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Mirrors > Home > MPE Home > Th. List > pco1 | Structured version Visualization version GIF version |
Description: The ending point of a path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.) |
Ref | Expression |
---|---|
pcoval.2 | ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) |
pcoval.3 | ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) |
Ref | Expression |
---|---|
pco1 | ⊢ (𝜑 → ((𝐹(*𝑝‘𝐽)𝐺)‘1) = (𝐺‘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 24982 | . . 3 ⊢ (𝜑 → (𝐹(*𝑝‘𝐽)𝐺) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))))) |
4 | 3 | fveq1d 6898 | . 2 ⊢ (𝜑 → ((𝐹(*𝑝‘𝐽)𝐺)‘1) = ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))))‘1)) |
5 | 1elunit 13482 | . . 3 ⊢ 1 ∈ (0[,]1) | |
6 | halflt1 12463 | . . . . . . . 8 ⊢ (1 / 2) < 1 | |
7 | halfre 12459 | . . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ | |
8 | 1re 11246 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
9 | 7, 8 | ltnlei 11367 | . . . . . . . 8 ⊢ ((1 / 2) < 1 ↔ ¬ 1 ≤ (1 / 2)) |
10 | 6, 9 | mpbi 229 | . . . . . . 7 ⊢ ¬ 1 ≤ (1 / 2) |
11 | breq1 5152 | . . . . . . 7 ⊢ (𝑥 = 1 → (𝑥 ≤ (1 / 2) ↔ 1 ≤ (1 / 2))) | |
12 | 10, 11 | mtbiri 326 | . . . . . 6 ⊢ (𝑥 = 1 → ¬ 𝑥 ≤ (1 / 2)) |
13 | 12 | iffalsed 4541 | . . . . 5 ⊢ (𝑥 = 1 → if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))) = (𝐺‘((2 · 𝑥) − 1))) |
14 | oveq2 7427 | . . . . . . . . 9 ⊢ (𝑥 = 1 → (2 · 𝑥) = (2 · 1)) | |
15 | 2t1e2 12408 | . . . . . . . . 9 ⊢ (2 · 1) = 2 | |
16 | 14, 15 | eqtrdi 2781 | . . . . . . . 8 ⊢ (𝑥 = 1 → (2 · 𝑥) = 2) |
17 | 16 | oveq1d 7434 | . . . . . . 7 ⊢ (𝑥 = 1 → ((2 · 𝑥) − 1) = (2 − 1)) |
18 | 2m1e1 12371 | . . . . . . 7 ⊢ (2 − 1) = 1 | |
19 | 17, 18 | eqtrdi 2781 | . . . . . 6 ⊢ (𝑥 = 1 → ((2 · 𝑥) − 1) = 1) |
20 | 19 | fveq2d 6900 | . . . . 5 ⊢ (𝑥 = 1 → (𝐺‘((2 · 𝑥) − 1)) = (𝐺‘1)) |
21 | 13, 20 | eqtrd 2765 | . . . 4 ⊢ (𝑥 = 1 → if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))) = (𝐺‘1)) |
22 | eqid 2725 | . . . 4 ⊢ (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)))) | |
23 | fvex 6909 | . . . 4 ⊢ (𝐺‘1) ∈ V | |
24 | 21, 22, 23 | fvmpt 7004 | . . 3 ⊢ (1 ∈ (0[,]1) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))))‘1) = (𝐺‘1)) |
25 | 5, 24 | ax-mp 5 | . 2 ⊢ ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))))‘1) = (𝐺‘1) |
26 | 4, 25 | eqtrdi 2781 | 1 ⊢ (𝜑 → ((𝐹(*𝑝‘𝐽)𝐺)‘1) = (𝐺‘1)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2098 ifcif 4530 class class class wbr 5149 ↦ cmpt 5232 ‘cfv 6549 (class class class)co 7419 0cc0 11140 1c1 11141 · cmul 11145 < clt 11280 ≤ cle 11281 − cmin 11476 / cdiv 11903 2c2 12300 [,]cicc 13362 Cn ccn 23172 IIcii 24839 *𝑝cpco 24971 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-2 12308 df-icc 13366 df-top 22840 df-topon 22857 df-cn 23175 df-pco 24976 |
This theorem is referenced by: pcohtpylem 24990 pcorevlem 24997 pcophtb 25000 om1addcl 25004 pi1xfrf 25024 pi1xfr 25026 pi1xfrcnvlem 25027 pi1coghm 25032 connpconn 34976 sconnpht2 34979 cvmlift3lem6 35065 |
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