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Mirrors > Home > MPE Home > Th. List > pcoval1 | Structured version Visualization version GIF version |
Description: Evaluate the concatenation of two paths on the first half. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.) |
Ref | Expression |
---|---|
pcoval.2 | ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) |
pcoval.3 | ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) |
Ref | Expression |
---|---|
pcoval1 | ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,](1 / 2))) → ((𝐹(*𝑝‘𝐽)𝐺)‘𝑋) = (𝐹‘(2 · 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 11260 | . . . . 5 ⊢ 0 ∈ ℝ | |
2 | 1re 11258 | . . . . 5 ⊢ 1 ∈ ℝ | |
3 | 0le0 12364 | . . . . 5 ⊢ 0 ≤ 0 | |
4 | halfre 12477 | . . . . . 6 ⊢ (1 / 2) ∈ ℝ | |
5 | halflt1 12481 | . . . . . 6 ⊢ (1 / 2) < 1 | |
6 | 4, 2, 5 | ltleii 11381 | . . . . 5 ⊢ (1 / 2) ≤ 1 |
7 | iccss 13451 | . . . . 5 ⊢ (((0 ∈ ℝ ∧ 1 ∈ ℝ) ∧ (0 ≤ 0 ∧ (1 / 2) ≤ 1)) → (0[,](1 / 2)) ⊆ (0[,]1)) | |
8 | 1, 2, 3, 6, 7 | mp4an 693 | . . . 4 ⊢ (0[,](1 / 2)) ⊆ (0[,]1) |
9 | 8 | sseli 3990 | . . 3 ⊢ (𝑋 ∈ (0[,](1 / 2)) → 𝑋 ∈ (0[,]1)) |
10 | pcoval.2 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) | |
11 | pcoval.3 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) | |
12 | 10, 11 | pcovalg 25058 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → ((𝐹(*𝑝‘𝐽)𝐺)‘𝑋) = if(𝑋 ≤ (1 / 2), (𝐹‘(2 · 𝑋)), (𝐺‘((2 · 𝑋) − 1)))) |
13 | 9, 12 | sylan2 593 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,](1 / 2))) → ((𝐹(*𝑝‘𝐽)𝐺)‘𝑋) = if(𝑋 ≤ (1 / 2), (𝐹‘(2 · 𝑋)), (𝐺‘((2 · 𝑋) − 1)))) |
14 | elii1 24977 | . . . . 5 ⊢ (𝑋 ∈ (0[,](1 / 2)) ↔ (𝑋 ∈ (0[,]1) ∧ 𝑋 ≤ (1 / 2))) | |
15 | 14 | simprbi 496 | . . . 4 ⊢ (𝑋 ∈ (0[,](1 / 2)) → 𝑋 ≤ (1 / 2)) |
16 | 15 | iftrued 4538 | . . 3 ⊢ (𝑋 ∈ (0[,](1 / 2)) → if(𝑋 ≤ (1 / 2), (𝐹‘(2 · 𝑋)), (𝐺‘((2 · 𝑋) − 1))) = (𝐹‘(2 · 𝑋))) |
17 | 16 | adantl 481 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,](1 / 2))) → if(𝑋 ≤ (1 / 2), (𝐹‘(2 · 𝑋)), (𝐺‘((2 · 𝑋) − 1))) = (𝐹‘(2 · 𝑋))) |
18 | 13, 17 | eqtrd 2774 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,](1 / 2))) → ((𝐹(*𝑝‘𝐽)𝐺)‘𝑋) = (𝐹‘(2 · 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ⊆ wss 3962 ifcif 4530 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 ℝcr 11151 0cc0 11152 1c1 11153 · cmul 11157 ≤ cle 11293 − cmin 11489 / cdiv 11917 2c2 12318 [,]cicc 13386 Cn ccn 23247 IIcii 24914 *𝑝cpco 25046 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8012 df-2nd 8013 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-2 12326 df-icc 13390 df-top 22915 df-topon 22932 df-cn 23250 df-pco 25051 |
This theorem is referenced by: pco0 25060 pcoass 25070 pcorevlem 25072 |
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