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Mirrors > Home > MPE Home > Th. List > pcovalg | Structured version Visualization version GIF version |
Description: Evaluate the concatenation of two paths. (Contributed by Mario Carneiro, 7-Jun-2014.) |
Ref | Expression |
---|---|
pcoval.2 | ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) |
pcoval.3 | ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) |
Ref | Expression |
---|---|
pcovalg | ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → ((𝐹(*𝑝‘𝐽)𝐺)‘𝑋) = if(𝑋 ≤ (1 / 2), (𝐹‘(2 · 𝑋)), (𝐺‘((2 · 𝑋) − 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pcoval.2 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) | |
2 | pcoval.3 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) | |
3 | 1, 2 | pcoval 23617 | . . 3 ⊢ (𝜑 → (𝐹(*𝑝‘𝐽)𝐺) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))))) |
4 | 3 | fveq1d 6674 | . 2 ⊢ (𝜑 → ((𝐹(*𝑝‘𝐽)𝐺)‘𝑋) = ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))))‘𝑋)) |
5 | breq1 5071 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ (1 / 2) ↔ 𝑋 ≤ (1 / 2))) | |
6 | oveq2 7166 | . . . . 5 ⊢ (𝑥 = 𝑋 → (2 · 𝑥) = (2 · 𝑋)) | |
7 | 6 | fveq2d 6676 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐹‘(2 · 𝑥)) = (𝐹‘(2 · 𝑋))) |
8 | 6 | fvoveq1d 7180 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐺‘((2 · 𝑥) − 1)) = (𝐺‘((2 · 𝑋) − 1))) |
9 | 5, 7, 8 | ifbieq12d 4496 | . . 3 ⊢ (𝑥 = 𝑋 → if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))) = if(𝑋 ≤ (1 / 2), (𝐹‘(2 · 𝑋)), (𝐺‘((2 · 𝑋) − 1)))) |
10 | eqid 2823 | . . 3 ⊢ (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)))) | |
11 | fvex 6685 | . . . 4 ⊢ (𝐹‘(2 · 𝑋)) ∈ V | |
12 | fvex 6685 | . . . 4 ⊢ (𝐺‘((2 · 𝑋) − 1)) ∈ V | |
13 | 11, 12 | ifex 4517 | . . 3 ⊢ if(𝑋 ≤ (1 / 2), (𝐹‘(2 · 𝑋)), (𝐺‘((2 · 𝑋) − 1))) ∈ V |
14 | 9, 10, 13 | fvmpt 6770 | . 2 ⊢ (𝑋 ∈ (0[,]1) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))))‘𝑋) = if(𝑋 ≤ (1 / 2), (𝐹‘(2 · 𝑋)), (𝐺‘((2 · 𝑋) − 1)))) |
15 | 4, 14 | sylan9eq 2878 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → ((𝐹(*𝑝‘𝐽)𝐺)‘𝑋) = if(𝑋 ≤ (1 / 2), (𝐹‘(2 · 𝑋)), (𝐺‘((2 · 𝑋) − 1)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ifcif 4469 class class class wbr 5068 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 0cc0 10539 1c1 10540 · cmul 10544 ≤ cle 10678 − cmin 10872 / cdiv 11299 2c2 11695 [,]cicc 12744 Cn ccn 21834 IIcii 23485 *𝑝cpco 23606 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-map 8410 df-top 21504 df-topon 21521 df-cn 21837 df-pco 23611 |
This theorem is referenced by: pcoval1 23619 pcoval2 23622 pcohtpylem 23625 |
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