Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 22619 | . . 3 ⊢ (𝜑 → (𝐹(*𝑝‘𝐽)𝐺) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))))) |
4 | 3 | fveq1d 6105 | . 2 ⊢ (𝜑 → ((𝐹(*𝑝‘𝐽)𝐺)‘1) = ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))))‘1)) |
5 | 1elunit 12162 | . . 3 ⊢ 1 ∈ (0[,]1) | |
6 | halflt1 11127 | . . . . . . . 8 ⊢ (1 / 2) < 1 | |
7 | halfre 11123 | . . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ | |
8 | 1re 9918 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
9 | 7, 8 | ltnlei 10037 | . . . . . . . 8 ⊢ ((1 / 2) < 1 ↔ ¬ 1 ≤ (1 / 2)) |
10 | 6, 9 | mpbi 219 | . . . . . . 7 ⊢ ¬ 1 ≤ (1 / 2) |
11 | breq1 4586 | . . . . . . 7 ⊢ (𝑥 = 1 → (𝑥 ≤ (1 / 2) ↔ 1 ≤ (1 / 2))) | |
12 | 10, 11 | mtbiri 316 | . . . . . 6 ⊢ (𝑥 = 1 → ¬ 𝑥 ≤ (1 / 2)) |
13 | 12 | iffalsed 4047 | . . . . 5 ⊢ (𝑥 = 1 → if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))) = (𝐺‘((2 · 𝑥) − 1))) |
14 | oveq2 6557 | . . . . . . . . 9 ⊢ (𝑥 = 1 → (2 · 𝑥) = (2 · 1)) | |
15 | 2t1e2 11053 | . . . . . . . . 9 ⊢ (2 · 1) = 2 | |
16 | 14, 15 | syl6eq 2660 | . . . . . . . 8 ⊢ (𝑥 = 1 → (2 · 𝑥) = 2) |
17 | 16 | oveq1d 6564 | . . . . . . 7 ⊢ (𝑥 = 1 → ((2 · 𝑥) − 1) = (2 − 1)) |
18 | 2m1e1 11012 | . . . . . . 7 ⊢ (2 − 1) = 1 | |
19 | 17, 18 | syl6eq 2660 | . . . . . 6 ⊢ (𝑥 = 1 → ((2 · 𝑥) − 1) = 1) |
20 | 19 | fveq2d 6107 | . . . . 5 ⊢ (𝑥 = 1 → (𝐺‘((2 · 𝑥) − 1)) = (𝐺‘1)) |
21 | 13, 20 | eqtrd 2644 | . . . 4 ⊢ (𝑥 = 1 → if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))) = (𝐺‘1)) |
22 | eqid 2610 | . . . 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 6113 | . . . 4 ⊢ (𝐺‘1) ∈ V | |
24 | 21, 22, 23 | fvmpt 6191 | . . 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 2660 | 1 ⊢ (𝜑 → ((𝐹(*𝑝‘𝐽)𝐺)‘1) = (𝐺‘1)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1475 ∈ wcel 1977 ifcif 4036 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 2c2 10947 [,]cicc 12049 Cn ccn 20838 IIcii 22486 *𝑝cpco 22608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-2 10956 df-icc 12053 df-top 20521 df-topon 20523 df-cn 20841 df-pco 22613 |
This theorem is referenced by: pcohtpylem 22627 pcorevlem 22634 pcophtb 22637 om1addcl 22641 pi1xfrf 22661 pi1xfr 22663 pi1xfrcnvlem 22664 pi1coghm 22669 conpcon 30471 sconpht2 30474 cvmlift3lem6 30560 |
Copyright terms: Public domain | W3C validator |