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Theorem pcoval 22619
 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 6102 . . . . . 6 (𝑓 = 𝐹 → (𝑓‘(2 · 𝑥)) = (𝐹‘(2 · 𝑥)))
43adantr 480 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓‘(2 · 𝑥)) = (𝐹‘(2 · 𝑥)))
5 fveq1 6102 . . . . . 6 (𝑔 = 𝐺 → (𝑔‘((2 · 𝑥) − 1)) = (𝐺‘((2 · 𝑥) − 1)))
65adantl 481 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑔‘((2 · 𝑥) − 1)) = (𝐺‘((2 · 𝑥) − 1)))
74, 6ifeq12d 4056 . . . 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 4673 . . 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 22618 . . 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 6577 . . . 4 (0[,]1) ∈ V
1110mptex 6390 . . 3 (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1)))) ∈ V
128, 9, 11ovmpt2a 6689 . 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 691 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 383   = wceq 1475   ∈ wcel 1977  ifcif 4036   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   · cmul 9820   ≤ cle 9954   − cmin 10145   / cdiv 10563  2c2 10947  [,]cicc 12049   Cn ccn 20838  IIcii 22486  *𝑝cpco 22608 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-top 20521  df-topon 20523  df-cn 20841  df-pco 22613 This theorem is referenced by:  pcovalg  22620  pco1  22623  pcocn  22625  copco  22626  pcopt  22630  pcopt2  22631  pcoass  22632
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