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Mirrors > Home > MPE Home > Th. List > pcoptcl | Structured version Visualization version GIF version |
Description: A constant function is a path from π to itself. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 19-Mar-2015.) |
Ref | Expression |
---|---|
pcopt.1 | β’ π = ((0[,]1) Γ {π}) |
Ref | Expression |
---|---|
pcoptcl | β’ ((π½ β (TopOnβπ) β§ π β π) β (π β (II Cn π½) β§ (πβ0) = π β§ (πβ1) = π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pcopt.1 | . . 3 β’ π = ((0[,]1) Γ {π}) | |
2 | iitopon 24087 | . . . 4 β’ II β (TopOnβ(0[,]1)) | |
3 | cnconst2 22479 | . . . 4 β’ ((II β (TopOnβ(0[,]1)) β§ π½ β (TopOnβπ) β§ π β π) β ((0[,]1) Γ {π}) β (II Cn π½)) | |
4 | 2, 3 | mp3an1 1448 | . . 3 β’ ((π½ β (TopOnβπ) β§ π β π) β ((0[,]1) Γ {π}) β (II Cn π½)) |
5 | 1, 4 | eqeltrid 2841 | . 2 β’ ((π½ β (TopOnβπ) β§ π β π) β π β (II Cn π½)) |
6 | 1 | fveq1i 6805 | . . 3 β’ (πβ0) = (((0[,]1) Γ {π})β0) |
7 | simpr 486 | . . . 4 β’ ((π½ β (TopOnβπ) β§ π β π) β π β π) | |
8 | 0elunit 13247 | . . . 4 β’ 0 β (0[,]1) | |
9 | fvconst2g 7109 | . . . 4 β’ ((π β π β§ 0 β (0[,]1)) β (((0[,]1) Γ {π})β0) = π) | |
10 | 7, 8, 9 | sylancl 587 | . . 3 β’ ((π½ β (TopOnβπ) β§ π β π) β (((0[,]1) Γ {π})β0) = π) |
11 | 6, 10 | eqtrid 2788 | . 2 β’ ((π½ β (TopOnβπ) β§ π β π) β (πβ0) = π) |
12 | 1 | fveq1i 6805 | . . 3 β’ (πβ1) = (((0[,]1) Γ {π})β1) |
13 | 1elunit 13248 | . . . 4 β’ 1 β (0[,]1) | |
14 | fvconst2g 7109 | . . . 4 β’ ((π β π β§ 1 β (0[,]1)) β (((0[,]1) Γ {π})β1) = π) | |
15 | 7, 13, 14 | sylancl 587 | . . 3 β’ ((π½ β (TopOnβπ) β§ π β π) β (((0[,]1) Γ {π})β1) = π) |
16 | 12, 15 | eqtrid 2788 | . 2 β’ ((π½ β (TopOnβπ) β§ π β π) β (πβ1) = π) |
17 | 5, 11, 16 | 3jca 1128 | 1 β’ ((π½ β (TopOnβπ) β§ π β π) β (π β (II Cn π½) β§ (πβ0) = π β§ (πβ1) = π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1087 = wceq 1539 β wcel 2104 {csn 4565 Γ cxp 5598 βcfv 6458 (class class class)co 7307 0cc0 10917 1c1 10918 [,]cicc 13128 TopOnctopon 22104 Cn ccn 22420 IIcii 24083 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 ax-pre-sup 10995 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-1st 7863 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-er 8529 df-map 8648 df-en 8765 df-dom 8766 df-sdom 8767 df-sup 9245 df-inf 9246 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-div 11679 df-nn 12020 df-2 12082 df-3 12083 df-n0 12280 df-z 12366 df-uz 12629 df-q 12735 df-rp 12777 df-xneg 12894 df-xadd 12895 df-xmul 12896 df-icc 13132 df-seq 13768 df-exp 13829 df-cj 14855 df-re 14856 df-im 14857 df-sqrt 14991 df-abs 14992 df-topgen 17199 df-psmet 20634 df-xmet 20635 df-met 20636 df-bl 20637 df-mopn 20638 df-top 22088 df-topon 22105 df-bases 22141 df-cn 22423 df-cnp 22424 df-ii 24085 |
This theorem is referenced by: pcopt 24230 pcopt2 24231 pcorevlem 24234 pi1grplem 24257 sconnpi1 33246 cvxsconn 33250 |
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