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Mirrors > Home > MPE Home > Th. List > phlsrng | Structured version Visualization version GIF version |
Description: The scalar ring of a pre-Hilbert space is a star ring. (Contributed by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
phlsrng.f | β’ πΉ = (Scalarβπ) |
Ref | Expression |
---|---|
phlsrng | β’ (π β PreHil β πΉ β *-Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . 3 β’ (Baseβπ) = (Baseβπ) | |
2 | phlsrng.f | . . 3 β’ πΉ = (Scalarβπ) | |
3 | eqid 2733 | . . 3 β’ (Β·πβπ) = (Β·πβπ) | |
4 | eqid 2733 | . . 3 β’ (0gβπ) = (0gβπ) | |
5 | eqid 2733 | . . 3 β’ (*πβπΉ) = (*πβπΉ) | |
6 | eqid 2733 | . . 3 β’ (0gβπΉ) = (0gβπΉ) | |
7 | 1, 2, 3, 4, 5, 6 | isphl 21055 | . 2 β’ (π β PreHil β (π β LVec β§ πΉ β *-Ring β§ βπ₯ β (Baseβπ)((π¦ β (Baseβπ) β¦ (π¦(Β·πβπ)π₯)) β (π LMHom (ringLModβπΉ)) β§ ((π₯(Β·πβπ)π₯) = (0gβπΉ) β π₯ = (0gβπ)) β§ βπ¦ β (Baseβπ)((*πβπΉ)β(π₯(Β·πβπ)π¦)) = (π¦(Β·πβπ)π₯)))) |
8 | 7 | simp2bi 1147 | 1 β’ (π β PreHil β πΉ β *-Ring) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3061 β¦ cmpt 5192 βcfv 6500 (class class class)co 7361 Basecbs 17091 *πcstv 17143 Scalarcsca 17144 Β·πcip 17146 0gc0g 17329 *-Ringcsr 20346 LMHom clmhm 20524 LVecclvec 20607 ringLModcrglmod 20675 PreHilcphl 21051 |
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-ext 2704 ax-nul 5267 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rab 3407 df-v 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-iota 6452 df-fv 6508 df-ov 7364 df-phl 21053 |
This theorem is referenced by: iporthcom 21062 ip0r 21064 ipdi 21067 ip2di 21068 ipassr 21073 ipassr2 21074 phlssphl 21086 cphcjcl 24570 |
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