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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 11141 | . . . . 5 ⊢ 0 ∈ ℝ | |
| 2 | 1re 11139 | . . . . 5 ⊢ 1 ∈ ℝ | |
| 3 | 0le0 12277 | . . . . 5 ⊢ 0 ≤ 0 | |
| 4 | halfre 12385 | . . . . . 6 ⊢ (1 / 2) ∈ ℝ | |
| 5 | halflt1 12389 | . . . . . 6 ⊢ (1 / 2) < 1 | |
| 6 | 4, 2, 5 | ltleii 11264 | . . . . 5 ⊢ (1 / 2) ≤ 1 |
| 7 | iccss 13362 | . . . . 5 ⊢ (((0 ∈ ℝ ∧ 1 ∈ ℝ) ∧ (0 ≤ 0 ∧ (1 / 2) ≤ 1)) → (0[,](1 / 2)) ⊆ (0[,]1)) | |
| 8 | 1, 2, 3, 6, 7 | mp4an 694 | . . . 4 ⊢ (0[,](1 / 2)) ⊆ (0[,]1) |
| 9 | 8 | sseli 3918 | . . 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 24993 | . . 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 594 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,](1 / 2))) → ((𝐹(*𝑝‘𝐽)𝐺)‘𝑋) = if(𝑋 ≤ (1 / 2), (𝐹‘(2 · 𝑋)), (𝐺‘((2 · 𝑋) − 1)))) |
| 14 | elii1 24916 | . . . . 5 ⊢ (𝑋 ∈ (0[,](1 / 2)) ↔ (𝑋 ∈ (0[,]1) ∧ 𝑋 ≤ (1 / 2))) | |
| 15 | 14 | simprbi 497 | . . . 4 ⊢ (𝑋 ∈ (0[,](1 / 2)) → 𝑋 ≤ (1 / 2)) |
| 16 | 15 | iftrued 4475 | . . 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 2772 | 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 1542 ∈ wcel 2114 ⊆ wss 3890 ifcif 4467 class class class wbr 5086 ‘cfv 6494 (class class class)co 7362 ℝcr 11032 0cc0 11033 1c1 11034 · cmul 11038 ≤ cle 11175 − cmin 11372 / cdiv 11802 2c2 12231 [,]cicc 13296 Cn ccn 23203 IIcii 24856 *𝑝cpco 24981 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7813 df-1st 7937 df-2nd 7938 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-er 8638 df-map 8770 df-en 8889 df-dom 8890 df-sdom 8891 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-icc 13300 df-top 22873 df-topon 22890 df-cn 23206 df-pco 24986 |
| This theorem is referenced by: pco0 24995 pcoass 25005 pcorevlem 25007 |
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