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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 24177 | . . . 4 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐺 ∈ (II Cn 𝐽) ∧ (𝐺‘0) = (𝐹‘1) ∧ (𝐺‘1) = (𝐹‘0))) |
3 | 2 | simp1d 1141 | . . 3 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐺 ∈ (II Cn 𝐽)) |
4 | eqid 2738 | . . . 4 ⊢ (𝑦 ∈ (0[,]1) ↦ (𝐺‘(1 − 𝑦))) = (𝑦 ∈ (0[,]1) ↦ (𝐺‘(1 − 𝑦))) | |
5 | eqid 2738 | . . . 4 ⊢ ((0[,]1) × {(𝐺‘1)}) = ((0[,]1) × {(𝐺‘1)}) | |
6 | 4, 5 | pcorev 24179 | . . 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 | iirev 24081 | . . . . . . 7 ⊢ (𝑦 ∈ (0[,]1) → (1 − 𝑦) ∈ (0[,]1)) | |
9 | oveq2 7277 | . . . . . . . . 9 ⊢ (𝑥 = (1 − 𝑦) → (1 − 𝑥) = (1 − (1 − 𝑦))) | |
10 | 9 | fveq2d 6772 | . . . . . . . 8 ⊢ (𝑥 = (1 − 𝑦) → (𝐹‘(1 − 𝑥)) = (𝐹‘(1 − (1 − 𝑦)))) |
11 | fvex 6781 | . . . . . . . 8 ⊢ (𝐹‘(1 − (1 − 𝑦))) ∈ V | |
12 | 10, 1, 11 | fvmpt 6869 | . . . . . . 7 ⊢ ((1 − 𝑦) ∈ (0[,]1) → (𝐺‘(1 − 𝑦)) = (𝐹‘(1 − (1 − 𝑦)))) |
13 | 8, 12 | syl 17 | . . . . . 6 ⊢ (𝑦 ∈ (0[,]1) → (𝐺‘(1 − 𝑦)) = (𝐹‘(1 − (1 − 𝑦)))) |
14 | ax-1cn 10918 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
15 | unitssre 13220 | . . . . . . . . . 10 ⊢ (0[,]1) ⊆ ℝ | |
16 | 15 | sseli 3918 | . . . . . . . . 9 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ ℝ) |
17 | 16 | recnd 10992 | . . . . . . . 8 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ ℂ) |
18 | nncan 11239 | . . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (1 − (1 − 𝑦)) = 𝑦) | |
19 | 14, 17, 18 | sylancr 587 | . . . . . . 7 ⊢ (𝑦 ∈ (0[,]1) → (1 − (1 − 𝑦)) = 𝑦) |
20 | 19 | fveq2d 6772 | . . . . . 6 ⊢ (𝑦 ∈ (0[,]1) → (𝐹‘(1 − (1 − 𝑦))) = (𝐹‘𝑦)) |
21 | 13, 20 | eqtrd 2778 | . . . . 5 ⊢ (𝑦 ∈ (0[,]1) → (𝐺‘(1 − 𝑦)) = (𝐹‘𝑦)) |
22 | 21 | mpteq2ia 5178 | . . . 4 ⊢ (𝑦 ∈ (0[,]1) ↦ (𝐺‘(1 − 𝑦))) = (𝑦 ∈ (0[,]1) ↦ (𝐹‘𝑦)) |
23 | iiuni 24033 | . . . . . 6 ⊢ (0[,]1) = ∪ II | |
24 | eqid 2738 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
25 | 23, 24 | cnf 22386 | . . . . 5 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐹:(0[,]1)⟶∪ 𝐽) |
26 | 25 | feqmptd 6831 | . . . 4 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐹 = (𝑦 ∈ (0[,]1) ↦ (𝐹‘𝑦))) |
27 | 22, 26 | eqtr4id 2797 | . . 3 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝑦 ∈ (0[,]1) ↦ (𝐺‘(1 − 𝑦))) = 𝐹) |
28 | 27 | oveq1d 7284 | . 2 ⊢ (𝐹 ∈ (II Cn 𝐽) → ((𝑦 ∈ (0[,]1) ↦ (𝐺‘(1 − 𝑦)))(*𝑝‘𝐽)𝐺) = (𝐹(*𝑝‘𝐽)𝐺)) |
29 | 2 | simp3d 1143 | . . . . 5 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐺‘1) = (𝐹‘0)) |
30 | 29 | sneqd 4575 | . . . 4 ⊢ (𝐹 ∈ (II Cn 𝐽) → {(𝐺‘1)} = {(𝐹‘0)}) |
31 | 30 | xpeq2d 5616 | . . 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 | eqtr4di 2796 | . 2 ⊢ (𝐹 ∈ (II Cn 𝐽) → ((0[,]1) × {(𝐺‘1)}) = 𝑃) |
