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| 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 25044 | . . 3 ⊢ (𝜑 → (𝐹(*𝑝‘𝐽)𝐺) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))))) | 
| 4 | 3 | fveq1d 6908 | . 2 ⊢ (𝜑 → ((𝐹(*𝑝‘𝐽)𝐺)‘1) = ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))))‘1)) | 
| 5 | 1elunit 13510 | . . 3 ⊢ 1 ∈ (0[,]1) | |
| 6 | halflt1 12484 | . . . . . . . 8 ⊢ (1 / 2) < 1 | |
| 7 | halfre 12480 | . . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ | |
| 8 | 1re 11261 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
| 9 | 7, 8 | ltnlei 11382 | . . . . . . . 8 ⊢ ((1 / 2) < 1 ↔ ¬ 1 ≤ (1 / 2)) | 
| 10 | 6, 9 | mpbi 230 | . . . . . . 7 ⊢ ¬ 1 ≤ (1 / 2) | 
| 11 | breq1 5146 | . . . . . . 7 ⊢ (𝑥 = 1 → (𝑥 ≤ (1 / 2) ↔ 1 ≤ (1 / 2))) | |
| 12 | 10, 11 | mtbiri 327 | . . . . . 6 ⊢ (𝑥 = 1 → ¬ 𝑥 ≤ (1 / 2)) | 
| 13 | 12 | iffalsed 4536 | . . . . 5 ⊢ (𝑥 = 1 → if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))) = (𝐺‘((2 · 𝑥) − 1))) | 
| 14 | oveq2 7439 | . . . . . . . . 9 ⊢ (𝑥 = 1 → (2 · 𝑥) = (2 · 1)) | |
| 15 | 2t1e2 12429 | . . . . . . . . 9 ⊢ (2 · 1) = 2 | |
| 16 | 14, 15 | eqtrdi 2793 | . . . . . . . 8 ⊢ (𝑥 = 1 → (2 · 𝑥) = 2) | 
| 17 | 16 | oveq1d 7446 | . . . . . . 7 ⊢ (𝑥 = 1 → ((2 · 𝑥) − 1) = (2 − 1)) | 
| 18 | 2m1e1 12392 | . . . . . . 7 ⊢ (2 − 1) = 1 | |
| 19 | 17, 18 | eqtrdi 2793 | . . . . . 6 ⊢ (𝑥 = 1 → ((2 · 𝑥) − 1) = 1) | 
| 20 | 19 | fveq2d 6910 | . . . . 5 ⊢ (𝑥 = 1 → (𝐺‘((2 · 𝑥) − 1)) = (𝐺‘1)) | 
| 21 | 13, 20 | eqtrd 2777 | . . . 4 ⊢ (𝑥 = 1 → if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1))) = (𝐺‘1)) | 
| 22 | eqid 2737 | . . . 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 6919 | . . . 4 ⊢ (𝐺‘1) ∈ V | |
| 24 | 21, 22, 23 | fvmpt 7016 | . . 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 2793 | 1 ⊢ (𝜑 → ((𝐹(*𝑝‘𝐽)𝐺)‘1) = (𝐺‘1)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 ifcif 4525 class class class wbr 5143 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 · cmul 11160 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 2c2 12321 [,]cicc 13390 Cn ccn 23232 IIcii 24901 *𝑝cpco 25033 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-2 12329 df-icc 13394 df-top 22900 df-topon 22917 df-cn 23235 df-pco 25038 | 
| This theorem is referenced by: pcohtpylem 25052 pcorevlem 25059 pcophtb 25062 om1addcl 25066 pi1xfrf 25086 pi1xfr 25088 pi1xfrcnvlem 25089 pi1coghm 25094 connpconn 35240 sconnpht2 35243 cvmlift3lem6 35329 | 
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