117   Cn ccn 20933  IIcii 22581  *𝑝cpco 22703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-map 7805  df-top 20616  df-topon 20618  df-cn 20936  df-pco 22708
This theorem is referenced by:  pcovalg  22715  pco1  22718  pcocn  22720  copco  22721  pcopt  22725  pcopt2  22726  pcoass  22727
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