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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 23333 | . . 3 ⊢ (𝜑 → (𝐹(*𝑝‘𝐽)𝐺) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))))) |
4 | 3 | fveq1d 6506 | . 2 ⊢ (𝜑 → ((𝐹(*𝑝‘𝐽)𝐺)‘1) = ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))))‘1)) |
5 | 1elunit 12678 | . . 3 ⊢ 1 ∈ (0[,]1) | |
6 | halflt1 11671 | . . . . . . . 8 ⊢ (1 / 2) < 1 | |
7 | halfre 11667 | . . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ | |
8 | 1re 10445 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
9 | 7, 8 | ltnlei 10567 | . . . . . . . 8 ⊢ ((1 / 2) < 1 ↔ ¬ 1 ≤ (1 / 2)) |
10 | 6, 9 | mpbi 222 | . . . . . . 7 ⊢ ¬ 1 ≤ (1 / 2) |
11 | breq1 4937 | . . . . . . 7 ⊢ (𝑥 = 1 → (𝑥 ≤ (1 / 2) ↔ 1 ≤ (1 / 2))) | |
12 | 10, 11 | mtbiri 319 | . . . . . 6 ⊢ (𝑥 = 1 → ¬ 𝑥 ≤ (1 / 2)) |
13 | 12 | iffalsed 4364 | . . . . 5 ⊢ (𝑥 = 1 → if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))) = (𝐺‘((2 · 𝑥) − 1))) |
14 | oveq2 6990 | . . . . . . . . 9 ⊢ (𝑥 = 1 → (2 · 𝑥) = (2 · 1)) | |
15 | 2t1e2 11616 | . . . . . . . . 9 ⊢ (2 · 1) = 2 | |
16 | 14, 15 | syl6eq 2832 | . . . . . . . 8 ⊢ (𝑥 = 1 → (2 · 𝑥) = 2) |
17 | 16 | oveq1d 6997 | . . . . . . 7 ⊢ (𝑥 = 1 → ((2 · 𝑥) − 1) = (2 − 1)) |
18 | 2m1e1 11579 | . . . . . . 7 ⊢ (2 − 1) = 1 | |
19 | 17, 18 | syl6eq 2832 | . . . . . 6 ⊢ (𝑥 = 1 → ((2 · 𝑥) − 1) = 1) |
20 | 19 | fveq2d 6508 | . . . . 5 ⊢ (𝑥 = 1 → (𝐺‘((2 · 𝑥) − 1)) = (𝐺‘1)) |
21 | 13, 20 | eqtrd 2816 | . . . 4 ⊢ (𝑥 = 1 → if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))) = (𝐺‘1)) |
22 | eqid 2780 | . . . 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 6517 | . . . 4 ⊢ (𝐺‘1) ∈ V | |
24 | 21, 22, 23 | fvmpt 6601 | . . 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 | syl6eq 2832 | 1 ⊢ (𝜑 → ((𝐹(*𝑝‘𝐽)𝐺)‘1) = (𝐺‘1)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1508 ∈ wcel 2051 ifcif 4353 class class class wbr 4934 ↦ cmpt 5013 ‘cfv 6193 (class class class)co 6982 0cc0 10341 1c1 10342 · cmul 10346 < clt 10480 ≤ cle 10481 − cmin 10676 / cdiv 11104 2c2 11501 [,]cicc 12563 Cn ccn 21551 IIcii 23201 *𝑝cpco 23322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-rep 5053 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 ax-cnex 10397 ax-resscn 10398 ax-1cn 10399 ax-icn 10400 ax-addcl 10401 ax-addrcl 10402 ax-mulcl 10403 ax-mulrcl 10404 ax-mulcom 10405 ax-addass 10406 ax-mulass 10407 ax-distr 10408 ax-i2m1 10409 ax-1ne0 10410 ax-1rid 10411 ax-rnegex 10412 ax-rrecex 10413 ax-cnre 10414 ax-pre-lttri 10415 ax-pre-lttrn 10416 ax-pre-ltadd 10417 ax-pre-mulgt0 10418 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-op 4451 df-uni 4718 df-iun 4799 df-br 4935 df-opab 4997 df-mpt 5014 df-id 5316 df-po 5330 df-so 5331 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-riota 6943 df-ov 6985 df-oprab 6986 df-mpo 6987 df-1st 7507 df-2nd 7508 df-er 8095 df-map 8214 df-en 8313 df-dom 8314 df-sdom 8315 df-pnf 10482 df-mnf 10483 df-xr 10484 df-ltxr 10485 df-le 10486 df-sub 10678 df-neg 10679 df-div 11105 df-2 11509 df-icc 12567 df-top 21221 df-topon 21238 df-cn 21554 df-pco 23327 |
This theorem is referenced by: pcohtpylem 23341 pcorevlem 23348 pcophtb 23351 om1addcl 23355 pi1xfrf 23375 pi1xfr 23377 pi1xfrcnvlem 23378 pi1coghm 23383 connpconn 32107 sconnpht2 32110 cvmlift3lem6 32196 |
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