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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > stoweidlem47 | Structured version Visualization version GIF version |
Description: Subtracting a constant from a real continuous function gives another continuous function. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
stoweidlem47.1 | β’ β²π‘πΉ |
stoweidlem47.2 | β’ β²π‘π |
stoweidlem47.3 | β’ β²π‘π |
stoweidlem47.4 | β’ π = βͺ π½ |
stoweidlem47.5 | β’ πΊ = (π Γ {-π}) |
stoweidlem47.6 | β’ πΎ = (topGenβran (,)) |
stoweidlem47.7 | β’ (π β π½ β Top) |
stoweidlem47.8 | β’ πΆ = (π½ Cn πΎ) |
stoweidlem47.9 | β’ (π β πΉ β πΆ) |
stoweidlem47.10 | β’ (π β π β β) |
Ref | Expression |
---|---|
stoweidlem47 | β’ (π β (π‘ β π β¦ ((πΉβπ‘) β π)) β πΆ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stoweidlem47.3 | . . 3 β’ β²π‘π | |
2 | stoweidlem47.5 | . . . . . . 7 β’ πΊ = (π Γ {-π}) | |
3 | 2 | fveq1i 6847 | . . . . . 6 β’ (πΊβπ‘) = ((π Γ {-π})βπ‘) |
4 | stoweidlem47.10 | . . . . . . . 8 β’ (π β π β β) | |
5 | 4 | renegcld 11590 | . . . . . . 7 β’ (π β -π β β) |
6 | fvconst2g 7155 | . . . . . . 7 β’ ((-π β β β§ π‘ β π) β ((π Γ {-π})βπ‘) = -π) | |
7 | 5, 6 | sylan 581 | . . . . . 6 β’ ((π β§ π‘ β π) β ((π Γ {-π})βπ‘) = -π) |
8 | 3, 7 | eqtrid 2785 | . . . . 5 β’ ((π β§ π‘ β π) β (πΊβπ‘) = -π) |
9 | 8 | oveq2d 7377 | . . . 4 β’ ((π β§ π‘ β π) β ((πΉβπ‘) + (πΊβπ‘)) = ((πΉβπ‘) + -π)) |
10 | stoweidlem47.6 | . . . . . . . 8 β’ πΎ = (topGenβran (,)) | |
11 | stoweidlem47.4 | . . . . . . . 8 β’ π = βͺ π½ | |
12 | stoweidlem47.8 | . . . . . . . 8 β’ πΆ = (π½ Cn πΎ) | |
13 | stoweidlem47.9 | . . . . . . . 8 β’ (π β πΉ β πΆ) | |
14 | 10, 11, 12, 13 | fcnre 43322 | . . . . . . 7 β’ (π β πΉ:πβΆβ) |
15 | 14 | ffvelcdmda 7039 | . . . . . 6 β’ ((π β§ π‘ β π) β (πΉβπ‘) β β) |
16 | 15 | recnd 11191 | . . . . 5 β’ ((π β§ π‘ β π) β (πΉβπ‘) β β) |
17 | 4 | recnd 11191 | . . . . . 6 β’ (π β π β β) |
18 | 17 | adantr 482 | . . . . 5 β’ ((π β§ π‘ β π) β π β β) |
19 | 16, 18 | negsubd 11526 | . . . 4 β’ ((π β§ π‘ β π) β ((πΉβπ‘) + -π) = ((πΉβπ‘) β π)) |
20 | 9, 19 | eqtrd 2773 | . . 3 β’ ((π β§ π‘ β π) β ((πΉβπ‘) + (πΊβπ‘)) = ((πΉβπ‘) β π)) |
21 | 1, 20 | mpteq2da 5207 | . 2 β’ (π β (π‘ β π β¦ ((πΉβπ‘) + (πΊβπ‘))) = (π‘ β π β¦ ((πΉβπ‘) β π))) |
22 | stoweidlem47.1 | . . . 4 β’ β²π‘πΉ | |
23 | nfcv 2904 | . . . . . 6 β’ β²π‘π | |
24 | stoweidlem47.2 | . . . . . . . 8 β’ β²π‘π | |
25 | 24 | nfneg 11405 | . . . . . . 7 β’ β²π‘-π |
26 | 25 | nfsn 4672 | . . . . . 6 β’ β²π‘{-π} |
27 | 23, 26 | nfxp 5670 | . . . . 5 β’ β²π‘(π Γ {-π}) |
28 | 2, 27 | nfcxfr 2902 | . . . 4 β’ β²π‘πΊ |
29 | stoweidlem47.7 | . . . . 5 β’ (π β π½ β Top) | |
