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Mirrors > Home > ILE Home > Th. List > ressipg | GIF version |
Description: The inner product is unaffected by restriction. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
Ref | Expression |
---|---|
resssca.1 | β’ π» = (πΊ βΎs π΄) |
ressip.2 | β’ , = (Β·πβπΊ) |
Ref | Expression |
---|---|
ressipg | β’ ((πΊ β π β§ π΄ β π) β , = (Β·πβπ»)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resssca.1 | . 2 β’ π» = (πΊ βΎs π΄) | |
2 | ressip.2 | . 2 β’ , = (Β·πβπΊ) | |
3 | ipslid 12648 | . 2 β’ (Β·π = Slot (Β·πβndx) β§ (Β·πβndx) β β) | |
4 | ipndxnbasendx 12649 | . 2 β’ (Β·πβndx) β (Baseβndx) | |
5 | simpl 109 | . 2 β’ ((πΊ β π β§ π΄ β π) β πΊ β π) | |
6 | simpr 110 | . 2 β’ ((πΊ β π β§ π΄ β π) β π΄ β π) | |
7 | 1, 2, 3, 4, 5, 6 | resseqnbasd 12551 | 1 β’ ((πΊ β π β§ π΄ β π) β , = (Β·πβπ»)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1364 β wcel 2160 βcfv 5231 (class class class)co 5891 βΎs cress 12481 Β·πcip 12560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7920 ax-resscn 7921 ax-1cn 7922 ax-1re 7923 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-addcom 7929 ax-addass 7931 ax-i2m1 7934 ax-0lt1 7935 ax-0id 7937 ax-rnegex 7938 ax-pre-ltirr 7941 ax-pre-lttrn 7943 ax-pre-ltadd 7945 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-iota 5193 df-fun 5233 df-fv 5239 df-ov 5894 df-oprab 5895 df-mpo 5896 df-pnf 8012 df-mnf 8013 df-ltxr 8015 df-inn 8938 df-2 8996 df-3 8997 df-4 8998 df-5 8999 df-6 9000 df-7 9001 df-8 9002 df-ndx 12483 df-slot 12484 df-base 12486 df-sets 12487 df-iress 12488 df-ip 12573 |
This theorem is referenced by: (None) |
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