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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 23862 | . . 3 ⊢ (𝜑 → (𝐹(*𝑝‘𝐽)𝐺) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))))) |
4 | 3 | fveq1d 6697 | . 2 ⊢ (𝜑 → ((𝐹(*𝑝‘𝐽)𝐺)‘1) = ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))))‘1)) |
5 | 1elunit 13023 | . . 3 ⊢ 1 ∈ (0[,]1) | |
6 | halflt1 12013 | . . . . . . . 8 ⊢ (1 / 2) < 1 | |
7 | halfre 12009 | . . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ | |
8 | 1re 10798 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
9 | 7, 8 | ltnlei 10918 | . . . . . . . 8 ⊢ ((1 / 2) < 1 ↔ ¬ 1 ≤ (1 / 2)) |
10 | 6, 9 | mpbi 233 | . . . . . . 7 ⊢ ¬ 1 ≤ (1 / 2) |
11 | breq1 5042 | . . . . . . 7 ⊢ (𝑥 = 1 → (𝑥 ≤ (1 / 2) ↔ 1 ≤ (1 / 2))) | |
12 | 10, 11 | mtbiri 330 | . . . . . 6 ⊢ (𝑥 = 1 → ¬ 𝑥 ≤ (1 / 2)) |
13 | 12 | iffalsed 4436 | . . . . 5 ⊢ (𝑥 = 1 → if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))) = (𝐺‘((2 · 𝑥) − 1))) |
14 | oveq2 7199 | . . . . . . . . 9 ⊢ (𝑥 = 1 → (2 · 𝑥) = (2 · 1)) | |
15 | 2t1e2 11958 | . . . . . . . . 9 ⊢ (2 · 1) = 2 | |
16 | 14, 15 | eqtrdi 2787 | . . . . . . . 8 ⊢ (𝑥 = 1 → (2 · 𝑥) = 2) |
17 | 16 | oveq1d 7206 | . . . . . . 7 ⊢ (𝑥 = 1 → ((2 · 𝑥) − 1) = (2 − 1)) |
18 | 2m1e1 11921 | . . . . . . 7 ⊢ (2 − 1) = 1 | |
19 | 17, 18 | eqtrdi 2787 | . . . . . 6 ⊢ (𝑥 = 1 → ((2 · 𝑥) − 1) = 1) |
20 | 19 | fveq2d 6699 | . . . . 5 ⊢ (𝑥 = 1 → (𝐺‘((2 · 𝑥) − 1)) = (𝐺‘1)) |
21 | 13, 20 | eqtrd 2771 | . . . 4 ⊢ (𝑥 = 1 → if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))) = (𝐺‘1)) |
22 | eqid 2736 | . . . 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 6708 | . . . 4 ⊢ (𝐺‘1) ∈ V | |
24 | 21, 22, 23 | fvmpt 6796 | . . 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 2787 | 1 ⊢ (𝜑 → ((𝐹(*𝑝‘𝐽)𝐺)‘1) = (𝐺‘1)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1543 ∈ wcel 2112 ifcif 4425 class class class wbr 5039 ↦ cmpt 5120 ‘cfv 6358 (class class class)co 7191 0cc0 10694 1c1 10695 · cmul 10699 < clt 10832 ≤ cle 10833 − cmin 11027 / cdiv 11454 2c2 11850 [,]cicc 12903 Cn ccn 22075 IIcii 23726 *𝑝cpco 23851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-2 11858 df-icc 12907 df-top 21745 df-topon 21762 df-cn 22078 df-pco 23856 |
This theorem is referenced by: pcohtpylem 23870 pcorevlem 23877 pcophtb 23880 om1addcl 23884 pi1xfrf 23904 pi1xfr 23906 pi1xfrcnvlem 23907 pi1coghm 23912 connpconn 32864 sconnpht2 32867 cvmlift3lem6 32953 |
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