34 | 7, 28, 33 | 3brtr3d 5106 | 1 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐹(*𝑝‘𝐽)𝐺)( ≃ph‘𝐽)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 {csn 4563 ∪ cuni 4841 class class class wbr 5075 ↦ cmpt 5158 × cxp 5584 ‘cfv 6428 (class class class)co 7269 ℂcc 10858 ℝcr 10859 0cc0 10860 1c1 10861 − cmin 11194 [,]cicc 13071 Cn ccn 22364 IIcii 24027 ≃phcphtpc 24121 *𝑝cpco 24152 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5210 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7580 ax-cnex 10916 ax-resscn 10917 ax-1cn 10918 ax-icn 10919 ax-addcl 10920 ax-addrcl 10921 ax-mulcl 10922 ax-mulrcl 10923 ax-mulcom 10924 ax-addass 10925 ax-mulass 10926 ax-distr 10927 ax-i2m1 10928 ax-1ne0 10929 ax-1rid 10930 ax-rnegex 10931 ax-rrecex 10932 ax-cnre 10933 ax-pre-lttri 10934 ax-pre-lttrn 10935 ax-pre-ltadd 10936 ax-pre-mulgt0 10937 ax-pre-sup 10938 ax-mulf 10940 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-iin 4929 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5486 df-eprel 5492 df-po 5500 df-so 5501 df-fr 5541 df-se 5542 df-we 5543 df-xp 5592 df-rel 5593 df-cnv 5594 df-co 5595 df-dm 5596 df-rn 5597 df-res 5598 df-ima 5599 df-pred 6197 df-ord 6264 df-on 6265 df-lim 6266 df-suc 6267 df-iota 6386 df-fun 6430 df-fn 6431 df-f 6432 df-f1 6433 df-fo 6434 df-f1o 6435 df-fv 6436 df-isom 6437 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-of 7525 df-om 7705 df-1st 7822 df-2nd 7823 df-supp 7967 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-2o 8287 df-er 8487 df-map 8606 df-ixp 8675 df-en 8723 df-dom 8724 df-sdom 8725 df-fin 8726 df-fsupp 9118 df-fi 9159 df-sup 9190 df-inf 9191 df-oi 9258 df-card 9686 df-pnf 11000 df-mnf 11001 df-xr 11002 df-ltxr 11003 df-le 11004 df-sub 11196 df-neg 11197 df-div 11622 df-nn 11963 df-2 12025 df-3 12026 df-4 12027 df-5 12028 df-6 12029 df-7 12030 df-8 12031 df-9 12032 df-n0 12223 df-z 12309 df-dec 12427 df-uz 12572 df-q 12678 df-rp 12720 df-xneg 12837 df-xadd 12838 df-xmul 12839 df-ioo 13072 df-icc 13075 df-fz 13229 df-fzo 13372 df-seq 13711 df-exp 13772 df-hash 14034 df-cj 14799 df-re 14800 df-im 14801 df-sqrt 14935 df-abs 14936 df-struct 16837 df-sets 16854 df-slot 16872 df-ndx 16884 df-base 16902 df-ress 16931 df-plusg 16964 df-mulr 16965 df-starv 16966 df-sca 16967 df-vsca 16968 df-ip 16969 df-tset 16970 df-ple 16971 df-ds 16973 df-unif 16974 df-hom 16975 df-cco 16976 df-rest 17122 df-topn 17123 df-0g 17141 df-gsum 17142 df-topgen 17143 df-pt 17144 df-prds 17147 df-xrs 17202 df-qtop 17207 df-imas 17208 df-xps 17210 df-mre 17284 df-mrc 17285 df-acs 17287 df-mgm 18315 df-sgrp 18364 df-mnd 18375 df-submnd 18420 df-mulg 18690 df-cntz 18912 df-cmn 19377 df-psmet 20578 df-xmet 20579 df-met 20580 df-bl 20581 df-mopn 20582 df-cnfld 20587 df-top 22032 df-topon 22049 df-topsp 22071 df-bases 22085 df-cld 22159 df-cn 22367 df-cnp 22368 df-tx 22702 df-hmeo 22895 df-xms 23462 df-ms 23463 df-tms 23464 df-ii 24029 df-htpy 24122 df-phtpy 24123 df-phtpc 24144 df-pco 24157 |
This theorem is referenced by: pcophtb 24181 pi1xfr 24207 pi1xfrcnvlem 24208 |
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