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Mirrors > Home > MPE Home > Th. List > cphsqrtcl | Structured version Visualization version GIF version |
Description: The scalar field of a subcomplex pre-Hilbert space is closed under square roots of nonnegative reals. (Contributed by Mario Carneiro, 8-Oct-2015.) |
Ref | Expression |
---|---|
cphsca.f | β’ πΉ = (Scalarβπ) |
cphsca.k | β’ πΎ = (BaseβπΉ) |
Ref | Expression |
---|---|
cphsqrtcl | β’ ((π β βPreHil β§ (π΄ β πΎ β§ π΄ β β β§ 0 β€ π΄)) β (ββπ΄) β πΎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sqrtf 15257 | . . . 4 β’ β:ββΆβ | |
2 | ffn 6672 | . . . 4 β’ (β:ββΆβ β β Fn β) | |
3 | 1, 2 | ax-mp 5 | . . 3 β’ β Fn β |
4 | inss2 4193 | . . . 4 β’ (πΎ β© (0[,)+β)) β (0[,)+β) | |
5 | rge0ssre 13382 | . . . . 5 β’ (0[,)+β) β β | |
6 | ax-resscn 11116 | . . . . 5 β’ β β β | |
7 | 5, 6 | sstri 3957 | . . . 4 β’ (0[,)+β) β β |
8 | 4, 7 | sstri 3957 | . . 3 β’ (πΎ β© (0[,)+β)) β β |
9 | simp1 1137 | . . . 4 β’ ((π΄ β πΎ β§ π΄ β β β§ 0 β€ π΄) β π΄ β πΎ) | |
10 | elrege0 13380 | . . . . . 6 β’ (π΄ β (0[,)+β) β (π΄ β β β§ 0 β€ π΄)) | |
11 | 10 | biimpri 227 | . . . . 5 β’ ((π΄ β β β§ 0 β€ π΄) β π΄ β (0[,)+β)) |
12 | 11 | 3adant1 1131 | . . . 4 β’ ((π΄ β πΎ β§ π΄ β β β§ 0 β€ π΄) β π΄ β (0[,)+β)) |
13 | 9, 12 | elind 4158 | . . 3 β’ ((π΄ β πΎ β§ π΄ β β β§ 0 β€ π΄) β π΄ β (πΎ β© (0[,)+β))) |
14 | fnfvima 7187 | . . 3 β’ ((β Fn β β§ (πΎ β© (0[,)+β)) β β β§ π΄ β (πΎ β© (0[,)+β))) β (ββπ΄) β (β β (πΎ β© (0[,)+β)))) | |
15 | 3, 8, 13, 14 | mp3an12i 1466 | . 2 β’ ((π΄ β πΎ β§ π΄ β β β§ 0 β€ π΄) β (ββπ΄) β (β β (πΎ β© (0[,)+β)))) |
16 | eqid 2733 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
17 | eqid 2733 | . . . . 5 β’ (Β·πβπ) = (Β·πβπ) | |
18 | eqid 2733 | . . . . 5 β’ (normβπ) = (normβπ) | |
19 | cphsca.f | . . . . 5 β’ πΉ = (Scalarβπ) | |
20 | cphsca.k | . . . . 5 β’ πΎ = (BaseβπΉ) | |
21 | 16, 17, 18, 19, 20 | iscph 24557 | . . . 4 β’ (π β βPreHil β ((π β PreHil β§ π β NrmMod β§ πΉ = (βfld βΎs πΎ)) β§ (β β (πΎ β© (0[,)+β))) β πΎ β§ (normβπ) = (π₯ β (Baseβπ) β¦ (ββ(π₯(Β·πβπ)π₯))))) |
22 | 21 | simp2bi 1147 | . . 3 β’ (π β βPreHil β (β β (πΎ β© (0[,)+β))) β πΎ) |
23 | 22 | sselda 3948 | . 2 β’ ((π β βPreHil β§ (ββπ΄) β (β β (πΎ β© (0[,)+β)))) β (ββπ΄) β πΎ) |
24 | 15, 23 | sylan2 594 | 1 β’ ((π β βPreHil β§ (π΄ β πΎ β§ π΄ β β β§ 0 β€ π΄)) β (ββπ΄) β πΎ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β© cin 3913 β wss 3914 class class class wbr 5109 β¦ cmpt 5192 β cima 5640 Fn wfn 6495 βΆwf 6496 βcfv 6500 (class class class)co 7361 βcc 11057 βcr 11058 0cc0 11059 +βcpnf 11194 β€ cle 11198 [,)cico 13275 βcsqrt 15127 Basecbs 17091 βΎs cress 17120 Scalarcsca 17144 Β·πcip 17146 βfldccnfld 20819 PreHilcphl 21051 normcnm 23955 NrmModcnlm 23959 βPreHilccph 24553 |
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-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 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 |
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-op 4597 df-uni 4870 df-iun 4960 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-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-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-sup 9386 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-n0 12422 df-z 12508 df-uz 12772 df-rp 12924 df-ico 13279 df-seq 13916 df-exp 13977 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-cph 24555 |
This theorem is referenced by: cphabscl 24572 cphsqrtcl2 24573 cphsqrtcl3 24574 cphnmf 24582 ipcau 24625 cphsscph 24638 |
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