| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 11263 | . . . . 5 ⊢ 0 ∈ ℝ | |
| 2 | 1re 11261 | . . . . 5 ⊢ 1 ∈ ℝ | |
| 3 | 0le0 12367 | . . . . 5 ⊢ 0 ≤ 0 | |
| 4 | halfre 12480 | . . . . . 6 ⊢ (1 / 2) ∈ ℝ | |
| 5 | halflt1 12484 | . . . . . 6 ⊢ (1 / 2) < 1 | |
| 6 | 4, 2, 5 | ltleii 11384 | . . . . 5 ⊢ (1 / 2) ≤ 1 |
| 7 | iccss 13455 | . . . . 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 3979 | . . 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 25045 | . . 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 24964 | . . . . 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 4533 | . . 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 2777 | 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 1540 ∈ wcel 2108 ⊆ wss 3951 ifcif 4525 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 · cmul 11160 ≤ cle 11296 − cmin 11492 / cdiv 11920 2c2 12321 [,]cicc 13390 Cn ccn 23232 IIcii 24901 *𝑝cpco 25033 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-2 12329 df-icc 13394 df-top 22900 df-topon 22917 df-cn 23235 df-pco 25038 |
| This theorem is referenced by: pco0 25047 pcoass 25057 pcorevlem 25059 |
| Copyright terms: Public domain | W3C validator |