30 | 11 | a1i 11 | . . . . 5 β’ (π β π = βͺ π½) |
31 | istopon 22284 | . . . . 5 β’ (π½ β (TopOnβπ) β (π½ β Top β§ π = βͺ π½)) | |
32 | 29, 30, 31 | sylanbrc 584 | . . . 4 β’ (π β π½ β (TopOnβπ)) |
33 | 13, 12 | eleqtrdi 2844 | . . . 4 β’ (π β πΉ β (π½ Cn πΎ)) |
34 | retopon 24150 | . . . . . . . 8 β’ (topGenβran (,)) β (TopOnββ) | |
35 | 10, 34 | eqeltri 2830 | . . . . . . 7 β’ πΎ β (TopOnββ) |
36 | 35 | a1i 11 | . . . . . 6 β’ (π β πΎ β (TopOnββ)) |
37 | cnconst2 22657 | . . . . . 6 β’ ((π½ β (TopOnβπ) β§ πΎ β (TopOnββ) β§ -π β β) β (π Γ {-π}) β (π½ Cn πΎ)) | |
38 | 32, 36, 5, 37 | syl3anc 1372 | . . . . 5 β’ (π β (π Γ {-π}) β (π½ Cn πΎ)) |
39 | 2, 38 | eqeltrid 2838 | . . . 4 β’ (π β πΊ β (π½ Cn πΎ)) |
40 | 22, 28, 1, 10, 32, 33, 39 | refsum2cn 43335 | . . 3 β’ (π β (π‘ β π β¦ ((πΉβπ‘) + (πΊβπ‘))) β (π½ Cn πΎ)) |
41 | 40, 12 | eleqtrrdi 2845 | . 2 β’ (π β (π‘ β π β¦ ((πΉβπ‘) + (πΊβπ‘))) β πΆ) |
42 | 21, 41 | eqeltrrd 2835 | 1 β’ (π β (π‘ β π β¦ ((πΉβπ‘) β π)) β πΆ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β²wnf 1786 β wcel 2107 β²wnfc 2884 {csn 4590 βͺ cuni 4869 β¦ cmpt 5192 Γ cxp 5635 ran crn 5638 βcfv 6500 (class class class)co 7361 βcc 11057 βcr 11058 + caddc 11062 β cmin 11393 -cneg 11394 (,)cioo 13273 topGenctg 17327 Topctop 22265 TopOnctopon 22282 Cn ccn 22598 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-inf2 9585 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-addf 11138 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-er 8654 df-map 8773 df-ixp 8842 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-fi 9355 df-sup 9386 df-inf 9387 df-oi 9454 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-q 12882 df-rp 12924 df-xneg 13041 df-xadd 13042 df-xmul 13043 df-ioo 13277 df-icc 13280 df-fz 13434 df-fzo 13577 df-seq 13916 df-exp 13977 df-hash 14240 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-clim 15379 df-sum 15580 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-starv 17156 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-hom 17165 df-cco 17166 df-rest 17312 df-topn 17313 df-0g 17331 df-gsum 17332 df-topgen 17333 df-pt 17334 df-prds 17337 df-xrs 17392 df-qtop 17397 df-imas 17398 df-xps 17400 df-mre 17474 df-mrc 17475 df-acs 17477 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-submnd 18610 df-mulg 18881 df-cntz 19105 df-cmn 19572 df-psmet 20811 df-xmet 20812 df-met 20813 df-bl 20814 df-mopn 20815 df-cnfld 20820 df-top 22266 df-topon 22283 df-topsp 22305 df-bases 22319 df-cn 22601 df-cnp 22602 df-tx 22936 df-hmeo 23129 df-xms 23696 df-ms 23697 df-tms 23698 |
This theorem is referenced by: stoweidlem62 44393 |
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