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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 24996 | . . . 4 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐺 ∈ (II Cn 𝐽) ∧ (𝐺‘0) = (𝐹‘1) ∧ (𝐺‘1) = (𝐹‘0))) |
3 | 2 | simp1d 1139 | . . 3 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐺 ∈ (II Cn 𝐽)) |
4 | eqid 2725 | . . . 4 ⊢ (𝑦 ∈ (0[,]1) ↦ (𝐺‘(1 − 𝑦))) = (𝑦 ∈ (0[,]1) ↦ (𝐺‘(1 − 𝑦))) | |
5 | eqid 2725 | . . . 4 ⊢ ((0[,]1) × {(𝐺‘1)}) = ((0[,]1) × {(𝐺‘1)}) | |
6 | 4, 5 | pcorev 24998 | . . 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 24894 | . . . . . . 7 ⊢ (𝑦 ∈ (0[,]1) → (1 − 𝑦) ∈ (0[,]1)) | |
9 | oveq2 7427 | . . . . . . . . 9 ⊢ (𝑥 = (1 − 𝑦) → (1 − 𝑥) = (1 − (1 − 𝑦))) | |
10 | 9 | fveq2d 6900 | . . . . . . . 8 ⊢ (𝑥 = (1 − 𝑦) → (𝐹‘(1 − 𝑥)) = (𝐹‘(1 − (1 − 𝑦)))) |
11 | fvex 6909 | . . . . . . . 8 ⊢ (𝐹‘(1 − (1 − 𝑦))) ∈ V | |
12 | 10, 1, 11 | fvmpt 7004 | . . . . . . 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 11198 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
15 | unitssre 13511 | . . . . . . . . . 10 ⊢ (0[,]1) ⊆ ℝ | |
16 | 15 | sseli 3972 | . . . . . . . . 9 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ ℝ) |
17 | 16 | recnd 11274 | . . . . . . . 8 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ ℂ) |
18 | nncan 11521 | . . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (1 − (1 − 𝑦)) = 𝑦) | |
19 | 14, 17, 18 | sylancr 585 | . . . . . . 7 ⊢ (𝑦 ∈ (0[,]1) → (1 − (1 − 𝑦)) = 𝑦) |
20 | 19 | fveq2d 6900 | . . . . . 6 ⊢ (𝑦 ∈ (0[,]1) → (𝐹‘(1 − (1 − 𝑦))) = (𝐹‘𝑦)) |
21 | 13, 20 | eqtrd 2765 | . . . . 5 ⊢ (𝑦 ∈ (0[,]1) → (𝐺‘(1 − 𝑦)) = (𝐹‘𝑦)) |
22 | 21 | mpteq2ia 5252 | . . . 4 ⊢ (𝑦 ∈ (0[,]1) ↦ (𝐺‘(1 − 𝑦))) = (𝑦 ∈ (0[,]1) ↦ (𝐹‘𝑦)) |
23 | iiuni 24845 | . . . . . 6 ⊢ (0[,]1) = ∪ II | |
24 | eqid 2725 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
25 | 23, 24 | cnf 23194 | . . . . 5 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐹:(0[,]1)⟶∪ 𝐽) |
26 | 25 | feqmptd 6966 | . . . 4 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐹 = (𝑦 ∈ (0[,]1) ↦ (𝐹‘𝑦))) |
27 | 22, 26 | eqtr4id 2784 | . . 3 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝑦 ∈ (0[,]1) ↦ (𝐺‘(1 − 𝑦))) = 𝐹) |
28 | 27 | oveq1d 7434 | . 2 ⊢ (𝐹 ∈ (II Cn 𝐽) → ((𝑦 ∈ (0[,]1) ↦ (𝐺‘(1 − 𝑦)))(*𝑝‘𝐽)𝐺) = (𝐹(*𝑝‘𝐽)𝐺)) |
29 | 2 | simp3d 1141 | . . . . 5 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐺‘1) = (𝐹‘0)) |
30 | 29 | sneqd 4642 | . . . 4 ⊢ (𝐹 ∈ (II Cn 𝐽) → {(𝐺‘1)} = {(𝐹‘0)}) |
31 | 30 | xpeq2d 5708 | . . 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 2783 | . 2 ⊢ (𝐹 ∈ (II Cn 𝐽) → ((0[,]1) × {(𝐺‘1)}) = 𝑃) |
34 | 7, 28, 33 | 3brtr3d 5180 | 1 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐹(*𝑝‘𝐽)𝐺)( ≃ph‘𝐽)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {csn 4630 ∪ cuni 4909 class class class wbr 5149 ↦ cmpt 5232 × cxp 5676 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 ℝcr 11139 0cc0 11140 1c1 11141 − cmin 11476 [,]cicc 13362 Cn ccn 23172 IIcii 24839 ≃phcphtpc 24939 *𝑝cpco 24971 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-fi 9436 df-sup 9467 df-inf 9468 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-ioo 13363 df-icc 13366 df-fz 13520 df-fzo 13663 df-seq 14003 df-exp 14063 df-hash 14326 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-starv 17251 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-hom 17260 df-cco 17261 df-rest 17407 df-topn 17408 df-0g 17426 df-gsum 17427 df-topgen 17428 df-pt 17429 df-prds 17432 df-xrs 17487 df-qtop 17492 df-imas 17493 df-xps 17495 df-mre 17569 df-mrc 17570 df-acs 17572 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-mulg 19032 df-cntz 19280 df-cmn 19749 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-cnfld 21297 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22893 df-cld 22967 df-cn 23175 df-cnp 23176 df-tx 23510 df-hmeo 23703 df-xms 24270 df-ms 24271 df-tms 24272 df-ii 24841 df-htpy 24940 df-phtpy 24941 df-phtpc 24962 df-pco 24976 |
This theorem is referenced by: pcophtb 25000 pi1xfr 25026 pi1xfrcnvlem 25027 |
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