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Mirrors > Home > MPE Home > Th. List > pcorev2 | Structured version Visualization version GIF version |
Description: Concatenation with the reverse path. (Contributed by Mario Carneiro, 12-Feb-2015.) |
Ref | Expression |
---|---|
pcorev2.1 | ⊢ 𝐺 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥))) |
pcorev2.2 | ⊢ 𝑃 = ((0[,]1) × {(𝐹‘0)}) |
Ref | Expression |
---|---|
pcorev2 | ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐹(*𝑝‘𝐽)𝐺)( ≃ph‘𝐽)𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pcorev2.1 | . . . . 5 ⊢ 𝐺 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥))) | |
2 | 1 | pcorevcl 23556 | . . . 4 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐺 ∈ (II Cn 𝐽) ∧ (𝐺‘0) = (𝐹‘1) ∧ (𝐺‘1) = (𝐹‘0))) |
3 | 2 | simp1d 1134 | . . 3 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐺 ∈ (II Cn 𝐽)) |
4 | eqid 2818 | . . . 4 ⊢ (𝑦 ∈ (0[,]1) ↦ (𝐺‘(1 − 𝑦))) = (𝑦 ∈ (0[,]1) ↦ (𝐺‘(1 − 𝑦))) | |
5 | eqid 2818 | . . . 4 ⊢ ((0[,]1) × {(𝐺‘1)}) = ((0[,]1) × {(𝐺‘1)}) | |
6 | 4, 5 | pcorev 23558 | . . 3 ⊢ (𝐺 ∈ (II Cn 𝐽) → ((𝑦 ∈ (0[,]1) ↦ (𝐺‘(1 − 𝑦)))(*𝑝‘𝐽)𝐺)( ≃ph‘𝐽)((0[,]1) × {(𝐺‘1)})) |
7 | 3, 6 | syl 17 | . 2 ⊢ (𝐹 ∈ (II Cn 𝐽) → ((𝑦 ∈ (0[,]1) ↦ (𝐺‘(1 − 𝑦)))(*𝑝‘𝐽)𝐺)( ≃ph‘𝐽)((0[,]1) × {(𝐺‘1)})) |
8 | iiuni 23416 | . . . . . 6 ⊢ (0[,]1) = ∪ II | |
9 | eqid 2818 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
10 | 8, 9 | cnf 21782 | . . . . 5 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐹:(0[,]1)⟶∪ 𝐽) |
11 | 10 | feqmptd 6726 | . . . 4 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐹 = (𝑦 ∈ (0[,]1) ↦ (𝐹‘𝑦))) |
12 | iirev 23460 | . . . . . . 7 ⊢ (𝑦 ∈ (0[,]1) → (1 − 𝑦) ∈ (0[,]1)) | |
13 | oveq2 7153 | . . . . . . . . 9 ⊢ (𝑥 = (1 − 𝑦) → (1 − 𝑥) = (1 − (1 − 𝑦))) | |
14 | 13 | fveq2d 6667 | . . . . . . . 8 ⊢ (𝑥 = (1 − 𝑦) → (𝐹‘(1 − 𝑥)) = (𝐹‘(1 − (1 − 𝑦)))) |
15 | fvex 6676 | . . . . . . . 8 ⊢ (𝐹‘(1 − (1 − 𝑦))) ∈ V | |
16 | 14, 1, 15 | fvmpt 6761 | . . . . . . 7 ⊢ ((1 − 𝑦) ∈ (0[,]1) → (𝐺‘(1 − 𝑦)) = (𝐹‘(1 − (1 − 𝑦)))) |
17 | 12, 16 | syl 17 | . . . . . 6 ⊢ (𝑦 ∈ (0[,]1) → (𝐺‘(1 − 𝑦)) = (𝐹‘(1 − (1 − 𝑦)))) |
18 | ax-1cn 10583 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
19 | unitssre 12873 | . . . . . . . . . 10 ⊢ (0[,]1) ⊆ ℝ | |
20 | 19 | sseli 3960 | . . . . . . . . 9 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ ℝ) |
21 | 20 | recnd 10657 | . . . . . . . 8 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ ℂ) |
22 | nncan 10903 | . . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (1 − (1 − 𝑦)) = 𝑦) | |
23 | 18, 21, 22 | sylancr 587 | . . . . . . 7 ⊢ (𝑦 ∈ (0[,]1) → (1 − (1 − 𝑦)) = 𝑦) |
24 | 23 | fveq2d 6667 | . . . . . 6 ⊢ (𝑦 ∈ (0[,]1) → (𝐹‘(1 − (1 − 𝑦))) = (𝐹‘𝑦)) |
25 | 17, 24 | eqtrd 2853 | . . . . 5 ⊢ (𝑦 ∈ (0[,]1) → (𝐺‘(1 − 𝑦)) = (𝐹‘𝑦)) |
26 | 25 | mpteq2ia 5148 | . . . 4 ⊢ (𝑦 ∈ (0[,]1) ↦ (𝐺‘(1 − 𝑦))) = (𝑦 ∈ (0[,]1) ↦ (𝐹‘𝑦)) |
27 | 11, 26 | syl6reqr 2872 | . . 3 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝑦 ∈ (0[,]1) ↦ (𝐺‘(1 − 𝑦))) = 𝐹) |
28 | 27 | oveq1d 7160 | . 2 ⊢ (𝐹 ∈ (II Cn 𝐽) → ((𝑦 ∈ (0[,]1) ↦ (𝐺‘(1 − 𝑦)))(*𝑝‘𝐽)𝐺) = (𝐹(*𝑝‘𝐽)𝐺)) |
29 | 2 | simp3d 1136 | . . . . 5 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐺‘1) = (𝐹‘0)) |
30 | 29 | sneqd 4569 | . . . 4 ⊢ (𝐹 ∈ (II Cn 𝐽) → {(𝐺‘1)} = {(𝐹‘0)}) |
31 | 30 | xpeq2d 5578 | . . 3 ⊢ (𝐹 ∈ (II Cn 𝐽) → ((0[,]1) × {(𝐺‘1)}) = ((0[,]1) × {(𝐹‘0)})) |
32 | pcorev2.2 | . . 3 ⊢ 𝑃 = ((0[,]1) × {(𝐹‘0)}) | |
33 | 31, 32 | syl6eqr 2871 | . 2 ⊢ (𝐹 ∈ (II Cn 𝐽) → ((0[,]1) × {(𝐺‘1)}) = 𝑃) |
34 | 7, 28, 33 | 3brtr3d 5088 | 1 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐹(*𝑝‘𝐽)𝐺)( ≃ph‘𝐽)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 {csn 4557 ∪ cuni 4830 class class class wbr 5057 ↦ cmpt 5137 × cxp 5546 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 ℝcr 10524 0cc0 10525 1c1 10526 − cmin 10858 [,]cicc 12729 Cn ccn 21760 IIcii 23410 ≃phcphtpc 23500 *𝑝cpco 23531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-icc 12733 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-mulg 18163 df-cntz 18385 df-cmn 18837 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-cn 21763 df-cnp 21764 df-tx 22098 df-hmeo 22291 df-xms 22857 df-ms 22858 df-tms 22859 df-ii 23412 df-htpy 23501 df-phtpy 23502 df-phtpc 23523 df-pco 23536 |
This theorem is referenced by: pcophtb 23560 pi1xfr 23586 pi1xfrcnvlem 23587 |
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