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Mirrors > Home > MPE Home > Th. List > sspba | Structured version Visualization version GIF version |
Description: The base set of a subspace is included in the parent base set. (Contributed by NM, 27-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sspba.x | β’ π = (BaseSetβπ) |
sspba.y | β’ π = (BaseSetβπ) |
sspba.h | β’ π» = (SubSpβπ) |
Ref | Expression |
---|---|
sspba | β’ ((π β NrmCVec β§ π β π») β π β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . . . . 6 β’ ( +π£ βπ) = ( +π£ βπ) | |
2 | eqid 2737 | . . . . . 6 β’ ( +π£ βπ) = ( +π£ βπ) | |
3 | eqid 2737 | . . . . . 6 β’ ( Β·π OLD βπ) = ( Β·π OLD βπ) | |
4 | eqid 2737 | . . . . . 6 β’ ( Β·π OLD βπ) = ( Β·π OLD βπ) | |
5 | eqid 2737 | . . . . . 6 β’ (normCVβπ) = (normCVβπ) | |
6 | eqid 2737 | . . . . . 6 β’ (normCVβπ) = (normCVβπ) | |
7 | sspba.h | . . . . . 6 β’ π» = (SubSpβπ) | |
8 | 1, 2, 3, 4, 5, 6, 7 | isssp 29708 | . . . . 5 β’ (π β NrmCVec β (π β π» β (π β NrmCVec β§ (( +π£ βπ) β ( +π£ βπ) β§ ( Β·π OLD βπ) β ( Β·π OLD βπ) β§ (normCVβπ) β (normCVβπ))))) |
9 | 8 | simplbda 501 | . . . 4 β’ ((π β NrmCVec β§ π β π») β (( +π£ βπ) β ( +π£ βπ) β§ ( Β·π OLD βπ) β ( Β·π OLD βπ) β§ (normCVβπ) β (normCVβπ))) |
10 | 9 | simp1d 1143 | . . 3 β’ ((π β NrmCVec β§ π β π») β ( +π£ βπ) β ( +π£ βπ)) |
11 | rnss 5899 | . . 3 β’ (( +π£ βπ) β ( +π£ βπ) β ran ( +π£ βπ) β ran ( +π£ βπ)) | |
12 | 10, 11 | syl 17 | . 2 β’ ((π β NrmCVec β§ π β π») β ran ( +π£ βπ) β ran ( +π£ βπ)) |
13 | sspba.y | . . 3 β’ π = (BaseSetβπ) | |
14 | 13, 2 | bafval 29588 | . 2 β’ π = ran ( +π£ βπ) |
15 | sspba.x | . . 3 β’ π = (BaseSetβπ) | |
16 | 15, 1 | bafval 29588 | . 2 β’ π = ran ( +π£ βπ) |
17 | 12, 14, 16 | 3sstr4g 3994 | 1 β’ ((π β NrmCVec β§ π β π») β π β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wss 3915 ran crn 5639 βcfv 6501 NrmCVeccnv 29568 +π£ cpv 29569 BaseSetcba 29570 Β·π OLD cns 29571 normCVcnmcv 29574 SubSpcss 29705 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
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-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-fo 6507 df-fv 6509 df-oprab 7366 df-1st 7926 df-2nd 7927 df-vc 29543 df-nv 29576 df-va 29579 df-ba 29580 df-sm 29581 df-nmcv 29584 df-ssp 29706 |
This theorem is referenced by: sspg 29712 ssps 29714 sspmlem 29716 sspmval 29717 sspz 29719 sspn 29720 sspimsval 29722 minvecolem1 29858 minvecolem2 29859 minvecolem3 29860 minvecolem4b 29862 minvecolem4 29864 minvecolem5 29865 minvecolem6 29866 minvecolem7 29867 |
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